From 387e1a2af577b9c39431472f085727da173d0f5d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?S=C3=A9bastien=20Miquel?= Date: Sat, 30 Nov 2024 16:25:34 +0100 Subject: [PATCH] =?UTF-8?q?Fourn=C3=A9e=202024?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Exercices 2023.org | 535 +- Exercices 2023.pdf | Bin 1143659 -> 446876 bytes Exercices 2024.org | 9557 ++++++++++++++++++++++++++++++++++++ Exercices 2024.pdf | Bin 0 -> 1155901 bytes Exercices XENS MP 2024.pdf | Bin 0 -> 632730 bytes 5 files changed, 9923 insertions(+), 169 deletions(-) create mode 100644 Exercices 2024.org create mode 100644 Exercices 2024.pdf create mode 100644 Exercices XENS MP 2024.pdf diff --git a/Exercices 2023.org b/Exercices 2023.org index f533587..038a91e 100644 --- a/Exercices 2023.org +++ b/Exercices 2023.org @@ -2,38 +2,24 @@ #+title: Exercices 2023 #+author: Sébastien Miquel #+date: 02-12-2023 -# Time-stamp: <14-07-24 13:19> +# Time-stamp: <30-11-24 16:51> #+OPTIONS: * Meta :noexport: + - $z^n\chi_A(1/n) = e^{- \sum \tr{A^n} \frac{z^n}{n}}$ ? + Si A est inversible : dérivée logarithmique, et décomposition en éléments simples. + ** Statistiques #+BEGIN_SRC emacs-lisp -(defun nb_unexed () - (let ((n 0)) - (save-excursion - (goto-char (point-min)) - (while (go-find-unexed-exo nil) - (setq n (1+ n)) - (forward-line 1)) - n))) - -(defun nb_todo () - (save-excursion - (goto-char (point-min)) - (let ((count 0)) - (while (re-search-forward "exercice.*:todo:" nil t) - (setq count (1+ count)) - (forward-line 1)) - count))) - -`(,(count-matches "\\?\\?") ,(1- (count-matches "!!")), (nb_todo), (nb_unexed)) +(my-stats-exo) #+END_SRC #+RESULTS: -| 5 | 11 | 14 | 673 | +| ? | ! | todo | unexed | +| 1 | 1 | 1 | 633 | #+BEGIN_SRC emacs-lisp (defun find_bad_hash () @@ -190,6 +176,13 @@ python3 -c "import torch; print(f'device name [0]:', torch.cuda.get_device_name( # #+select_tags: mines # #+export_file_name: Exercices Mines 2023 +*** Mines Centrale + +# #+select_tags: mines cent +# #+options: toc:2 +# #+export_file_name: Exercices Mines Centrale 2023 + + *** todoes #+options: title:nil nopage:t tags:nil @@ -197,7 +190,18 @@ python3 -c "import torch; print(f'device name [0]:', torch.cuda.get_device_name( #+export_file_name: Exercices 2023 todo #+relocate_tags: todo +*** autre +# #+options: title:nil nopage:t tags:nil +# #+select_tags: autre +# #+export_file_name: Exercices XENS 2023 autres +# #+relocate_tags: todo + + + +* 2024 + +#+INCLUDE: "./2024/Exercices 2024.org::*Christophe" * ENS MP-MPI :xens: @@ -600,7 +604,7 @@ On considère $\phi\colon\left(\R^4\right)^2 \ra \M_4(\R)$ qui à $(u, v)$ assoc 2. Réciproquement, la seule difficulté est de montrer que $\phi$ est non nulle, pour $(u,v)$ libre. Mais si elle était nulle, pour tout $w, w'$, $\det( u', v', w,w')$ serait nul. 3. $\Rightarrow$ : On peut vérifier que transformer $u, v$ en $Pu, Pv$ transforme $A$ en $P A P^T$, on est donc ramené aux cas où $u = (1,0, 0, 0)$ et $v = (0, 1, 0, 0)$. - $\Leftarrow$ : Réduction d'une matrie antisymétrique ? Prendre $x$, $Ax$ (ils sont orthogonaux), puis une BON du reste, alors dans cette base $A$ doit être antisymétrique, et le premier ??. Plutôt, partir du noyau, qui est de dimension $2$ et stable, et prendre un orthogonal du noyau. + $\Leftarrow$ : Réduction d'une matrie antisymétrique ? Prendre $x$, $Ax$ (ils sont orthogonaux), puis une BON du reste, alors dans cette base $A$ doit être antisymétrique, et le premier ?. Plutôt, partir du noyau, qui est de dimension $2$ et stable, et prendre un orthogonal du noyau. 4. Le noyau est $\vect (u,v)^{\bot}$, l'image est $\vect (u, v)$, car les deux sont orthogonaux. #+END_proof @@ -964,16 +968,21 @@ On obtient l'autre inégalité en passant à l'opposé. ou $XAX = B$, qui se met sous la forme $\sqrt{A} X \sqrt{A} \sqrt{A} X \sqrt{A} = \sqrt{A}B\sqrt{A}$ : on a trouvé $X$. #+END_proof -#+begin_exercice [ENS 2023 # 72] :todo: +# ID:7683 +#+begin_exercice [ENS 2023 # 72] Soit $A\in{\cal S}_n({\R})$. On définit $p(A)$ comme la dimension maximale d'un sous-espace $V$ sur lequel $\forall x\in V\setminus\{0\},\,\langle Ax,x\rangle\gt 0$. On définit de même $q(A)$ avec la condition $\langle Ax,x\rangle\lt 0$. - Montrer que $p(A)+q(A)=\mbox{rg}\,A$. - Montrer que, si $A$ est inversible, alors $p$ et $q$ sont constantes sur un voisinage de $A$ dans ${\cal S}_n({\R})$. - Soit $B\in{\cal S}_n({\R})$, on suppose que $f\colon t\mapsto\det(A+tB)$ n'a que des racines simples sur ${\R}$. Montrer que $f$ admet au moins $|p(B)-q(B)|$ racines dans ${\R}$. #+end_exercice -#+BEGIN_proof :todo: +#+BEGIN_proof - Simple. - Elles sont localement croissante : si on a un témoin $V$, il suffit de témoigner sur l'intersection de $V$ avec la sphère unité. - Si $B$ est définie positive, par coréduction ça marche. Sinon ? + + Simplement, on regarde les valeurs propres de $A+tB$, quand $t\ra +\i$, elles ont le même signe que celles de $B$, et quand $t\ra -\i$, les signes opposés. Si $B$ est inversible, on conçoit bien que pour passer d'un état à un autre, il faut que le nombre annoncé de racines changent de signe (ici, continuité des valeurs propres), qui se faut via la question précédente. + + Si $B$ n'est pas inversible. #+END_proof @@ -1067,6 +1076,17 @@ Soit $\lN\cdot\rN$ une norme multiplicative sur $\M_n(\R)$. Si $\lN A\rN\neq \lN ** Analyse +# ID:7354 +#+BEGIN_exercice +Soit $k\geq 1$ et $f\in\mc C^k(\R,\R)$ telle que $\sum_{j=0}^k f^{(j)}(x)$ admette une limite quand $x\ra +\i$. Peut-on en déduire que $f$ admet une limite en $+\i$, selon la valeur de $k$ ? +#+END_exercice +#+BEGIN_proof +Si $\sum_{j=0}^k f^{(j)}(x) = k$, on regarde l'équation homogène, dont l'équation caractéristique est $1 + X + \dots + X^k = 0$, et dont le comportement est décidé par le signe de la partie réelle des racines. Si $k = 1$ c'est bon, si $k = 2$, c'est bon aussi, pour $k=3$, on a $\pm i$ comme racines, donc ce n'est plus le cas. + +Reste à traiter les cas $k = 1, 2$ en général, par des variations de la constante. +#+END_proof + + # ID:7160 #+BEGIN_exercice [ENS 2023 # 79] Soit $p\gt 1$. On pose, pour $x\in\R^n$, $\lN x\rN_{p} = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}$. @@ -1257,13 +1277,18 @@ On considère une suite $a\in\{2,3\}^{\N^*}$ telle que $a_1=2$ et, pour tout $n\ + (La solution de $\l = \frac{1}{4 - \l}$ est irrationnelle.) + La suite $p_{n^{(k)}_1}$ converge vers $\l$. + Un peu galère, mais on peut en déduire que $(p_{n_k})$ converge vers $l$, donc que $(p_n)$ converge vers $\l$ (les $n_k$ sont distants d'au plus $3$). + +En fait, on a $n_k = (a_0 + 1) + \dots + (a_{k-1} + 1)$ , donc $n_k = 4k - \#\{n_i \in \db{0,k-1}\}$. #+END_proof -#+BEGIN_exercice [ENS 2023 # 97] :todo: +# ID:7588 +#+BEGIN_exercice [ENS 2023 # 97] On considère une suite $a \in\{2,3\}^{\N^*}$ telle que $a_1=2$ et, pour tout $n \geq 1$, le nombre de $3$ apparaissant dans la suite $a$ entre la $n$-ième occurrence de 2 et la $(n+1)$-ième occurrence de $2$ soit égal à $a_n$. Montrer qu'il existe un unique irrationnel $\alpha$ tel que les indices $n \geq 1$ tels que $a_n=2$ soient exactement les entiers de la forme $\lfloor m \alpha\rfloor+1$ pour un $m \in \N$. #+END_exercice -#+BEGIN_proof :todo: -On sait que la proportion de terme qui vaut $2$ tend vers la solution à $\l = \frac{1}{4 - \l}$. +#+BEGIN_proof +On sait que la proportion de terme qui vaut $2$ tend vers la solution à $\l = \frac{1}{4 - \l}$, c'est-à-dire $2 + \sqrt{3}$. + +On vérifie que ça marche : il s'agit de justifier qu'en notant $n_k = \lfloor k \a\rfloor + 1$, $n_k$ est bien égal à $4k - \#\{n_i \in \db{0,k-1}\}$. #+END_proof @@ -1741,15 +1766,19 @@ Pour $x$ réel, on pose $J(x)=\int_0^{\pi}\cos(x\sin t)\,dt$. Soient $f$ et $g$ deux fonctions de classe $\mc C^{\i}$ de $\R^+$ dans $\R$. On pose $f\star g\colon x\in\R_+\mapsto\int_0^xf(t) g(x-t) dt$. Montrer que $f\star g$ est dérivable et donner une expression de sa derivée. #+end_exercice -#+begin_exercice [ENS 2023 # 136] -Soit $f:]0,1[\to\R$ continue. Pour $n\geq 1$ et $s\lt t$ dans $]0,1[$, on pose -$a_n(f,s,t)=\frac{2}{t-s}\int_s^tf(u)\cos\left(\frac{2n\pi}{t-s}(u-s) \right)\,du$. +# ID:7424 +#+begin_exercice [ENS 2023 # 136] +Soit $f\colon ]0,1[\to\R$ continue. Pour $n\geq 1$ et $s\lt t$ dans $]0,1[$, on pose $a_n(f,s,t)=\frac{2}{t-s}\int_s^tf(u)\cos\left(\frac{2n\pi}{t-s}(u-s) \right)\,du$. - On suppose $f$ strictement convexe. Montrer que $a_1(f,s,t)\gt 0$ pour tous $s\lt t$ dans $]0,1[$. - On suppose $f$ strictement convexe. Montrer que $a_n(f,s,t)\gt 0$ pour tous $s\lt t$ dans $]0,1[$ et tout $n\in\N^*$. - Réciproquement, on suppose $f$ de classe $\mc C^2$ et $a_1(f,s,t)\gt 0$ pour tous $s\lt t$ dans $]0,1[$. Montrer que $f$ est strictement convexe. - #+end_exercice +#+BEGIN_proof + - On obtient $\int_0^{\frac{\pi}{2}}\cos t \big(f(t) - f(\pi - t) - f(\pi + 1) + f(2\pi - t)\big)\dt$. Faire le dessin des quatre valeurs dans $[0,2\pi]$, c'est la différence de deux différences (à même distance) de valeurs de $f$. Si on avait une FAF, on conclurait, mais sans, c'est une inégalité des pentes (avec des pentes intermédiaires). + - Découper. + - Quant $t\ra s$, l'intégrale est équivalente à $f''(s)$. +#+END_proof # ID:6895 #+begin_exercice [ENS 2023 # 137] @@ -1814,15 +1843,22 @@ Soient $A$ une application continue de $\R_+$ dans $\M_n(\R)$, $M$ l'unique appl #+END_proof -#+begin_exercice [ENS 2023 # 142] :todo: +# ID:7473 +#+begin_exercice [ENS 2023 # 142] # Critère de stabilité de Lyapounov, pour l'équation de Hill Soit $p\colon\R\to\R$ une fonction continue, non identiquement nulle, $\pi$-périodique et telle que $\int_0^{\pi}p(t)dt\geq 0$ et $\int_0^{\pi}|p(t)|dt\leq\frac{\pi}{4}$. Montrer que l'équation $u''+pu=0$ n'admet pas de solution $u$ non nulle sur $\R$ telle qu'il existe $\lambda\in\R^*$ tel que $\forall t\in\R$, $u(t+\pi)=\lambda\,u(t)$. #+end_exercice -#+BEGIN_proof :todo: -On pose $v = u e^{-t\tau}$, de sorte que $v$ soit périodique, on obtient $v' = u'e^{-t\tau} - \tau v$, $v'' = u'' e^{-t\tau} - 2\tau e^{-t\tau} u' + \tau^2 e^{-t\tau} u$, - donc $v'' = (3\tau^2 - p)v - 2\tau v'$, je crois. +#+BEGIN_proof +Remarque : l'application $u\mapsto u(\cdot +\pi)$ a une matrice de déterminant $1$ (à cause du Wronskien). On veut savoir si elle admet une valeur propre réelle. + +Si la solution $u$ ne s'annule pas, on à $\int \frac{u''}{u} = \int p$, mais une IPP donne une contradiction avec la positivité. + +Si la solution s'annule, les zéros sont isolés, on en considère deux consécutifs $a\lt b$, avec $u$ positive, et on montre que $\int \left|\frac{u''}{u}\right| \gt \frac{4}{b-a}$. + +Pour cela, on majore par $u_{max}^{-1} \max |u'(x) - u'(y)|$, et on applique Rolle entre $a$ et le max, et le max et $b$, puis IAG. #+END_proof + # ID:7220 #+begin_exercice [ENS 2023 # 143] Soit $A_0\in\M_n(\R)$ telle que $\text{Sp}(A_0+A_0^T)\subset\R_-$. @@ -2079,38 +2115,70 @@ Le faire pour $m = p$, puis lemme Chinois. # ID:7044 #+begin_exercice [ENS 2023 # 166] Deux joueurs $A$ et $B$ lancent une pièce truquée donnant pile avec une probabilité égale à $5/9$. Les règles de gain sont les suivantes : pile rapporte $5$ euros et face $4$ euros. Pour $n\in\N^*$, chacun des joueurs effectue $9n$ lancers indépendants ; on note $A_n$ (resp. $B_n$) la variable aléatoire donnant le gain du joueur $A$ (resp. $B$). - - Trouver un équivalent, lorsque $n$ tend vers $+\i$, de $\mathbf{P}\left(A_n=B_n\right)$. + - Trouvre la limite de $\mathbf{P}\left(A_n=B_n\right)$ quand $n\ra +\i$. - Montrer que $\mathbf{P}\left(A_n\geq B_n\right)\geq\frac{1}{2}$. - Vers quoi tend $\mathbf{P}\left(A_n\lt B_n\right)$ ? + - S Trouver un équivalent, quand $n\ra +\i$, de $\mathbf{P}\left(A_n=B_n\right)$. #+end_exercice #+BEGIN_proof - - IDK, Cela ne dépend que du nombre de Pile obtenus, pas du gain… Éventuellement, on tend vers une loi normale… + - - C'est clair. - Découle des questions précédentes. + - C'est $\sum_{k=0}^{9n} a_k^2 = \sum_{k=0}^{9n} a^{2k} b^{2(9n-k)} {9 n \choose k} {9 n \choose 9n - k}$. + + Pour calculer un équivalent de $\sum_{k=0}^{9n} {9n \choose k} {9 n \choose 9n - k}$ sans la formule de Van der Monde, on l'écrit comme $\int_0^{2\pi} \left(1 + e^{i\theta})^{9n}(1 + e^{-i\theta}\right)^{9n}$, puis CVD. Avec un facteur $q^2$, cela fonctionne également. + + Pour trouver l'équivalent de $\sum {n\choose k}^2$ : on trouve l'élément maximal, en $m = \frac{n}{2}$, on factorise par ça, puis $\l$ termes à droite/ à gauche, on a $\frac{(m-1)\dots(m-\l)}{(m+1)\dots (m+\l)}$. On prend le logarithme, on utilise $|\ln (1+x) - x| \leq \frac{x^2}{2}$. On obtient l'ordre de grandeur précis du terme, on peut sommer et comparer à une intégrale. #+END_proof -# Relier à l'exercice d'espérance avec des 6 -#+begin_exercice [ENS 2023 # 167, 177] :todo: +# ID:7466 +#+begin_exercice [ENS 2023 # 167, 177] On joue à pile ou face avec une pièce pipée qui donne pile avec probabilité $p\lt \frac{1}{2}$. On lance la pièce $2n$ fois et on compte le nombre de «Piles». Déterminer l'entier $n$ qui maximise la probabilité d'avoir compté au moins $n+1$ «Piles». #+end_exercice -#+BEGIN_proof :todo: -On a $P(S_{2n} = n+k)\leq P(S_{2n} = n-k)$, puis on montre que $P(S_{2n}\geq n+1) + \frac{1}{2}P(S_{2n} = n)$ est décroissante. Mais on connaît $P(S_{2n} = n)$, et il suffit de voir quand elle devient plus petite que les premières valeurs de $P(S_{2n} \geq n+1)$. +#+BEGIN_proof +On écrit $$P(S_{2(n+1)}\geq n+2) = P(S_{2n}\geq n+2) + P(S_{2n}= n+1) \big(1 - (1-p)^2\big) + P(S_{2n} = n)p^2 = P(S_{2n}\geq n+1) + P(S_{2n} = n)p^2 - P(S_{2n} = n+1)(1-p)^2,$$ +puis on compare ces deux, et on obtient $n\leq \frac{p}{1-2p}$. #+END_proof -#+BEGIN_exercice [ENS 2023 # 168] :todo: +# ID:7486 +#+BEGIN_exercice [ENS 2023 # 168] Soit $X$ une variable aléatoire à valeurs dans $\N$ telle que $\mathbf{E}(X)=1$, $\mathbf{E}\left(X^2\right)=2$ et $\mathbf{E}\left(X^3\right)=5$. Quelle est la valeur minimale de $\mathbf{P}(X=0)$ ? #+END_exercice -#+BEGIN_proof :todo: +#+BEGIN_proof On a $E(X) E(X^3)\geq E(X^2)^2$. En notant $e = P(X=1)$, on a $E(X 1_{X\gt 1}) E(X^3 1_{X\gt 1})\geq E(X^2 1_{X\gt 1})^2$, donc $(*)$ $(1-e)(5-e) \geq (2-e)^2$, qui donne $e\leq \frac{1}{2}$. Comme $E(X) = 1$, on doit avoir $P(X=0)\geq \frac{1}{4}$, mais le cas d'égalité ne donne pas les bonnes valeurs : mais $E(X) = 1$, $E(X^2) = \frac{3}{2}$ et $E(X^3) = \frac{5}{2}$. Si on suppose que $e = \frac{1}{2}$, on doit avoir égalité dans Cauchy-Schwarz qui donne $(*)$, donc $X$ ne prend qu'une seule valeur $\gt 1$. On peut prendre $Y$ qui vaut $3$ avec probabilité $\frac{1}{6}$ et $0$ avec probabilité $\frac{1}{3}$, et on a les bonnes valeurs (et c'est la seule façon). -On a aussi $E(X 1_{X\geq 1})^2\leq E(1_{X\geq 1}) E(X^2)$, donc $(1 - r)\geq \frac{1}{2}$, $r\leq \frac{1}{2}$ +On montre que l'on ne peut pas faire mieux. -!! Manque : on ne peut pas faire mieux… +Stratégie : + + On montre que si $X$ prend $5$ valeurs (avec des probabilités $\gt 0$), alors on peut toujours diminuer la probabilité que $P(X=0)$. + + Si $X$ prend $4$ valeurs, le système de Van der Monde a une unique solution (pas forcément positive…). On montre que la valeur $p_0$ est minimale si les quatre valeurs sont $0, 1, 2, 3$. + + Comment ? En diminuant continûment une valeur sinon. + +On sait que $X$ doit prend la valeur $0$, car $E(X) = 1$ (non…). Si on suppose que $X$ ne prend pas la valeur $1$, alors $P(X = 0)\geq \frac{3}{8}$ (via $E(X^3) = 5$), qui est plus grand que le $\frac{1}{3}$ trouvé. + +On note $v_2,v_3$ les autres valeurs. ($v_0 = 0$, $v_1 = 1$) + +Les $p_i$ vérifient $V = \begin{pmatrix}1 & 1 & 1 & 1 \\ 0 & 1 & v_2 & v_3 \\ 0 & 1 & v_2^2 & v_3^2 \\ 0 & 1 & v_2^3 & v_3^3 \end{pmatrix} \vvvv{p_0}{p_1}{p_2}{p_3} = \vvvv{1}{1}{2}{5}$. +On cherche la première ligne de $V^{-1}$. $V^T$ est la matrice de Vandermonde, qui à $P$ associe $(P(0), P(1), P(v_2), P(v_3))$. On cherche la première colonne de son inverse, c'est-à-dire les coefficients du polynôme $\frac{(X-1)(X-v_2)(X-v_3)}{- v_2 v_3} = 1 + X\big(-1 - \frac{1}{v_2} - \frac{1}{v_3}\big) + X^2\big(\frac{1}{v_2} + \frac{1}{v_3} + \frac{1}{v_2 +v_3}\big) - \frac{1}{v_2v_3} X^3$ +Puis, le coefficient en $p_0$ est le produit avec $(1\, 1\, 2\, 5)$, donc $-\frac{1}{v_2} - \frac{1}{v_3} + 2 \big(\frac{1}{v_2} + \frac{1}{v_3} + \frac{1}{v_2v_3}\big) - \frac{5}{v_2v_3}$ $= \frac{1}{v_2} + \frac{1}{v_3} - \frac{3}{v_2v_3}$ + +On veut montrer qu'en diminuant $v_3$, il diminue, mais c'est $\frac{1}{v_3} \big(1 - \frac{3}{v_2}\big)$, donc c'est le cas si $v_2 \gt 3$. Autrement dit, si l'un des termes est $\geq 3$, on peut diminuer l'autre. + +Il faudrait montrer que pour $v_2 = 2$, il y a toujours une solution positive. + +Plutôt : on les diminues ensembles, la différentielle est : $-\frac{1}{(v_2)^2} - \frac{1}{(v_3)^2} + \frac{3}{v_2 v_3^2} + \frac{3}{v_3 v_2^2}$ + +Si $v_2 = 2$, on veut montrer qu'une solution n'est pas possible, sauf $v_3 = 3$. Pour cela, on différencie la seconde (troisième) coordonnée selon $v_3$. + +Le polynôme est $\frac{X(X-1)(X-v_3)}{2 (2-v_3)} = \frac{X^3 - (1+v_3)X^2 + v_3 X}{2(2-v_3)}$, sa troisième coordonnée est $-\frac{1+v_3}{2(2-v_3)}$, qui est décroissante, donc si on passe de $(\cdot, \cdot, 2, v_3)$ à $(\cdot, \cdot, 2, 3)$, la partie en $2$ a augmenté, et a atteint $0$, impossible ! + +Donc la seule solution avec un $2$ est $(\cdot, \cdot, 2, 3)$. Puis si on a une solution où les deux termes sont $\geq 3$, on peut se ramener à $(\cdot, \cdot, 3, 4)$, puis $(\cdot, \cdot, 3, 3)$, qui existe bien. D'où le résultat. #+END_proof @@ -2125,6 +2193,7 @@ Pour les autres valeurs que $0$ modulo $n$, il faut prendre $X^k G_m(X)$, cela m #+END_proof +# ID:7375 #+begin_exercice [ENS 2023 # 170] Pour $\sigma\in\mc{S}_n$ on note $I(\sigma)$ le nombre d'inversions de $\sigma$ c'est-a-dire le nombre de couples $(i,j)$ avec $i\lt j$ et $\sigma(i)\gt \sigma(j)$. - Montrer que $P_n=\sum_{\sigma\in\mc{S}_n}X^{I(\sigma)}=\prod_{k=1}^{n-1}(1+X+ \cdots+X^k)$. @@ -2134,12 +2203,13 @@ Pour $\sigma\in\mc{S}_n$ on note $I(\sigma)$ le nombre d'inversions de $\sigma$ #+BEGIN_proof - $\sigma$ est déterminé par : le nombre d'inversion avec $(n-1)$ (une ou $0$), le nombre d'inversions avec $n-2$, etc. En effet, pour chaque $\sigma$, on peut donner ces nombres, et réciproquement, si on les connaît, on connaît l'image de $(n-1)$ par rapport à l'image de $n$, puis l'image de $n-2$ par rapport à $(n-1)$ et $n$, etc, donc $\sigma$ est entièrement déterminée. - $f(n)$ est la somme des coefficients en $X^{k(n+1)}$ de $P_n$, donc $\frac{P(1) + P(\om) + \dots + P(\om^n)}{n+1}$. - - Avec les questions précédentes, on trouve $f(p-1) = \frac{1}{p}\left((p-1)! + \sum_{\om \in \m U_p\setminus 1} \frac{p}{(1-\om)^2}\right)$. + - Avec les questions précédentes, on trouve $f(p-1) = \frac{1}{p}\left((p-1)! + \sum_{\om \in \m U_p\setminus 1} \frac{p}{(1-\om)^p}\right)$, en utilisant $1+X + \dots + X^{k-1} = \frac{1 - X^k}{1-X}$. - Cette somme, s'obtient en prenant $Q = 1 + X + \dots + X^{p-1}$, $\frac{Q'}{Q} = \sum \frac{1}{X - \om}$, donc $\frac{Q''Q - (Q')^2}{Q^2} = - \sum \frac{1}{(X-\om)^2}$ - En évaluant en $1$, on obtient un dénominateur de $p^2$, et un numérateur $p \sum_{k=2}^{p-1} k (k-1)- \big(\frac{(p-1)p}{2}\big)^2 = \frac{p^2}{3}(p-2)(p-1) - \big(\frac{(p-1)p}{2}\big)^2$, qui a un signe constant APCR… + Par ailleurs, $(1+\om)^p = 2^p \left(\cos \big(\frac{\theta}{2}\big)\right)^p e^{i \frac{p\theta}{2}}$, la partie en $e^{i\dots}$ valant $\pm 1$. - J'ai confirmé numériquement que $\sum \frac{1}{(1-\om)^2}$ est négatif… !! + La somme porte donc sur $p-1$ éléments, qui parcourent le demi-cercle supérieur. Comme $p$ est impair, les signes à droite et à gauche sont les mêmes (car le $\pm 1$ alterne, un nombre impair de fois, et le cos change de signe). + + Comme $p-1$ est pair, il n'y a pas de terme au milieu. Une fois regroupé sur le quadrant en haut à droite, la somme est alternée. Son signe est donnée par la parité de $\frac{p-1}{2}$. #+END_proof @@ -2161,7 +2231,8 @@ Dans tout l'exercice, les variables aléatoires considérées sont supposées r 2. Donner un exemple de couple $(X, Y)$ pour lequel $X \leq_c Y$ mais $X \neq Y$. 3. Montrer que si $X \leq_c Y$ alors $\mathbf{E}(X)=\mathbf{E}(Y)$ et $\mathbf{V}(X) \leq \mathbf{V}(Y)$. 4. Montrer que $X \leq_c Y$ si et seulement si $\mathbf{E}(X)=\mathbf{E}(Y)$ et -$$ \forall a \in \R, \int_a^{+\i} \mathbf{P}(X \geq x)\dx \leq \int_a^{+\i} \mathbf{P}(Y \geq x)\dx.$$ + $$ \forall a \in \R, \int_a^{+\i} \mathbf{P}(X \geq x)\dx \leq \int_a^{+\i} \mathbf{P}(Y \geq x)\dx.$$ + 5. Montrer que $X + E(X)\leq 2X$. #+END_exercice #+BEGIN_proof 1. Convexité @@ -2192,42 +2263,45 @@ On fixe $N \in \N^*$. On choisit de façon équiprobable $u_1 \in \db{1, N}$, pu 3. Semble facile. #+END_proof -#+begin_exercice [ENS 2023 # 174] :todo: +# ID:7348 +#+begin_exercice [ENS 2023 # 174] Dans tout l'énoncé, on fixe un entier $p\geq 1$. - Développper $(x_1+\cdots+x_N)^p$ pour toute liste $(x_1,\ldots,x_N)$ de nombres réels. - Soient $X_1,\ldots,X_n$ des variables aléatoires i.i.d. suivant la loi uniforme sur $\{-1,1\}$. Soit $(a_1,\ldots,a_n)\in\R^n$. On pose $X=\sum_{i=1}^na_iX_i$. Montrer que $\mathbf{E}(X^{2p})\leq(2p)^p(\mathbf{E}(X^2))^p$. - Montrer que $\mathbf{E}(X^{2p})\leq p^p(\mathbf{E}(X^2))^p$. -- Soit $(a_k)_{k\geq 1}$ une suite réelle telle que $\sum_{k=1}^{+\i}a_k^2=1$. Soient $x\in\R$ et $Y_x=\sum_{k=1}^na_k\cos(kx)\,X_i$. + - Soit $(a_k)_{k\geq 1}$ une suite réelle telle que $\sum_{k=1}^{+\i}a_k^2=1$. Soient $x\in\R$ et $Y_x=\sum_{k=1}^na_k\cos(kx)\,X_i$. - Montrer que $\omega\mapsto\int_0^{2\pi}Y_x(\omega)^{2p} \dx$ prend au moins une valeur inférieure ou égal a $2\pi p^p$. + Montrer que $\omega\mapsto\int_0^{2\pi}Y_x(\omega)^{2p} \dx$ prend au moins une valeur inférieure ou égal a $2\pi p^p$. #+end_exercice -#+BEGIN_proof :todo: - - +#+BEGIN_proof + - - Si on développe, seuls les termes d'exposants tous pairs restent. On veut alors $$\sum_{n_1 + \dots + n_n = p} {2p \choose 2n_1,\dots, 2n_n} a_1^{2n_1}\dots a_n^{2n_p}\leq (2p)^p \sum_{n_1 + \dots + n_n = p} {n\choose n_1,\dots, n_n} a_1^{2n_1}\dots a_n^{2n_p}.$$ - Il suffit de le montrer pour chaque multinôme. Pour $n = 2$ c'est déjà pas si clair… + Il suffit de le montrer pour chaque multinôme. - !! - Si $g\colon \db{1,2p}\ra \db{1,n}$ est une application telle que $1$ ait $2n_1$ antécédents, …, $n$ en ait $2n_n$, on peut lui associé le couple formé de $h_1\colon \db{1,p}\ra \db{1,n}$ définie en ne gardant que les $n_1$ premiers antécédents de $1$, …, les $n_n$ premiers antécédents de $n_n$ et $h_2\db{1,2p}\ra \db{1,p}$ qui à $i$ associe le premier élément qui a la même image que lui. NOPE. + Pour $n = 2$ : On veut montrer que ${2p\choose 2k}\leq (2p)^p {p \choose k}$, c'est-à-dire $\frac{(2p) \dots (2p-2k+1)}{(2k)!} \leq (2p)^p \frac{(p) \dots (p-k+1)}{k!}$. Ce qui se fait assez bien : on majore les premiers facteurs par $2p$, et s'il en restent, par le reste, ce qui est possible car $2k - p \leq k$. - L'application $h\colon (h_1,h_2)$ est injective, et le nombre de $h_2$ possible est $\leq (2p)^p$. - - + La même démonstration marche assez bien dans le cas général. + - Pour $n = 2$ : on veut $\frac{(2p) \dots (2p-2k+1)}{(2k)!} \leq (p)^p \frac{(p) \dots (p-k+1)}{k!}$, cela revient à $\frac{(2p) \dots (2p-2k+1)}{(p) \dots (p-k+1)} \frac{k!}{(2k)!}\leq p^p$, et en groupant les termes avec le prochain, on obtient $\frac{(2p) (2p-1)}{(2k) (2k-1)} k = \frac{p (2p-1)}{2k-1}\leq p$, d'où la conclusion. + + J'imagine que ça marche aussi en général. - Pas de difficulté. #+END_proof -#+begin_exercice [ENS 2023 # 175] :todo: +# ID:7489 +#+begin_exercice [ENS 2023 # 175] suivant la loi uniforme sur $\{1,-1\}$. Soient $X_1,\ldots,X_n$ des variables aléatoires i.i.d. suivant la loi de Rademacher, et $a_1,\ldots,a_n$ des réels. On pose $Y=\sum_{k=1}^na_kX_k$. 1. Montrer que $\mathbf{E}(|Y|)^2\leq\mathbf{E}(Y^2)$. 2. Montrer que $\mathbf{E}(Y^2)=\sum_{k=1}^na_k^2$. - 3. Montrer que si $\sum_{k=1}^na_k^2=1$ alors $\mathbf{E}(Y^2)\leq e\,\mathbf{E}(|Y|)^2$. + 3. On suppose que $\sum_{k=1}^na_k^2=1$. En considérant la variable complexe $Z = \prod_{k=1}^n (1 + i X_k)$, montrer que $\mathbf{E}(Y^2)\leq e\,\mathbf{E}(|Y|)^2$. 4. Montrer que $\mathbf{E}(Y^2)\leq e\,\mathbf{E}(|Y|)^2$ en toute généralite. #+end_exercice -#+BEGIN_proof :todo: +#+BEGIN_proof 1. 2. - 3. On veut $E(|Y|)^2 \geq \frac{1}{e}$, sachant $\sum a_k^2 = 1$, !! + 3. La variable $G$ vérifie $|G|^2 = \prod (1 + a_k^2)\leq e^{\sum a_k^2} = e^1$, et l'espérance de $YG$ vaut $1$. #+END_proof @@ -2287,7 +2361,7 @@ C'est la probabilité d'extinction. # ID:7329 -#+begin_exercice [ENS 2023 # 182] +#+begin_exercice [ENS 2023 # 182] On construit itérativement et aléatoirement un arbre aléatoire sur l'ensemble de sommets $\db{1,n}$ (graphe orienté) selon le procédé suivant : à l'étape $k$, on choisit aléatoirement un point dans $\db{1,k}$ (avec probabilité uniforme) et on rajoute une arete orientée de ce point vers $k+1$. Ces choix s'effectuent de maniere indépendante les uns des autres. - On note $X_n$ la variable aléatoire donnant le nombre d'arêtes partant du point $1$. Déterminer l'espérance et la variance de $X_n$. - On suppose $n\geq 2$. On note $S_n$ la variable aléatoire donnant le nombre de descendants (directs ou non) du sommet $2$. Déterminer la loi de $S_n$. @@ -2435,11 +2509,14 @@ $a_{i,j}=\begin{cases}1\ \ \text{si }i+1=j\\ -c_{i-1}\ \ \text{si }j=n\end{cases - La matrice $\left(\begin{array}{cc}-1&0\\ 0&-4\end{array}\right)$ est-elle le carré d'une matrice réelle? #+end_exercice +# ID:7388 #+begin_exercice [ENS PSI 2023 # 196] Soit $A\in{\cal M}_n({\R})$. Montrer qu'il existe $P\in{\R}[X]$ tel que $P(A)={\rm Com}(A)^T$. - -_Ind._ Commencera par $A$ inversible. #+end_exercice +#+BEGIN_proof +Pour $A$ inversible, c'est le morceau du polynôme caractéristique, sans le terme constant, qui vaut d'office $\det A$. Par continuité des coefficients du polynôme caractéristique, et des coefficients de la comatrice, c'est encore le cas si $A$ non inversible. +#+END_proof + #+begin_exercice [ENS PSI 2023 # 197] Soient $E$ un ${\R}$-espace vectoriel de dimension $d\in{\N}^*$ et $f\in{\cal L}(E)$ telle que $f\circ f=-$id. @@ -2523,16 +2600,35 @@ On note $(e_1,\ldots,e_n)$ la base canonique de ${\cal M}_{n,1}(\R)$. Soit $(c_0 - Montrer que $A$ s'écrit comme le produit de deux matrices symétriques. #+end_exercice + +# ID:7425 #+begin_exercice [ENS PSI 2023 # 205] -_a) i)_ Soit $m$ un entier $\geq 2$. Montrer que $\int_1^{m-1}\frac{{\rm d}x}{\sqrt{x(m-x)}}\leq\sum_{k=1}^{m-1}\frac{1} {\sqrt{k(m-k)}}$. - - Calculer $\int_1^{m-1}\frac{{\rm d}x}{\sqrt{x(m-x)}}$ l'aide du changement de variables $x=\frac{m}{1+t^2}$. + - + - Soit $m$ un entier $\geq 2$. Montrer que $\int_1^{m-1}\frac{{\rm d}x}{\sqrt{x(m-x)}}\leq\sum_{k=1}^{m-1}\frac{1} {\sqrt{k(m-k)}}$. + - Calculer $\int_1^{m-1}\frac{\dx}{\sqrt{x(m-x)}}$ à l'aide du changement de variables $x=\frac{m}{1+t^2}$. - Soit $A_n\in{\cal M}_n(\R)$ la matrice de terme général $\frac{1}{i+j-1}$. - - Montrer que $A_n\in S_n^{++}(\R)$. - - Soit $\lambda_n$ la plus petite des valeurs propres de $A_n$. Montrer qu'il existe $a,b\gt 0$ tels que $\forall n\geq 1,\,0\leq\lambda_n\leq\frac{1}{n}\big{(}a+b \ln(n)\big{)}$. - - Soient $\mu_n$ la plus grande valeur propre de $A_n$ et $X=(1/\sqrt{1},1/\sqrt{2},\ldots,1/\sqrt{n})^T\in\R^n$. Montrer que $\langle A_nX,X\rangle\geq 2\sum_{i=1}^n\frac{1}{i}\arctan(\sqrt{i-1})$ ou $\langle\,\ \rangle$ designe le produit scalaire canonique sur $\R^n$. - - Montrer que, pour tout $P\in\R[X]$, $\int_{-1}^1P(t)\,{\rm d}t=-i\int_0^{\pi}P(e^{i\theta})e^{i\theta}\,{\rm d}\theta$. En déduire que, pour tout $Q=\sum_{k=0}^da_kX^k\in\R[X]$, $\int_0^1Q^2(t)\,{\rm d}t\leq\int_{-1}^1Q^2(t)\,{\rm d}t \leq\pi\sum_{k=0}^da_k^2$. - - En déduire que $\lim_{n\to+\i}\mu_n=\pi$. + - Montrer que $A_n\in S_n^{++}(\R)$. + - Soit $\lambda_n$ la plus petite des valeurs propres de $A_n$. Montrer $\la_n = O\big(\frac{1}{n}\big)$. + - Soient $\mu_n$ la plus grande valeur propre de $A_n$ et $X=(1/\sqrt{1},1/\sqrt{2},\ldots,1/\sqrt{n})^T\in\R^n$. + - Montrer que + $$\langle A_nX,X\rangle\geq 2\sum_{i=1}^n\frac{1}{i}\arctan(\sqrt{i-1}),$$ + - Montrer que, pour tout $P\in\R[X]$, $\int_{-1}^1P(t)\dt=-i\int_0^{\pi}P(e^{i\theta})e^{i\theta}\d\theta$. En déduire que, pour tout $Q=\sum_{k=0}^da_kX^k\in\R[X]$, $\int_0^1Q^2(t)\dt\leq\int_{-1}^1Q^2(t)\dt \leq\pi\sum_{k=0}^da_k^2$. + - En déduire que $\lim_{n\to+\i}\mu_n=\pi$. #+end_exercice +#+BEGIN_proof + - + - Trivial. + - Simple. À écrire, on trouve du $\arctan$. + - + - Pas simple… + - On a $\langle AU, U\rangle = \int_0^1 \big(\sum u_i t^{i-1}\big)^2\dt$. On veut montrer que $\la_n = O\big(\frac{\ln n}{n}\big)$, il suffit de trouver $U$ tel que $\frac{\langle AU, U\rangle}{\lN U\rN^2} = O\big(\frac{\ln n}{n}\big)$. + + Si on prend $U$ avec des coefficients $\pm 1$, on obtient $O\big(\frac{1}{n}\big)$. L'énoncé d'origine annonçait $O\big(\frac{\ln n}{n}\big)$. + - + - C'est les préliminaires : sans passer par l'intégrale, on fait apparaître la somme, et on applique les deux questions. + - Pas de difficulté. + - C'est Cesàro, et l'inégalité précédente. +#+END_proof #+begin_exercice [ENS PSI 2023 # 206] On munit $\R^n$ de sa structure euclidienne canonique. On considère des réels $\lambda_1,\ldots,\lambda_n$ tels que : $0\lt \lambda_1\leq\lambda_2\leq\cdots\leq\lambda_n$, et, pour tout entier $i$ tel que $1\leq i\leq n$, on pose $M_i=(\lambda_i,\lambda_i^{-1})$.On considère $y=(y_1,\ldots,y_n)\in\R^n$ tel que $\|y\|_2=1$ et on note $M$ le barycentre des $M_i$ pondere par les coefficients $y_i^2$. @@ -2653,22 +2749,37 @@ _c) i)_ Trouver toutes les solutions ${\cal C}^2$ de l'équation d'onde a variab - Soit $n\in\N$. On pose : $g:x\mapsto\sum_{k=1}^na_k\sin(k\pi x)$ et $h=0$. Déterminer $u(x,t)$. #+end_exercice +# ID:7423 #+begin_exercice [ENS PSI 2023 # 215] -On munit $\R^d$ de sa structure euclidienne canonique. On dit que $f$ est differentiable sur l'ouvert $\Omega$ si $\nabla f$ existe et est continu. - - Soient $C$ ouvert convexe non vide de $\R^d$, $f:C\to\R$ differentiable. On suppose que $\nabla f$ est $L$-lipschitzien. Soient $w,v\in C$ et $g:t\mapsto f(v+t(w-v))$. - - Experimer $g'(t)$. - - Montrer que $f(w)-f(v)=\int_0^1\left\langle\nabla f(v+t(w-v)),w-v\right\rangle{\rm d}t$. - - Montrer que $f(w)\leq f(v)+\left\langle\nabla f(v),w-v\right\rangle+\frac{L}{2}\left\| w-v\right\|$. - - Soit $f\colon\R^d\to\R$ differentiable. Montrer que $f$ est convexe si et seulement si - -$\forall w,v\in\R^d$, $f(w)\geq f(v)+\left\langle\nabla f(v),w-v\right\rangle$. Ind. Commencer par $d=1$. - - Soit $f\colon\R^d\to\R$ differentiable. On pose $v_0=0$ et $v_{n+1}=v_n-\frac{1}{2L}\|\nabla f(v_n)\|^2$ pour $n\in\N$. Montrer que $f(v_{n+1})\leq f(v_n)-\frac{1}{2L}\|\nabla f(v_n)\|^2$ pour $n\in\N$. - - On suppose de plus $f$ convexe. - - Montrer que $\forall w\in\R^d$, $f(v_{n+1})\leq f(w)+\left\langle\nabla f(v_n),v_n-w\right\rangle- \frac{1}{2L}\|\nabla f(v_n)\|^2$. - - Montrer que $f(v_n)-f(w)\leq\frac{L}{2}(\|v_n-w\|^2-\|v_{n+1}-w\|^2)$. - - Montrer que $f(v_n)-f(w)\leq\frac{L}{2n}\|w\|^2$. - - Soit $v_*$ un point critique de $f$. Montrer que $v_*$ est un minimum local de $f$ et que la suite $(v_n)$ converge vers $v_*$. +On munit $\R^d$ de sa structure euclidienne canonique. On dit que $f$ est différentiable sur l'ouvert $\Omega$ si $\nabla f$ existe et est continu. + 1. Soient $C$ ouvert convexe non vide de $\R^d$, $f\colon C\to\R$ différentiable. On suppose que $\nabla f$ est $L$-lipschitzien. + 1. Soient $w,v\in C$ et $g:t\mapsto f(v+t(w-v))$. Exprimer $g'(t)$. + 2. Montrer que $f(w)-f(v)=\int_0^1\left\langle\nabla f(v+t(w-v)),w-v\right\rangle{\rm d}t$. + 3. Montrer que $f(w)\leq f(v)+\left\langle\nabla f(v),w-v\right\rangle+\frac{L}{2}\left\| w-v\right\|$. + 2. Soit $f\colon\R^d\to\R$ différentiable. Montrer que $f$ est convexe si et seulement si + $\forall w,v\in\R^d,\, f(w)\geq f(v)+\left\langle\nabla f(v),w-v\right\rangle$. + 3. Soit $f\colon\R^d\to\R$ différentiable. On pose $v_0=0$ et $v_{n+1}=v_n-\frac{1}{2L}\|\nabla f(v_n)\|^2$ pour $n\in\N$. + 1. Montrer que $f(v_{n+1})\leq f(v_n)-\frac{1}{L}\nabla f(v_n)$ pour $n\in\N$. + 2. On suppose dorénavant que plus $f$ est convexe. On considère $w\in\R^d$. + 1. Montrer que$f(v_{n+1})\leq f(w)+\left\langle\nabla f(v_n),v_n-w\right\rangle- \frac{1}{2L}\|\nabla f(v_n)\|^2$. + 2. Montrer que $f(v_n)-f(w)\leq\frac{L}{2}(\|v_n-w\|^2-\|v_{n+1}-w\|^2)$. + 3. Montrer que $f(v_n)-f(w)\leq\frac{L}{2n}\|w\|^2$. + 3. On suppose $f$ strictement convexe. Soit $v_*$ un point critique de $f$. Montrer que $v_*$ est un minimum local de $f$ et que la suite $(v_n)$ converge vers $v_*$. #+end_exercice +#+BEGIN_proof + 1. + 1. Simple, + 2. Simple. + 3. Utilise le caractère $L$-lip. + 2. Si $f$ est convexe, utiliser la question b), et réciproquement, on obtient la croissance de $g'$. + 3. + 1. Ok. + 2. + 1. + 2. + 3. + 3. +#+END_proof *** Probabilités @@ -2715,9 +2826,14 @@ Soient $X_1,\ldots,X_n$ des variables aléatoires a valeurs réelles, identiquem ${}^{\bigstar}$ Soit $A$ une partie de cardinal $n$ de $\R$. On pose $B=A+A=\{a+a',\ a,a'\in A\}$. Montrer que $2n-1\leq\mathrm{card}(B)\leq\dfrac{n(n+1)}{2}$. Généraliser a $B=kA=A+A+\cdots+A$ ($k$ fois). #+end_exercice +# ID:7380 #+begin_exercice [ENS PC 2023 # 220] Soient $a,b\in\Z$ deux entiers distincts. Trouver tous les polynômes $P\in\Z[X]$ tels que $P(a)=b$ et $P(b)=a$. #+end_exercice +#+BEGIN_proof +On peut appliquer $P\mapsto Q(X-a) + a$, pour supposer que $a = 0$, alors on a $P(b) = 0$ et $P(0) = b$, donc $P = (X-b) Q$ tel que $Q(0) = 1$ (et à coefficients entiers). +#+END_proof + #+begin_exercice [ENS PC 2023 # 221] ${}^{\bigstar}$ Soient $P_1,P_2,P_3,P_4\in\R[X]$. Montrer qu'il n'existe aucun voisinage ouvert de $0$ sur lequel on ait simultanement i) $\forall x\lt 0,\ P_1(x)\lt P_2(x)\lt P_3(x)\lt P_4(x)$ @@ -3682,15 +3798,14 @@ Soit $d\in\N^*$. On munit $\R^d$ de la structure euclidienne canonique. On défi #+END_proof +# ID:7421 #+begin_exercice [X MP 2023 # 328] On définit la longueur d'un intervalle borné $I$ de bornes $a$ et $b$ par $\ell(I)=|b-a|$. - Soient $N\in\N^*$, $I_1,\ldots,I_N$ des intervalles bornes de $\R$ tels que $[0,1]\subset\bigcup_{i=1}^NI_i$. Que peut-on dire de $\sum_{i=1}^N\ell(I_i)$? - - Soit $\delta\colon [0,1]\to\R^{+*}$. Montrer qu'il existe $p\in\N^*$, $0\leq x_1\lt x_2\lt \cdots\lt x_p=1$, $t_1,\ldots,t_p\in\R$ tels que, pour tout $k\in\db{1,p}$, $x_{q-1}\leq t_q\leq x_q$ et $x_q-x_{q-1}\leq\delta(t_q)$. - Soit $(I_n)_{n\geq 1}$ une suite d'intervalles bornes de $\R$ telle que $[0,1]\subset\bigcup_{n=1}^{+\i}I_n$. Que peut-on dire de $\sum_{n=1}^{+\i}\ell(I_n)$? #+end_exercice #+BEGIN_proof - - - Incompréhensible ??. Quel sens pour $x_1$ ? Il faudrait que $\delta$ soit continue ? - Si $\sum \ell(I_n)\lt 1$, on montre que ce n'est pas possible. On considère une suite $(\eps_n)$ telle que $\sum \ell(I_n) + \eps_n \lt 1$. On choisit $x_0 = 0$, puis le plus grand intervalle restant qui contient (n'existe pas …) $x_0$, puis $\l(I_{n_0}) \lt x_1\lt \l(I_{n_0}) + \eps_{n_0}$, puis le plus grand qui le contient etc. @@ -3910,11 +4025,12 @@ Que dire d'une fonction $f\colon \R \ra \R$ continue, $1$-périodique et $\sqrt{ Easy. #+END_proof +# ID:nil #+begin_exercice [X MP 2023 # 347] Trouver les fonctions $f\colon\R\to\R$ de classe $\mc C^1$ telles que $|f'|+|f+1|\leq 1$. #+end_exercice #+BEGIN_proof -?? On obtient $f\leq 0$, $f= 0\rightarrow f' = 0$, la fonction est coincée entre $-2$ et $0$. +On obtient $f\leq 0$, $f= 0\rightarrow f' = 0$, la fonction est coincée entre $-2$ et $0$. On peut juste poser $g = f+1$, auquel cas $|g| + |g'|\leq 1$. La fonction $g$ peut osciller tranquillement… @@ -4009,6 +4125,7 @@ Justifier l'existence et calculer $\int_0^1\frac{dt}{2+\lfloor\frac{1}{t}\rfloor Facile #+END_proof +# ID:7453 #+BEGIN_exercice [X 2023 # 356] Soit $f\colon x \in \R \mapsto e^{\frac{x^2}{2}} \int_x^{+\i} e^{-\frac{t^2}{2}}\dt$. 1. Montrer que $f(x)\lt \frac{1}{x}$ pour tout $x\gt 0$. @@ -4017,10 +4134,10 @@ Soit $f\colon x \in \R \mapsto e^{\frac{x^2}{2}} \int_x^{+\i} e^{-\frac{t^2}{2}} #+END_exercice #+BEGIN_proof 1. On a $xf(x)\lt e^{x^2/2} \int_x^{+\i} t e^{-\frac{t^2}{2}}\dt$, ou IPP. - 2. L'IPP donne $\frac{1}{x} - e^{x^2/2} \int_x^{+\i} \frac{e^{-t^2/2}}{t^2}$. Une autre donnera $\geq \frac{1}{x} - \frac{1}{x^3}$, ce qui ne permet pas de conclure. + 2. L'IPP donne une minoration, insuffisante (cf question suivante). - L'inégalité revient à montrer que $f(x)^2 + x f(x) - 1 \geq 0$. !! - 3. IPP successives semblent fonctionner. + L'inégalité revient à montrer que $f(x)^2 + x f(x) - 1 \geq 0$, c'est-à-dire $f(x)\big(f(x) + x\big)\gt 1$. En dérivant $\big(x e^{-x^2/2}\big)' = e^{-x^2/2}\big(1 - x^2\big)$, donc $f(x) + x = e^{x^2/2}\int_x^{+\i} t^2 e^{-t^2/2}\dt$. Puis Cauchy-Schwarz sur le produit $f(x)\big(f(x) + x\big)$ donne le résultat. + 3. L'IPP donne $\frac{1}{x} - e^{x^2/2} \int_x^{+\i} \frac{e^{-t^2/2}}{t^2}$, et on réitère. #+END_proof @@ -4082,20 +4199,32 @@ Pour $f\in F$ et $n\in\N$, soit $L_n(f)\colon x\in[0,1]\mapsto\inf\limits_{y\in[ #+END_proof +# ID:7374 #+begin_exercice [X MP 2023 # 361] Soient $a\in\R^{+*}$ et $f\colon\R^+\to\R^{+*}$ de classe $C^1$ telle que $\dfrac{f'(x)}{f(x)}\sim\dfrac{a}{x}$ quand $x\to+\i$. 1. s Rappeler le théorème d'intégration des relations de comparaison. 2. Donner un équivalent de $\ln f(x)$ quand $x\to+\i$. 3. Déterminer le domaine de définition de la fonction $u\colon x\mapsto\sum_{n=0}^{+\i}f(n)e^{-nx}$. 4. Déterminer les limites de $u$ aux bornes de son intervalle de définition. - 5. Montrer qu'il existe une constante $C\gt 0$ telle que $u(x)\sim\dfrac{C}{x}u\left(\dfrac{1}{x}\right)$ quand $x\to+\i$. + 5. Montrer qu'il existe une constante $C\gt 0$ telle que $u(x)\sim\frac{C}{x^{\a + 1}}u\left(\dfrac{1}{x}\right)$ quand $x\to+\i$. #+end_exercice #+BEGIN_proof 1. 2. $\ln f(x) \sim a \ln x$. 3. Tout $x\gt 0$. 4. L'infini et $0$. - 5. !! + 5. On regarde le produit $x u(x) u\big(\frac{1}{x}\big)$. Si $f(x) = x^{a}$, alors $u(x) = \sum n^{\a} e^{-n x}$. + + Pour $\a$ entier, c'est la dérivée $\a$-ième de $\frac{1}{1-e^{-x}}$. + + On prend $a = 1$, c'est $\frac{e^x}{\big(e^x-1\big)^2}$, équivalent à $e^{-x}$ en $+\i$, + et en $0$, on est équivalent à $\frac{1}{x^2}$. + + Problème d'énoncé. + + En général, $u(x)$ va être équivalent, en $+\i$, à son premier terme. + + En $0$, les premiers termes sont toujours négligeables, donc on peut remplacer $f(n)$ par $n^{\a}$, puis dans l'intégrale obtenue, il suffit de faire le changement de variable : $\int_0^{+\i} t^{\a} e^{-tx}\dt$. #+END_proof # ID:7314 @@ -4219,18 +4348,26 @@ Soit $f$ continue sur $[0,1]$ et $g\colon x\mapsto\int_0^1\frac{f(t)}{1+xt}\dt$ # ID:7322 #+begin_exercice [X MP 2023 # 372] - Calculer $\int_0^{+\i}e^{-t}\sin(xt)\,dt$ pour tout réel $x$. - - On pose $F:x\mapsto\int_0^{+\i}\frac{\sin(xt)}{t\,(1+t^2)}\,dt$. Montrer que $F$ est de classe $\mc C^2$ sur $\R^{+*}$ et que $\forall x\gt 0,\ F^{''}(x)=F(x)-\int_0^{+\i}\frac{\sin t}{t}\, dt$ + - On pose $F:x\mapsto\int_0^{+\i}\frac{\sin(xt)}{t\,(1+t^2)}\,dt$. Montrer que $F$ est de classe $\mc C^2$ sur $\R^{+*}$ et que $\forall x\gt 0,\ F''(x)=F(x)-\int_0^{+\i}\frac{\sin t}{t}\, dt$ - Donner une expression simplifiée de $F$. #+end_exercice -#+begin_exercice [X MP 2023 # 373] :todo: +# ID:7504 +#+begin_exercice [X MP 2023 # 373] Soit $f\in\mc C^0(\R_+^{*},\R)$ de carré intégrable. On pose $S_f\colon x\in\R_+^{*}\mapsto\int_0^{+\i}\frac{f(y)}{x+y}dy$. - Justifier la bonne définition de $S_f$. - Montrer que $S_f$ est de carré intégrable. #+end_exercice -#+BEGIN_proof :todo: +#+BEGIN_proof - CS - - !! + - On a, par Cauchy-Schwarz, en multipliant et divisant par du $y^{1/4}$, + $$S_f(x)^2 = \left(\int_0^{+\i} \frac{f(y) \sqrt{y}}{\sqrt{x+y}} \times \frac{1}{\sqrt{y} (x+y)}\right)^2 \leq \int \frac{f^2 y}{x+y} \dy \int \frac{1}{y (x+y)\dy},$$ + + $$S_f(x)^2 \leq \int \frac{1}{x+y} \frac{\dy}{\sqrt{y}} \int \frac{f(y)^2 \sqrt{y}}{x+y}\dy.$$ + + Le premier facteur est en $x^{-1/2}$, puis $x^{-1/2}\int \frac{f(y)^2 \sqrt{y}}{x+y}\dy = x^{-1/2}$, si on admet Fubini (positif), $\int \frac{\sqrt{y}}{\sqrt{x} (x+y)}\dx = \pi$, d'où l'intégrabilité en $x$ de ce qui précède. + + - R L'opérateur est connu sous le nom de «Hilbert-Hankel», ou «Carleman». C'est le carré de l'opérateur de Laplace $L_f \colon x\mapsto \int_{0}^{+\i} e^{-xy} f(y)\dy$. La démonstration de la bonne définition de $L_f$ est la même #+END_proof @@ -4276,6 +4413,7 @@ Soient $f$ et $g$ deux fonctions de classe $\mc C^1$ de $\R^+$ dans $\R^{+*}$. S #+END_proof +# ID:7422 #+begin_exercice [X MP 2023 # 377] Soient $v\colon \R\to\R$ une fonction continue à support compact et $\omega\in\R^{+*}$. On considère l'équation différentielle $y''+\omega^2 y=v(t)$ dont on note $\mc{S}_E$ l'ensemble des solutions. - Montrer que, pour tout $(a,b)\in\R^2$, il existe une unique solution $f^+_{a,b}$ (resp. $f^-_{a,b}$) de $(E)$ telle que $f^+_{a,b}(t)=a\cos(\omega t)+b\sin(\omega t)$ pour tout $t$ dans un voisinage de $+\i$, (resp. $f^-_{a,b}(t)=a\cos(\omega t)+b\sin(\omega t)$ pour tout $t$ dans un voisinage de $-\i$. @@ -4286,8 +4424,8 @@ Soient $v\colon \R\to\R$ une fonction continue à support compact et $\omega\in\ #+BEGIN_proof 1. Appliquer les conditions aux bords du compact. 2. Pas de difficulté. - 3. Méthode de variation de la constante je pense, à écrire. - 4. !! + 3. Intégrer l'équation différentielle, une fois multipliée par $\cos (\om t)$, faire deux IPP. Pour simplifier, se placer en un multiple de $2\pi$, en dehors du support. + 4. Ce serait le cas si $\forall \om,\, c(\om) = s(\om) = 0$. On peut ou bien appliquer le théorème de Weierstrass trigonométrique, ou bien dériver successivement, et obtenir que $\int P(t) v(t) \cos (\om t) \dt = 0$, et en prenant $P$ une approximation à $\eps$ près de $v$, on conclut. #+END_proof @@ -4581,18 +4719,27 @@ Une urne contient $a$ boules jaunes et $b$ boules rouges. On effectue une succes #+END_proof -#+BEGIN_exercice [X 2023 # 396] :todo: +# ID:7349 +#+BEGIN_exercice [X 2023 # 396] Soient $n \geq 1$ et $A, B, C$ des variables aléatoires indépendantes uniformément distribuées sur $\{0,1\}^n$. 1. Pour $n \geq 2$, calculer la probabilité $p_n$ que $A B C$ soit un triangle équilatéral. 2. Déterminer un équivalent de $p_n$. #+END_exercice -#+BEGIN_proof :todo: +#+BEGIN_proof 1. On prend $A = \vec 0$. Alors on veut $B,C$ avec autant de termes $1$, et autant de différences entre les deux. On considère les ensembles $B\subset \db{1,n}$, $C\db{1,n}$, et $B\oplus C$. - Les parties $U = B\setminus C$, $V = C\setminus B$ et $W = B\cap C$ vérifient $u + w = v + w = u+v$, donc ils sont de même cardinaux, et disjoints. - 2. Super dur, fait dans la RMS. + Les parties $U = B\setminus C$, $V = C\setminus B$ et $W = B\cap C$ vérifient $u + w = v + w = u+v$, donc ils sont de même cardinaux, et disjoints, et cela revient à les choisir. + + On obtient $sum_{k} {n\choose k}{n-k\choose k} {n-2k \choose k}$. + 2. Par exemple, dans le cas plus simple $\sum_k {n\choose k}$, on sait que le poids est aux environs de $k = \frac{n}{2}$, avec de l'ordre de $\sqrt{n}$ autour. + + Pour $\sum_k {n\choose k} {n-k \choose k} = \sum_k \frac{n!}{k! k! (n-2k)!}$, le terme est maximal lorsque $k^2 = (n - 2k)(n-2k -1)$, c'est-à-dire $k\simeq \frac{n}{3}$. + + La RMS remarque que $p_n$, à un facteur près, est égal à l'intégrale double + $$I_n = \int_{-\pi}^{\pi}\int_{-\pi}^{\pi} \big(1 + e^{i\theta} + e^{i \theta'} + e^{-i(\theta + \theta')}\big)^n \d\theta \d \theta'.$$ + Puis on écrit la fonction intégrée comme $1 - \frac{1}{4}\big(\theta^2 + \theta'^2 + \theta \theta'\big) + o(\theta^2 + \theta'^2)$. On change de variable en $\sqrt{n}$, on fait de la convergence dominée, on diagonalise la forme quadratique (dans une BON). #+END_proof # ID:nil @@ -4631,22 +4778,26 @@ Soit $M=\begin{pmatrix}a&-b&-c&-d\\ b&a&d&-c\\ c&-d&a&b\\ d&c&-b&a\end{pmatrix}$ - On suppose que $X-Y$ et $Y$ sont indépendantes. Déterminer la loi de $Y$, puis celle de $X$. #+end_exercice -#+BEGIN_exercice [X MP 2023 # 400] :todo: +# ID:7350 +#+BEGIN_exercice [X MP 2023 # 400] Soit $n\geq 3$ un entier. Si $k\in\Z$, on note $\overline{k}$ la reduction de $k$ modulo $n$. Soient $X_1,\ldots,X_n$ des variables aléatoires indépendantes à valeurs dans $\Z/n\Z$ telles que, pour tout $k\in\db{1,n}$, $X_k$ suit la loi uniforme sur $\{\overline{1},\overline{2},\overline{3}\}$. Soit $F$ l'application aléatoire de $\Z/n\Z$ dans lui-même telle que, pour tout $k\in\db{1,n}$, $F(\overline{k})=\overline{k}+X_k$. Calculer la probabilité que $F$ soit bijective. #+END_exercice -#+BEGIN_proof :todo: +#+BEGIN_proof Quand $X_k$ vaut $2$, le prochain ne doit pas valoir $1$, et quand $X_k$ vaut $3$ le prochain ne doit pas valoir $2$, et celui d'après ne doit pas valoir $1$. On interdit les pattern 21, 32, et 3*1. -Si c'est une bijection, alors en itérant, ou bien on obtient deux $\frac{n}{2}$-cycles, ou bien $3$, $\frac{n}{3}$-cycles, ou bien ? - + S'il y a au moins trois cycles, il doivent trois alternés, donc ils sont tous de longueur $3$. Facile de les compter. - + S'il y a un unique cycle, on peut compter le nombre de tours qu'il fait. S'il fait un seul tour, c'est que des $+1$. +Plutôt, pour un interstice entre $i$ et $i+1$, on regarde le nombre de truc qui lui saute au dessus. S'il y en a $3$, alors cela implique de même pour le prochain interstice. S'il y en a $1$ seul, de même. - S'il fait trois tours, ils doivent $3$-alterner, donc c'est que des $+3$. +S'il y en a $2$, alors ou bien les sauts se croisent, ou il ne se croisent pas, et il peut en être de même pour les interstices d'à côté. Je dirais donc qu'il y a $2^n$ possibilités, plus $2$. - S'il fait deux tours : alors il peut juste jamais faire 1, 1, attention également à la fin du tour (compter aussi le premier pas, et le dernier qui arrive en n-1). - + S'il y a deux cycles, ils sont fortement liés, et l'un détermine entièrement le second. Il faut simplement que l'un ne fasse jamais deux sauts de $1$. +Vérifions : pour $n = 2$. Soit $+1, +1$, soit $+2, + 2$, soit $+3, +3$, soit $+1, +3$, $+3, +1$. Bon ça ne marche pas. + +Pour $n = 3$ : $(1, 1, 1)$ (chaque interstice est sautée une fois). $(3, 3, 3)$ (chaque interstice est sautée trois fois). $(2, 2, 2)$ donne tous croisés, et il se trouve que si une interstice est parallèle, les deux autres doivent être croisées. Finalement, $6$ possibilités. + +En fait, autour d'une parallèle, on doit être croisé. Le nombre de cercle de $n$ termes $0, 1$ sans paires de $1$ consécutifs est : $F_{n-1} + F_{n-3}$. + +Pour $n = 4$, on prédit donc $1 + 1 + 5 + 2 = 9$, ce qui fonctionne. #+END_proof @@ -4683,6 +4834,7 @@ Soient $x\in\R^{+*}$, $(X_k)_{k\geq 1}$ une suite i.i.d. de variables aléatoire ** X PSI :autre: *** Algèbre +# ID:7381 #+begin_exercice [X PSI 2023 # 404] Pour $n\geq 2$ on pose $P_n=(X+1)^n+X^n+1$ et $Q(X)=(X^2+X+1)^2$. @@ -4758,11 +4910,13 @@ Calculer le volume du parallelepipede engendre par les vecteurs $\overrightarrow #+end_exercice *** Géométrie + +# ID:7382 #+begin_exercice [X PSI 2023 # 418] Soient $abc$ un vrai triangle du plan complexe, $\alpha$ (resp. $\beta$, resp. $\gamma$) a rotation de centre $a$ (resp. $b$, resp. $c$) et d'angle $\dfrac{2\pi}{3}$. - - Montrere que le centre de $\alpha\circ\beta$ appartient a l'intersection des trisectrices du triangles. - - Montrere que $\alpha^3\circ\beta^3\circ\gamma^3$ est l'identite du plan. - - Montrere que les points d'intersection des trisectrices forment un triangle equlateral. + - Montrer que le centre de $\alpha\circ\beta$ appartient à l'intersection des trisectrices du triangle. + - Montrer que $\alpha^3\circ\beta^3\circ\gamma^3$ est l'identité du plan. + - Montrer que les points d'intersection des trisectrices forment un triangle équilatéral. #+end_exercice *** Probabilités @@ -4794,6 +4948,7 @@ On dispose de $n$ objets differents. On effectue des tirages aléatoires indépe ** X PC :autre: *** Algèbre +# ID:7383 #+begin_exercice [X PC 2023 # 424] Montrer que, pour tout $n\in\N^*$, il existe $m\in\N^*$ et $\eps_1,\ldots,\eps_m\in\{-1,1\}$ tels que $n=\sum_{k=1}^m\eps_kk^2$. #+end_exercice @@ -4989,9 +5144,14 @@ Soit $E$ un espace vectoriel norme de dimension finie. Soient $p,q\in\mc{L}(E)$ - Montrer que $u=pq+(\mathrm{id}-p)(\mathrm{id}-q)$ est inversible et que $p=uqu^{-1}$. #+end_exercice +# ID:7387 #+begin_exercice [X PC 2023 # 467] Soit $x\geq 0$. Donner un équivalent de la suite de terme général $u_n=\prod_{i=1}^n(x+i)$. #+end_exercice +#+BEGIN_proof +Quotienter par $n!$, passer au logarithme, et utiliser $\big|\ln (1+x) - x\big|\leq x^2$. +#+END_proof + #+begin_exercice [X PC 2023 # 468] Montrer que, pour tout $n\in\N^*$, $\sum_{k=0}^{n-1}|\cos(k)|\geq\frac{4n}{10}$. @@ -5072,9 +5232,14 @@ Soient $f$ et $g\colon\R\to\R$ continues et croissantes. Soit $\lambda\gt 0$. Mo Soit $f:[0,1]\to[0,1]$ une fonction croissante. Montrer que $f$ admet un point fixe. #+end_exercice +# ID:7384 #+begin_exercice [X PC 2023 # 483] -Soit $f:[0,1]\mapsto\R$ de classe $C^1$ telle que $f(0)=f(1)=0$. Montrer que, pour tout $a\in\R$, $f'+af$ s'annule sur $]0,1[$. +Soit $f\colon [0,1]\mapsto\R$ de classe $C^1$ telle que $f(0)=f(1)=0$. Montrer que, pour tout $a\in\R$, $f'+af$ s'annule sur $]0,1[$. #+end_exercice +#+BEGIN_proof +C'est la dérivée de… +#+END_proof + #+begin_exercice [X PC 2023 # 484] - Soit $f\colon\R^{+*}\to\R$ une fonction $C^{\i}$. Montrer que pour tout $n\gt 0$ et pour tout $x\gt 0$ il existe $c\in]x,x+n[$ tel que $\sum_{k=0}^n\binom{k}{n}(-1)^{n-k}f(x+k)=f^{(n)}(c)$. @@ -5098,12 +5263,19 @@ Soit $f\colon\R\to\R$ continue. Montrer que les propositions suivantes sont équ - pour tout intervalle $I\subset\R$ ouvert, pour toute $\phi\in\mc C^{\i}\left(I,\R\right)$, pour tout $x_0\in I$, si $f-\phi$ admet un minimum local en $x_0$, alors $\phi'\left(x_0\right)\geq 0$. #+end_exercice +# ID:7385 #+begin_exercice [X PC 2023 # 488] Soit $g\in\mc C^3([0,2],\R)$ telle que $g(0)=g(1)=g(2)=0$. - Montrer : $\forall x\in[0,2]$, $\exists c\in[0,2]$, $g(x)=\dfrac{1}{6}x(x-1)(x-2)g^{(3)}(c)$. - Montrer que $\int_0^2|g(x)|\ dx\leq\dfrac{1}{12}\|g^{(3)}\|_{\i}$. - Montrer que $\left|\int_0^2g(x)\ dx\right|\leq\dfrac{1}{24}\left[\sup \left(g^{(3)}\right)-\inf\left(g^{(3)}\right)\right]$. #+end_exercice +#+BEGIN_proof + - + - + - Retirer à $g$ un polynôme de la forme $\la (x-1)(x-2)x$, qui a une intégrale nulle, de sorte que son sup soit égal à l'opposé de son inf. +#+END_proof + #+begin_exercice [X PC 2023 # 489] Soient $(a,b)\in\R^2$ avec $a\lt b$, et $f,g\in\mc C^0([a,b],\R^{+*})$. @@ -5118,41 +5290,41 @@ $\forall x\in\left[0,1\right]$, $ f\left(x\right)=\int_0^1K(x,z)g(z)\,dz$ et $ g #+end_exercice #+begin_exercice [X PC 2023 # 491] -Soit $E$ l'espace des fonctions $ f\in\mc C^2(\R,\R)$ telles que +Soit $E$ l'espace des fonctions $f\in\mc C^2(\R,\R)$ telles que $\sup\limits_{x\in\R}\left(1+x^2\right)\bigl{(}\left|f(x)\right|+ \left|f'(x)\right|+\left|f^{''}(x)\right|\bigr{)}\lt +\i$. -Pour $(t,x)\in\R^2$, on définit $ A_t(f)(x)=txf(x)+f'(x)$ et $ A_t^*(f)(x)=txf(x)-f'(x)$. +Pour $(t,x)\in\R^2$, on définit $A_t(f)(x)=txf(x)+f'(x)$ et $A_t^*(f)(x)=txf(x)-f'(x)$. Montrer que $\forall t\in\R$, $\forall f\in E$, $\int_{-\i}^{+\i}A_t^*(A_t(f))(x)\,f(x)\,dx\geq 0$. #+end_exercice #+begin_exercice [X PC 2023 # 492] Soient $a,b\in\R$ avec $a\lt b$. - - Soient $ f_1,\dots,f_n\in\R^{[a,b]}$. Montrer que $(f_1,\dots,f_n)$ est libre si et seulement s'il existe $ x_1,\dots,x_n\in[a,b]$ tels que la matrice $(f_i(x_j))_{1\leq i,j\leq n}$ soit inversible. + - Soient $f_1,\dots,f_n\in\R^{[a,b]}$. Montrer que $(f_1,\dots,f_n)$ est libre si et seulement s'il existe $x_1,\dots,x_n\in[a,b]$ tels que la matrice $(f_i(x_j))_{1\leq i,j\leq n}$ soit inversible. - Soit $E=\text{Vect}(f_1,\dots,f_n)$. Montrer que toute limite simple de fonctions de $E$ est encore dans $E$. - - La convergence est-elle uniforme? + - La convergence est-elle uniforme ? #+end_exercice #+begin_exercice [X PC 2023 # 493] -Posons $ A=\Q\cap\left[\,0\,;\,1\,\right]$. Existe-t-il une suite $(f_n)$ de fonctions de $A$ dans $A$, continues sur $A$ et qui converge simplement sur $A$ vers une fonction $f$ qui n'est continue en aucun point de $A$? La convergence peut-elle être uniforme? +Posons $A=\Q\cap\left[\,0\,;\,1\,\right]$. Existe-t-il une suite $(f_n)$ de fonctions de $A$ dans $A$, continues sur $A$ et qui converge simplement sur $A$ vers une fonction $f$ qui n'est continue en aucun point de $A$? La convergence peut-elle être uniforme? #+end_exercice #+begin_exercice [X PC 2023 # 494] -On considère l'ensemble $E$ des applications continues $ f\colon\R\mapsto\R$ telles qu'il existe $ M\gt 0$ vérifiant $\forall x,y\in\R,\left|f(x+y)-f(x)-f(y)\right|\leq M$. +On considère l'ensemble $E$ des applications continues $f\colon\R\mapsto\R$ telles qu'il existe $M\gt 0$ vérifiant $\forall x,y\in\R,\left|f(x+y)-f(x)-f(y)\right|\leq M$. - Montrer que $E$ est un espace vectoriel contenant le sous-espace des applications linéaires et celui des applications bornées. - - Soit $ f\in E$. Pour $n\in\N$, on pose $ g_n:x\in\R\mapsto 2^{-n}f(2^nx)$. Montrer que la suite $(g_n)$ converge uniformément vers une application linéaire $g$. En déduire que $f$ s'écrit, de facon unique, comme somme d'une application linéaire et d'une application bornée. + - Soit $f\in E$. Pour $n\in\N$, on pose $g_n:x\in\R\mapsto 2^{-n}f(2^nx)$. Montrer que la suite $(g_n)$ converge uniformément vers une application linéaire $g$. En déduire que $f$ s'écrit, de facon unique, comme somme d'une application linéaire et d'une application bornée. #+end_exercice #+begin_exercice [X PC 2023 # 495] On considère une suite $(f_n)_{n\geq 0}$ d'applications de $[0,1]$ dans $\R$ qui converge simplement sur $[0,1]$ vers une application continue $f$. - - On suppose les $f_n$ de classe $C^1$ et de derivées uniformément bornées, c'est-a-dire qu'il existe $ C\geq 0$ tel que $\forall n,\ \ \|f_n'\|_{\i}\leq C$. Montrer que la convergence de $(f_n)$ vers $f$ est uniforme sur $[0,1]$. - - On suppose maintenant les $f_n$ de classe $C^k$ pour un entier $ k\in\N^*$ et de derivées $k$-ièmes uniformément bornées. La convergence de la suite $(f_n)$ est-elle toujours uniforme sur $[0,1]$? + - On suppose les $f_n$ de classe $C^1$ et de derivées uniformément bornées, c'est-a-dire qu'il existe $C\geq 0$ tel que $\forall n,\ \ \|f_n'\|_{\i}\leq C$. Montrer que la convergence de $(f_n)$ vers $f$ est uniforme sur $[0,1]$. + - On suppose maintenant les $f_n$ de classe $C^k$ pour un entier $k\in\N^*$ et de derivées $k$-ièmes uniformément bornées. La convergence de la suite $(f_n)$ est-elle toujours uniforme sur $[0,1]$? #+end_exercice #+begin_exercice [X PC 2023 # 496] -Pour $ n\in\N^*$ et $ x\in\R^+$, on pose $ f_n(x)=\cos\biggl{(}\dfrac{x}{\sqrt{n}}\biggr{)}\,\mathbf{1}_{\left[1,\frac{ \pi\sqrt{n}}{2}\right]}(x)$. - - Montrer que $(f_n)$ converge simplement vers une fonction $ f$ que l'on precisera. - Montrer qu'il existe $C\gt 0$ tel que $\forall x\in\R^+,\ \forall n\in\N^*,\ |f_n(x)-f(x)|\leq\frac{C}{\sqrt{n}}$. +Pour $n\in\N^*$ et $x\in\R^+$, on pose $f_n(x)=\cos\biggl{(}\dfrac{x}{\sqrt{n}}\biggr{)}\,\mathbf{1}_{\left[1,\frac{ \pi\sqrt{n}}{2}\right]}(x)$. + - Montrer que $(f_n)$ converge simplement vers une fonction $f$ que l'on precisera. - Montrer qu'il existe $C\gt 0$ tel que $\forall x\in\R^+,\ \forall n\in\N^*,\ |f_n(x)-f(x)|\leq\frac{C}{\sqrt{n}}$. #+end_exercice #+begin_exercice [X PC 2023 # 497] @@ -5162,7 +5334,7 @@ Soit $(f_n)_{n\in\N}$ une suite de fonctions appartenant a $\mc C^3(\R,\R)$ et $ #+end_exercice #+begin_exercice [X PC 2023 # 498] -On note $E=\mc C^0([0,1],\R)$. Si $f\in E$, on définit la fonction $T(f)$ par $T(f)(0)=f(0)$ et $ T(f)(x)=\frac{1}{x}\int_0^xf(t)dt$ pour $x\in\,]0,1]$. +On note $E=\mc C^0([0,1],\R)$. Si $f\in E$, on définit la fonction $T(f)$ par $T(f)(0)=f(0)$ et $T(f)(x)=\frac{1}{x}\int_0^xf(t)dt$ pour $x\in\,]0,1]$. On définit par recurrence sur $n\in\N$, $T^{n+1}(f)=T(T^n(f))$. - Montrer que $T$ est bien définie comme fonction de $E$ dans lui-meme. @@ -5171,30 +5343,30 @@ On définit par recurrence sur $n\in\N$, $T^{n+1}(f)=T(T^n(f))$. #+end_exercice #+begin_exercice [X PC 2023 # 499] -Soit $ F:x\mapsto\sum_{k=0}^{+\i}\left(-1\right)^kx^{2^k}$. +Soit $F:x\mapsto\sum_{k=0}^{+\i}\left(-1\right)^kx^{2^k}$. - Déterminer le domaine de définition de $F$. - Trouver une relation entre $F\left(x\right)$ et $F\left(x^2\right)$. -On pose $ G:x\mapsto\sum_{k=0}^{+\i}x^{4^k}\left(1-x^{4^k}\right)$. - - Montrer que $ G\left(x\right)$ converge pour tout $x\in\,]0,1[$. +On pose $G:x\mapsto\sum_{k=0}^{+\i}x^{4^k}\left(1-x^{4^k}\right)$. + - Montrer que $G\left(x\right)$ converge pour tout $x\in\,]0,1[$. - Trouver une relation entre $F$ et $G$. #+end_exercice #+begin_exercice [X PC 2023 # 500] -Soient $\alpha\gt 0$ et, pour $n\in\N^*$, $ f_n:x\mapsto\frac{\sin nx}{n^{\alpha}}$. La série $\sum f_n$ converge-t-elle simplement sur $\R$? Pour quels $\alpha$ a-t-on convergence uniforme? +Soient $\alpha\gt 0$ et, pour $n\in\N^*$, $f_n:x\mapsto\frac{\sin nx}{n^{\alpha}}$. La série $\sum f_n$ converge-t-elle simplement sur $\R$? Pour quels $\alpha$ a-t-on convergence uniforme? #+end_exercice #+begin_exercice [X PC 2023 # 501] -On pose $ g:x\mapsto\frac{1}{x}-\sum_{n=1}^{+\i}\frac{2x}{n^2-x^2}$. +On pose $g:x\mapsto\frac{1}{x}-\sum_{n=1}^{+\i}\frac{2x}{n^2-x^2}$. - Montrer que $g$ est définie et continue sur $\R\setminus\Z$. - Montrer que $g$ est 1-périoddique. - - Etablir une relation entre $ g\left(\frac{x}{2}\right)$, $ g\left(\frac{x+1}{2}\right)$ et $ g(x)$ des que les termes font sens. + - Etablir une relation entre $g\left(\frac{x}{2}\right)$, $g\left(\frac{x+1}{2}\right)$ et $g(x)$ des que les termes font sens. - En déduire que $\pi\,\mathrm{cotan}(\pi x)=g(x)$ pour tout $x\in\R\setminus\Z$. #+end_exercice #+begin_exercice [X PC 2023 # 502] Soit $(a_n)$ une suite de réels positifs tels que $\sum a_n$ diverge. - - Montrer que, pour tout intervalle de longueur non nulle $I$, il existe $x\in I$ tel que la série $\sum a_n\cos(nx)$ ne converge pas absolument. On pourra d'abord montrer que, pour tout $ a\lt b$ et tout $N$ il existe $M\in\N$ et $ x\in[a,b]$ tel que $\sum_{n=0}^Ma_n\cos^2(nx)\gt N$. - Existe-t-il des exemples ou la série converge sur un intervalle non trivial? + - Montrer que, pour tout intervalle de longueur non nulle $I$, il existe $x\in I$ tel que la série $\sum a_n\cos(nx)$ ne converge pas absolument. On pourra d'abord montrer que, pour tout $a\lt b$ et tout $N$ il existe $M\in\N$ et $x\in[a,b]$ tel que $\sum_{n=0}^Ma_n\cos^2(nx)\gt N$. - Existe-t-il des exemples ou la série converge sur un intervalle non trivial? #+end_exercice #+begin_exercice [X PC 2023 # 503] @@ -5262,9 +5434,14 @@ On dispose de $N$ pieces equilibrées. On lance les $N$ pieces de maniere indép - Soit $T$ l'instant ou l'on n'a plus de piece. Calculer $\mathbf{E}\left(T\right)$ dans le cas ou $N=4$. #+end_exercice +# ID:7386 #+begin_exercice [X PC 2023 # 514] -Soient $X$ et $Y$ deux variables aléatoires discrètes indépendantes telles que $Y$ prenne un nombre fini de valeurs, et $\mathbf{E}(Y)=0$. On suppose que $|X|$ admet une espérance. Montrer que $\mathbf{E}(|X-Y|)\geq\mathbf{E}(|X|)$. +Soient $X$ et $Y$ deux variables aléatoires discrètes indépendantes telles que $Y$ prenne un nombre fini de valeurs, et $\mathbf{E}(Y)=0$. On suppose que $|X|$ admet une espérance. Montrer que $\mathbf{E}(|X+Y|)\geq\mathbf{E}(|X|)$. #+end_exercice +#+BEGIN_proof +Conditionner sur les valeurs de $X$, c'est de la convexité de la valeur absolue. +#+END_proof + #+begin_exercice [X PC 2023 # 515] On tire une piece $n$ fois indépendamment avec probabilité de faire pile $1/n$. Soit $p_n$ la probabilité d'obtenir un nombre impair de fois pile. Étudier le comportement de $p_n$. @@ -5468,9 +5645,14 @@ Montrer que, pour tout $q\in\N^*$, $P_q=\prod_{i=1}^n(X-\lambda_i^q)$ est a coef Soit $\mathbb{K}=\Q+\sqrt{2}\Q+\sqrt{3}\Q+\sqrt{6}\Q$. Montrer que $\mathbb{K}$ est un $\Q$-sous-espace vectoriel de $\R$ et que $(1,\sqrt{2},\sqrt{3},\sqrt{6})$ est une base de $\mathbb{K}$. #+end_exercice +# ID:7528 #+begin_exercice [Mines 2023 # 547] -Quelle est la dimension du $\Q$-sous-espace de $\R$ engendre par $\mathbb{U}_5$? +Quelle est la dimension du $\Q$-sous-espace de $\R$ engendré par $\mathbb{U}_5$? #+end_exercice +#+BEGIN_proof +On a $1$, on a $\cos \frac{2\pi}{5} = \frac{1}{4}\big(\sqrt{5} - 1\big)$, ce que l'on trouve ou bien en écrivant $\cos (2\theta) = \cos (3\theta)$ et en linéarisant, ou bien en écrivant $\cos \frac{2\pi}{5} = z + z^{-1}$. Comme $1 + z + \dots + z^4 = 0$, on peut trouver une équation quadratique vérifiée par $\cos \theta$. +#+END_proof + #+begin_exercice [Mines 2023 # 548] Soient $x,y,z$ des rationnels non nuls. Montrer que la matrice $\left(\begin{array}{ccc}x&y&z\\ 2y&z&2x\\ z&x&2y\end{array}\right)$ est inversible. @@ -5480,9 +5662,16 @@ Soient $x,y,z$ des rationnels non nuls. Montrer que la matrice $\left(\begin{arr Soient $x,y\in\R$ et $D=\begin{vmatrix}1&0&1&0&0\\ x&1&y&1&0\\ x^2&2x&y^2&2y&2\\ x^3&3x^2&y^3&3y^2&6y\\ x^4&4x^3&y^4&4y^3&12y^2\end{vmatrix}$. Montrer que $D=0$ si et seulement si $x=y$. #+end_exercice +# ID:7529 #+begin_exercice [Mines 2023 # 550] Soit $A\in\M_n(\mathbb{K})$ dont on note $C_1,\ldots,C_n$ les colonnes. Soit $B$ la matrice dont les colonnes sont $C'_1,\ldots,C'_n$ avec : $C'_j=\sum_{i\neq j}C_i$. Déterminer $\det B$ en fonction de $\det A$. #+end_exercice +#+BEGIN_proof +Depuis $C'$, on retire chaque colonne à la précédente, on obtient + $C_2-C_1, C_3-C_2, \dots,C_n - C_{n-1}, \sum_{i\neq n} C_i$. En ajoutant des premières à la dernière, on obtient $(n-1)C_n$ comme dernière colonne. + + Par ailleurs, à partir de $(C_1,\dots, C_n)$, $(-C_1, -C_2,\dots, -C_n)$, puis $(C_2 - C_1, C_3 - C_2, \dots, C_n-C_{n-1}, -C_n)$, puis +#+END_proof #+begin_exercice [Mines 2023 # 551] - Soient $p\in\N^*$, $a_1,\ldots,a_p\in\R$ non tous nuls et $b_1,\ldots,b_p\in\R$ avec $b_1\lt \cdots\lt b_p$. Montrer que $f_p:x\in\R\mapsto\sum_{i=1}^pa_ie^{b_ix}$ s'annule au plus $p-1$ fois sur $\R$. @@ -5570,12 +5759,16 @@ Montrer $\forall k\geq 1$, $\mathrm{tr}(A^k)+\mathrm{tr}(B^k)=\mathrm{tr}\left(( Soit $f\colon\M_n(\mathbb{K})\to\mathbb{K}$ non constante telle que : $\forall A,B\in\M_n(\mathbb{K})$, $f(AB)=f(A)f(B)$. Montrer que $A\in\mathrm{GL}_n(\mathbb{K})\Longleftrightarrow f(A)\neq 0$. #+end_exercice +# ID:7516 #+begin_exercice [Mines 2023 # 565] Soient $A,B$ dans $\M_n(\R)$. Montrer que $\mathrm{Ker}\,A=\mathrm{Ker}\,B$ si et seulement s'il existe $P$ inversible telle que $B=PA$. #+end_exercice +# ID:7517 #+begin_exercice [Mines 2023 # 566] -Soient $E$ un $\mathbb{K}$-espace vectoriel de dimension finie et $u\in\mc{L}(E)$. Montrer l'equivalence entre : i) $u^2=0$ et $\exists v\in\mc{L}(E),\,u\circ v+v\circ u=\mathrm{id}$, ii) $\mathrm{Im}\,u=\mathrm{Ker}\,u$. +Soient $E$ un $\mathbb{K}$-espace vectoriel de dimension finie et $u\in\mc{L}(E)$. Montrer l'equivalence entre : + + $u^2=0$ et $\exists v\in\mc{L}(E),\,u\circ v+v\circ u=\op{id}$ + + $\op{Im} u=\op{Ker} u$. #+end_exercice #+begin_exercice [Mines 2023 # 567] @@ -5650,6 +5843,7 @@ Soit $(M_{i,j})$ une base de $\M_n(\mathbb{K})$ vérifiant : $\forall(i,j,k,\ell - Expliciter les automorphismes de l'algèbre $\M_n(\mathbb{K})$. #+end_exercice +# ID:7551 #+begin_exercice [Mines 2023 # 578] Soit $U$ une partie de $\M_n(\C)$ non vide, finie et stable par produit. Montrer qu'il existe $M\in U$ tel que $\mathrm{tr}\,M\in\{0,\ldots,n\}$. #+end_exercice @@ -5721,6 +5915,7 @@ Soit $E=\M_n(\R)$. Soient $A\in E$ et $u_A:M\in E\mapsto AM$. - Montrer que $u_A$ est diagonalisable si et seulement si $A$ est diagonalisable. #+end_exercice +# ID:7552 #+begin_exercice [Mines 2023 # 591] Soient $A,B\in\M_n(\R)$ non nulles et $f:M\in\M_n(\R)\mapsto M+\op{tr}(AM)B$. - Déterminer un polynôme de degré $2$ annulateur de $f$. @@ -5794,10 +5989,9 @@ On suppose dans la suite que $\mathbb{K}=\C$ et que $E$ est de dimension $n\in\N - Soient $f_1,\dots,f_n\in\mc{L}(E)$ nilpotents qui commutent. Calculer $f_1\circ\dots\circ f_n$. #+end_exercice +# ID:7553 #+begin_exercice [Mines 2023 # 600] - Soit $A\in{\cal M}_n({\C})$. Montrer que $A$ est diagonalisable si et seulement si - -$\forall P\in{\C}[X],\ P(A)$ nilpotent $\Rightarrow P(A)=0$. +Soit $A\in{\cal M}_n({\C})$. Montrer que $A$ est diagonalisable si et seulement si $\forall P\in{\C}[X],\ P(A)$ nilpotent $\Rightarrow P(A)=0$. #+end_exercice #+begin_exercice [Mines 2023 # 601] @@ -6030,13 +6224,14 @@ $\Phi(P,Q)=\int_0^{+\i}P(t)\,Q(t)\,e^{-t}dt$. - Calculer la distance de $X^3$ a $\R_2[X]$. #+end_exercice -#+begin_exercice [Mines 2023 # 637] :todo: -Soit $n\in\N^*$. Montrer que : $\forall P\in\R_{n-1}[X],\int_0^{+\i}e^{-x}\left(P(x)+x^n\right)^{2}\,dx\geq(n!)^2$. +# ID:7590 +#+begin_exercice [Mines 2023 # 637] +Soit $n\in\N^*$. Montrer que : $\forall P\in\R_{n-1}[X],\int_0^{+\i}e^{-x}\left(P(x)+x^n\right)^{2} \dx\geq(n!)^2$. #+end_exercice -#+BEGIN_proof :todo: -!! - + $\frac{(-1)^n}{n!} e^x \big(e^{-x} x^n\big)^{(n)}$ est une BON. - + Ou, comme quotient de deux déterminants de Gram… +#+BEGIN_proof +On cherche le projeté orthogonal de $-X^n$ sur $\R_{n-1}[X]$. On le note $Q = \sum a_i X^i$. On écrit $\langle Q + X^n, X^k \rangle = 0$, on obtient des relations sur les $a_i$, et en les divisant par $i!$, cela donne des racines d'un polynôme $T = X(X+1)\dots (X+n-1) + \sum a_i X(X+1)\dots (X+i)$, ou un truc du genre. + +Rmq : $\frac{(-1)^n}{n!} e^x \big(e^{-x} x^n\big)^{(n)}$ est une BON. #+END_proof @@ -6418,6 +6613,7 @@ Soit $f:[0,2]\to\R$ une fonction $C^1$. On pose $u_n=\frac{1}{n}\sum_{k=1}^nf\le Étudier la convergence de la suite $(u_n)_{n\geq 1}$. #+end_exercice +# ID:7530 #+begin_exercice [Mines 2023 # 695] Pour $n\in\N^*$, on pose $u_n=\sum_{k=1}^n\sin\left(\frac{\sqrt{k}}{n}\right)$. Déterminer un équivalent de $u_n$. #+end_exercice @@ -6428,21 +6624,24 @@ Soit $\mc{B}$ le sous-espace de $\C^{\Z}$ formé des suites $(u_n)_{n\in\Z}$ bor - Déterminer les sous-espaces de dimension finie de $\mc{B}$ stables par $T$. #+end_exercice +# ID:7531 #+begin_exercice [Mines 2023 # 697] -Étudier les suites définies par $u_1,v_1$ réels et - -$\forall n\in\N^*$, $u_{n+1}=u_n+v_n\arctan\left(\frac{1}{n^2}\right)$ et $v_{n+1}=v_n-u_n\arctan\left(\frac{1}{n^2}\right)$. +Étudier les suites définies par $u_1,v_1$ réels et $\forall n\in\N^*$, $u_{n+1}=u_n+v_n\arctan\left(\frac{1}{n^2}\right)$ et $v_{n+1}=v_n-u_n\arctan\left(\frac{1}{n^2}\right)$. #+end_exercice +#+BEGIN_proof +Il suffit de montrer que les suites sont bornées. Pour ça, on passe en valeur absolue, et en sommant, $|u_n| + |v_n|$ reste bornée. +#+END_proof + #+begin_exercice [Mines 2023 # 698] -?? $(d_n)_{n\geq 1}$ est-elle convergente? - La suite $(d_n)_{n\geq 1}$ est-elle bornée? + $(d_n)_{n\geq 1}$ est-elle convergente? - La suite $(d_n)_{n\geq 1}$ est-elle bornée? #+end_exercice +# ID:7532 #+begin_exercice [Mines 2023 # 699] Soit $(b_n)_{n\in\N}$ une suite strictement positive, croissante et non majorée. - Montrer que, si $(a_n)_{n\in\N}$ est une suite réelle convergente de limite $\ell$, alors - -$$\frac{1}{b_n}\sum_{k=0}^{n-1}(b_{k+1}-b_k)a_k\underset{n\to+\i}{ \longrightarrow}\ell.$$ + $$\frac{1}{b_n}\sum_{k=0}^{n-1}(b_{k+1}-b_k)a_k\underset{n\to+\i}{ \longrightarrow}\ell.$$ - Soit $(a_n)_{n\in\N}$ une suite réelle. Montrer que, si la suite $\left(\frac{a_{n+1}-a_n}{b_{n+1}-b_n}\right)_{n\in\N}$ converge vers $\ell\in\R$, alors $\frac{a_n}{b_n}\to\ell$ quand $n\to+\i$. - La réciproque de la propriété précédente est-elle vraie? #+end_exercice @@ -6495,10 +6694,11 @@ Pour tout $n\in\N^*$, on pose $P_n=\prod_{i=0}^n(X-i)$. - Déterminer un équivalent simple de $r_n$. #+end_exercice +# ID:7533 #+begin_exercice [Mines 2023 # 707] Soit $(u_n)$ la suite définie par $u_0\geq 0$ et, pour tout $n\in\N$, $u_{n+1}=\sqrt{n+u_n}$ - Montrer que $u_n\to+\i$. - - Donner un développement asymptotique a trois termes de $u_n$. + - Donner un développement asymptotique à trois termes de $u_n$. #+end_exercice #+begin_exercice [Mines 2023 # 708] @@ -7636,20 +7836,17 @@ Soit $\alpha\gt 1$. On munit $\N^*$ de la loi de probabilité $\mathbf{P}_{\alph - En déduire la formule d'Euler $\zeta(\alpha)=\prod_{k=1}^{+\i}\left(1-\dfrac{1}{p_k^{\alpha}} \right)^{-1}$. #+end_exercice -#+begin_exercice [Mines 2023 # 906] :todo: -Soient $X$ et $Y$ deux variables aléatoires discrètes strictement positives, de même loi et d'espérance finie. Montrer que $\mathbf{E}(X/Y)\geq 1$. Ind. Commencer par le cas où $X$ et $Y$ sont indépendantes. +# ID:7356 +#+begin_exercice [Mines 2023 # 906] +Soient $X$ et $Y$ deux variables aléatoires discrètes strictement positives, de même loi et d'espérance finie. + 1. On suppose $X$ et $Y$ indépendantes. Montrer que $\mathbf{E}(X/Y)\geq 1$. + 2. Même question, sans l'hypothèse d'indépendance. #+end_exercice -#+BEGIN_proof :todo: -Si $X,Y$ sont indépendantes, c'est du Cauchy-Schwarz. Sinon, on a $\sum_y \sum_x \frac{x}{y} P(X = x, Y = y) = \sum_y \frac{1}{y} E(X \mid Y = y) P(Y = y)$, sachant $\sum E(X \mid Y = y) P(Y = y) = E(X)$, +#+BEGIN_proof + 1. Si $X,Y$ sont indépendantes, c'est du Cauchy-Schwarz. + 2. C'est du réordonnement. Si $X$ est à valeurs finies, de plus grande valeur $a_1$, et si l'évènement $(X = a_1, Y = a_2)$ est de probabilité $\gt 0$, alors il existe $a_3$ tel que $(X = a_3, Y = a_1)$ soit possible, en notant $p$ la plus petite des deux probabilités, on peut annuler l'un des évènements en changeant la répartition (en ajoutant du $(X = x_3, Y = a_2)$). Cela diminue l'espérance, du moment que $(a_1 - a_2) (a_1 - a_3)\geq 0$. -C'est une inégalité de concavité pour $x\mapsto \frac{1}{x}$ : - - $$\sum_y \frac{1}{y}\sum_{x} x P(X = x, Y = y) \geq \left(\sum_{y, x} x P(X=x, Y=y)\right) \frac{1}{\sum_y y \sum_x x P(X=x, Y = y)} = \frac{E(X)}{E(XY)}$$ - -Ne permet pas de conclure. On perd le cas d'égalité $X = Y$. !! - -If we drag a value to $1$ : On a $\sum_{x, y} \frac{x}{y} P(X = x, Y = y)$, et on le considère comme une fonction de $X(\Om)^2$. On fixe une valeur $x\gt 1$, et on la remplace par $1$. On a alors transformé - $x \sum_{y} \frac{1}{y}P(X = x, Y = y) + \frac{1}{x} \sum_y y P(Y = x, X = y)$ en $\sum_y \frac{1}{y} P(X=x, Y = y) + y P(Y = x, X = y)$ + On peut obtenir le cas $X,Y$ à valeurs dénombrable par limite… #+END_proof @@ -7775,7 +7972,7 @@ Soit $P\in\Z[X]$ tel que $\forall k\in\Z$, $P(k)$ est premier. Montrer que $P$ e #+begin_exercice [Mines PSI 2023 # 924] Montrer qu'il existe $(a_0,\ldots,a_{n-1})\in\R^n$ tel que -$\forall P\in\R_{n-1}[X]$, $ P(X+n)+\sum_{k=0}^{n-1}a_kP(X+k)=0$. +$\forall P\in\R_{n-1}[X]$, $P(X+n)+\sum_{k=0}^{n-1}a_kP(X+k)=0$. #+end_exercice #+begin_exercice [Mines PSI 2023 # 925] diff --git a/Exercices 2023.pdf b/Exercices 2023.pdf index ff5d97243f45909c23a0ee96a3973d89c7279ee0..26201140e0e6d1c0385c83706ba5b1ae4a641e15 100644 GIT binary patch delta 358878 zcmV(=K-s_R=0%*N8xSQ>L`E$!E;R}z(A34)3Ne=fwgwXdHZqss%?2u$Uq1l_e>akT z*RSA&AH)uewYU#yyb+Q{Gcgk$+p-iJjW>c-Q*<_oCSKeR037`%D%sSYlEJy4< zNMsj59En8cmlsGTw-Yk?$0O45`r_@Q|MfPIaHQ$^3IHQD+=|6trfAGmnPd9h&-F{mxlXkvyTUf=E>zhnwR2NE$(C@2{ z%k7n!F3QF7YQ`j)`a$43BKnT7g%T9L>c8^i525A1zYDL7Fdp805ni~V@A4_gs5@;7 z!C2khcbmgDj44dih|$7WQ#V?N7LrCSOreFY8~j9=ta^!uqNI?tX~e3vf5k0CVNJ8N zCt(4bb$DHG_Df`H+!T~FoqkufuoG2S&5diJ3T_n;x)4?*3rOiuSSlt_qgAyB_ij6v z+tO_)+q?a#KDc>&SVv7ZtKV-o>*_P#?DnuKVMG0cIYti&TqbEokYw(GAiPji4o!U) 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+** Options + +#+OPTIONS: latex:verbatim +#+OPTIONS: toc:t +#+exclude_types: proof + +*** All + +#+OPTIONS: toc:t +#+export_file_name: Exercices 2024 + +*** XENS + +# #+select_tags: xens +# #+export_file_name: Exercices XENS 2024 + +*** XENS MP + +# #+select_tags: xens +# #+exclude_tags: autre +# #+exclude_types: proof +# #+export_file_name: Exercices XENS MP 2024 + +*** Centrale + +# #+select_tags: cent +# #+export_file_name: Exercices Centrale 2024 + +*** Mines + +# #+select_tags: mines +# #+export_file_name: Exercices Mines 2024 + +*** Mines Centrale + +# #+select_tags: mines cent +# #+options: toc:2 +# #+export_file_name: Exercices Mines Centrale 2024 + +*** todoes + +# #+options: title:nil nopage:t tags:nil +# #+select_tags: todo +# #+export_file_name: Exercices 2024 todo +# #+relocate_tags: todo + +*** autre + +# #+options: title:nil nopage:t tags:nil +# #+select_tags: autre +# #+export_file_name: Exercices XENS 2024 autres +# #+relocate_tags: todo + + + + + +* ENS MP 2024 :xens: + +** Algèbre + +# ID:7636 +#+begin_exercice [ENS MP 2024 # 1] +Soit $E$ un ensemble fini non vide. Pour tout $(x_1,x_2,x_3)\in E^3$ et tout $\sigma\in\mathfrak{S}_3$, on note $\sigma\cdot(x_1,x_2,x_3)=(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)})$. Soit $E^{3*}=\{(x,y,z)\in E^3\;;\;x,y,z\;\text{sont distincts}\}$. Soit $S\subset E^{3*}$ tel que + + $\forall\sigma\in\mathfrak{S}_3$, si $\eps(\sigma)=-1$ alors $\sigma\cdot(S)=\{\sigma\cdot x\;;\;x\in S\}=E^{3*}\setminus S$, + + $\forall a,b,c,d\in E$, si $(a,b,c)\in S$ et $(a,c,d)\in S$, alors $(a,b,d)\in S$ et $(b,c,d)\in S$. + +Montrer qu'il existe $g\colon E\ra\R$ injective telle que $\forall(a,b,c)\in E^3$, $g(a)\lt g(b)\lt g(c)\Rightarrow(a,b,c)\in S$. +#+end_exercice +#+BEGIN_proof +En pratique les éléments de $S$ seront les triplets où $a\lt b\lt c$, ou $c\lt b \lt a$. + + + C'est clair pour $n = 3$. + + Si on suppose que c'est vrai au rang $n=4$, on peut passer au rang suivant : si on retire l'élément $a$, l'ensemble des triplets sans $a$ vérifie les mêmes hypothèses. Donc il existe une fonction $g_a$. + + + Étant donné $g_a$ et $g_b$, il reste au moins trois autres éléments, et pour chaque triplet d'éléments, les fonctions $g_a,g_b$ restreintes à ceux-ci doivent être compatibles, donc $g_a$ et $g_b$ sont compatibles (quitte à prendre l'opposé de l'une). + + Toutes les fonctions doivent être compatibles, d'où le résultat. + + Reste à traiter le cas $n = 4$, où $E = \{a, b, c, d\}$ + + On montre qu'il n'est pas possible que chaque terme apparaisse au + milieu d'un triplet : On fixe un triplet $a\lt b \lt c$. + Alors il existe $e \lt a \lt f$, et un de $e, f$ est égal à $b$ ou $c$, et s'il est égale à $b$ on peut le remplacer par $c$. + + Donc on a $e \lt a \lt c$, et $e = d$ (on exclu $e = b$ via les propriétés) + + Mais on a aussi $k\lt e \lt l$. Si l'un de $k$ ou $\l$ vaut $c$, on peut le remplacer par $b$, et on peut remplacer $b$ par $a$, on obtient $a\lt e \lt a$, contradiction. + + Alors un élément n'est jamais au milieu, et cela suffit (appliquer la récurrence aux trois autres). +#+END_proof + + +# ID:7652 +#+begin_exercice Théorème d'Ostrowski [ENS MP 2024 # 2] +Soit $N$ une application de $\Q$ vers $\R^+$ vérifiant : + + $N(xy)=N(x)N(y)$ pour tous $x,y\in\Q$, + + $N(x+y)\leq N(x)+N(y)$ pour tous $x,y\in\Q$, + + pour tout $x\in\Q$, $N(x)=0\Rightarrow x=0$, + + il existe $n\in\N$ tel que $N(n)\gt 1$. +Montrer qu'il existe $\lambda\in\left]0,1\right]$ tel que $N(x)=|x|^{\lambda}$ pour tout $x\in\Q$. +#+end_exercice +#+BEGIN_proof +On a $N(1) = 1$, puis $N(\frac{1}{p}) = \frac{1}{N(p)}$. La multiplicativité permet de justifier que l'exposant $\la$ est le même pour tout nombre premier : encadrer $p^n$ par deux puissances de $q$ et utiliser la seconde I.T. Puis pour que l'inégalité triangulaire soit vérifiée, il faut que $\la \leq 1$. +#+END_proof + +# ID:nil # Classique +#+begin_exercice [ENS MP 2024 # 3] +On etend de facon naturelle la valuation $2$-adique $v_2$ à $\Q^*$. Pour $n\in\N^*$, soit $H_n=\sum_{k=1}^n\dfrac{1}{k}$. Calculer $v_2(H_n)$. +#+end_exercice + + +# ID:7637 +#+begin_exercice [ENS MP 2024 # 4] Congruences sur les coefficients binomiaux +Soit $(m,n,p)\in\left(\N^*\right)^3$, avec $p$ premier supérieur ou egal à 5, $m$ et $p$ premiers entre eux. + - Montrer que $\begin{pmatrix}np\\ m\end{pmatrix}\equiv 0\left[p\right]$. + - Montrer que $\begin{pmatrix}np\\ mp\end{pmatrix}=\sum_{k=0}^p\begin{pmatrix}p(n-1)\\ mp-k\end{pmatrix}\begin{pmatrix}p\\ k\end{pmatrix}$. + - Montrer que $\begin{pmatrix}np\\ mp\end{pmatrix}\equiv\begin{pmatrix}n\\ m\end{pmatrix}[p^2]$. + - On veut montrer que $\begin{pmatrix}2p\\ p\end{pmatrix}\equiv 2\left[p^3\right]$. + - Montrer que $\forall k\in\db{1,p},\, {p-1 \choose k-1}\equiv \pm 1 [p]$. + - Montrer que $\sum_{k=1}^{p-1} \left(\frac{(p-1)!}{k}\right)^2 \equiv 0[p]$. + - Conclure +#+end_exercice +#+BEGIN_proof + - Formule du capitaine. + - Faire le dessin, appliquer Pascal, c'est du binôme, par récurrence. + - Par récurrence sur $n$, utiliser les deux questions précédentes. + - + - Écrire le quotient, les numérateur et dénominateur sont congrus. + - $\sum \frac{1}{k^2} = \sum k^2 = \frac{n(n+1)(2n+1)}{6}$. + - On applique Q2, on a ${2p \choose p} = \sum_{k=0}^p {p \choose p-k} {p\choose k}$, faire deux capitaines, plus question précédente. +#+END_proof + +# ID:7651 +#+BEGIN_exercice Détermination de $\left(\frac{2}{p}\right)$ [ENS MP 2024 # 5] +Soit $p$ un nombre premier impair. + - Déterminer $\op{Card} \{x^2,\, x\in\Z/p\Z^*\}$. + - Démontrer l'équivalence : $a^{\frac{p-1}{2}}\equiv 1[p]$ si et seulement si $a$ est un carré non nul modulo $p$. + - On pose $a = \prod_{k=1}^{\frac{p-1}{2}}(2k)$. Montrer que + - Si $p\equiv 1[4]$, alors $a\equiv (-1)^{\frac{p-1}{4}}\big(\frac{p-1}{2}\big)! [p]$ + - Si $p\equiv -1[4]$, alors $a\equiv (-1)^{\frac{p+1}{4}}\big(\frac{p-1}{2}\big)! [p]$ + - CNS pour que $2$ soit un carré modulo $p$. +#+END_exercice +#+BEGIN_proof + - + - + - Chaque $2k$ s'écrit $\eps_k k'$, avec $k' \in \db{1,\frac{p-1}{2}}$ et $\eps_k = \pm 1$, et $k\mapsto k'$ est bijective. En comptant le nombre, on trouve ce qu'on veut. +#+END_proof + + +# ID:7638 +#+BEGIN_exercice [ENS MP 2024 # 6] +On considère l'équation $2^a + 3^b = 5^c$, où $(a,b,c)\in\N^3$. + - Résoudre l'équation dans le cas $a = b = c$. + - Traiter le cas $b$ impair. + - Traiter le cas $c$ impair. + - Traiter le cas général. +#+END_exercice +#+BEGIN_proof + - Analyse. + - L'étude modulo $4$ donne $a = 1$, et écrire $2 = 1 + 1$. + - En regardant mod $6$, on trouve $a$ impair, donc $a\geq 3$, donc $3^b\equiv 5^c [8]$, donc $b$ est impair. + - On a $b,c$ pair, donc une différence de carrés, on obtient $2^a = (3^b - 5^c)(3^b + 5^c)$ et on peut descendre. +#+END_proof + + +# Clarifier, mettre une suite, sommes de Gauss +#+BEGIN_exercice [ENS MP 2024 # 7] :todo: +Soit $p$ un nombre premier impair. + - Quel est le cardinal du groupe des inversibles de $\Z/p\Z$ ? + - Montrer que l'équation $x^2 = 1$ possède exactement deux solutions dans $\Z/p\Z$. + - En déduire $\op{Card} \{x^2,\, x\in\Z/p\Z\}$. + - Soit $\chi\colon\Z\ra\{-1,0,1\}$ telle que : $\chi(n)=1$ si $n\wedge p=1$ et si $n$ est un carré modulo $p$ ; $\chi(n)=-1$ si $n\wedge p=1$ et si $n$ n'est pas un carré modulo $p$ ; $\chi(n)=0$ si $p\mid n$. + - [[ref]] Déterminer $\sum_{k=0}^{p-1}\chi(k)$. + - + - s Montrer que le produit d'un carré et d'un non carré est un non carré. + - En utilisant le caractère bijectif de $x\mapsto ax$ dans $(\Z/p\Z)^*$, montrer que : $\forall(a,b)\in\Z^2,\chi(ab)=\chi(a)\chi(b)$. + - Déduire de <> une majoration de $\left|\sum_{k=0}^N\chi(k)\right|$ pour $N\in\N$. + - On pose $\xi=e^{2i\pi/p}$. Montrer que + $$\chi(n)=\frac{1}{p}\sum_{k=0}^{p-1}\sum_{a=0}^{p-1}\chi(a)\xi^{k(a-n)}.$$ + - Pour $k\in\db{1,p-1}$, on note $S_k(N)=\sum_{n=0}^N\xi^{-kn}$. + - Montrer que $\forall N\geq 0$, $|S_k(N)|\leq\frac{1}{|\sin(k\pi/p)|}$. + - En déduire que, pour $k\lt p/2$, $|S_k(N)|\leq p/2k$. + - Trouver une majoration similaire pour $k\gt p/2$. + - On pose $G_k=\sum_{a=0}^{p-1}\chi(a)\xi^{ka}$. + - Montrer que, pour $k\in\db{1,p-1}$, $|G_k|=\sqrt{p}$. + - Montrer que $G_k$ est réel ou imaginaire pur. + - On suppose que $G_1$ est réel, montrer que $G_1\geq 0$. ?? Trouver des questions pour finir… +#+END_exercice +#+BEGIN_proof + - + - + - + - + - Il s'agit de montrer que le produit de deux non carrés est un carré. Cela qui découle de propriété de cardinal car un carré fois un non carré est un non carré. + - $0$ : autant de carrés que de non carrés, se fait sans ce qui précède. + - Simple. + - Simple. + - + - Somme géométrique. + - Inégalité de convexité. + - Simple. + - + - Considérer $|G_k|^2$, et changement de variable. + - Si $-1$ est un carré, $G_k$ est réel, et si $-1$ n'est pas un carré, $\ol{G_k} = - G_k$. + - Rajouter $\sum \chi(a)$, puis séparer la somme. La somme des termes pour les carrés $\leq \sqrt{n}$ + + On a $\chi(n) = \frac{1}{p}\sum_{k=0}^{p-1} G_k \xi^{-nk}$ +#+END_proof + + + +# ID:7639 +#+begin_exercice Anneaux euclidiens [ENS MP 2024 # 8] +On dit que $A$ est un anneau euclidien si $A$ est un anneau intègre (donc commutatif) et qu'il existe $t\colon A\setminus\{0\}\ra\N$ vérifiant : + - pour tout $(a,b)\in A\times(A\setminus\{0\})$, il existe $(q,r)\in A^2$ tel que $a=bq+r$ avec $r=0$ ou $t(r)\lt t(b)$, + - $\forall(a,b)\in(A\setminus\{0\})^2,t(ab)\geq t(a)$. + - Les anneaux $\Z$ et $\R[X]$ sont-ils euclidiens ? Montrer qu'un corps est un anneau euclidien. + - Soient $A$ un anneau euclidien et $I$ un idéal de $A$. Montrer qu'il existe $x\in A$ tel que $I=xA$. Y a-t-il unicité de $x$? + - Dans cette question, on se donne $A$ un anneau euclidien tel que $t(1)=1$. Soit $x\in A$. Montrer que $x$ est inversible si et seulement si $t(x)=1$. +#+end_exercice + + +# ID:259 +#+begin_exercice [ENS MP 2024 # 9] +Soit $A$ l'ensemble des fonctions de $\N^*$ dans $\C$. + +Pour $f,g\in A$, on pose $(f*g)(n)=\sum_{d\mid n}f(d)\,g(n/d)$ pour tout $n\in\N^*$. + - Montrer que $(A,+,*)$ est un anneau commutatif intègre. + - Caractériser les inversibles de l'anneau $A$. + - Résoudre l'équation $ax^2+bx+c=0$ dans l'anneau $A$ avec $a$ et $b^2-4ac$ inversibles. +#+end_exercice + + +# ID:7666 +#+begin_exercice [ENS MP 2024 # 10] + - Montrer que les sous-groupes de $\Z/n\Z$ sont cycliques. + - <> Alice et Barbara jouent à un jeu. Elles choisissent à tour de role un élément de $\Z/n\Z$ sans remise qu'elles ajoutent à un ensemble $S$. Le jeu s’arrête quand $S$ engendre $\Z/n\Z$ et la joueuse ayant tiré le dernier numero perd. Selon $n$, y a-t-il une stratégie gagnante pour la première joueuse ? + - Même question si à chaque étape, on ne peut pas retirer un élément de $\langle S\rangle$. + - Reprendre [[ref1]] avec le groupe ${\cal S}_n$. +#+end_exercice +#+BEGIN_proof + - + - Avec l'interprétation de l'énoncé, s'il existe un sous-groupe maximal de $G$ de cardinal impair, Alice l'engendre, puis ils restent dedans, donc Alice gagne. Sinon, quoi que Alice fasse, Bob choisi un sous-groupe maximal de cardinal impair qui contient l'élément d'Alice. + - Avec la version où il faut augmenter $\langle S\rangle$, Alice gagne en engendrant un sous-groupe maximal directement. + - + + Pour $n = 2$ : Alice gagne, en jouant l'identité. + + Pour $n = 3$ : Une transposition + n'importe quoi d'autre que l'identité engendre tout le groupe. + + + Si Alice joue l'identité, Bob la transposition, Alice perd. + + Si Alice joue une transposition, Bob l'identité, Alice perd. + + Donc Alice joue un 3-cycle (donc dans le groupe Alterné). Si Bob joue dans $A_3$, Alice peut le faire encore une fois, et elle gagne. Si Bob joue en dehors, il perd. + + Pour $n \geq 4$ : Alice joue n'importe quoi. Si c'est d'ordre pair, Bob reste dans un sous-groupe maximal de cardinal pair. Sinon, Bob joue un élément d'ordre pair qui n'engendre pas tout $\mc S_n$. C'est possible, car même si Alice joue un $n$-cycle, avec $n$ impair, Bob peut jouer une bi-transposition, et rester dans $\mc A_n$. + + Pour l'autre interprétation : probablement pas faisable. +#+END_proof + + +# ID:7641 +#+begin_exercice [ENS MP 2024 # 11] + - Soient $\sigma\in S_n$ et $c_1\circ\cdots\circ c_r$ sa décomposition en produit de cycles à supports disjoints. Calculer l'ordre de $\sigma$ dans le groupe $S_n$. + - On note $g(n)$ l'ordre maximal d'une permutation de $S_n$. Montrer que $g$ est croissante et $n\leq g(n)\leq n!$ + - Trouver $n$ minimal tel que $g(n)\gt n$. + - On note $(p_k)_{k\in\N^*}$ la suite strictement croissante des nombres premiers. Montrer que : + + $n\geq\sum_{i=1}^rp_i^{\alpha_i}\implies g(n)\geq \prod_{i=1}^rp_i^{\alpha_i}$. + - On suppose que $g(n)=\prod_{i=1}^rp_i^{\alpha_i}$. Montrer que : $n\geq\sum_{i=1}^rp_i^{\alpha_i}$. + - Montrer que $\forall\eps\gt 0$, $\exists C\gt 0,\,\forall n\in\N^*,\,g(n)\leq Ce^{ \eps n}$. +#+end_exercice +#+BEGIN_proof +Pas de difficulté. + - + - + - +#+END_proof + + +# ID:nil # Classique +#+begin_exercice [ENS MP 2024 # 12] +Lorsque $\sigma\in S_n$, on note $n_k(\sigma)$ le nombre de $k$-cycles dans la décomposition de $\sigma$ en produit de cycles à supports disjoints. Ainsi $n_1(\sigma)$ est le nombre de points fixes de $\sigma$. On note egalement $m(\sigma)=\sum_{k=1}^nn_k(\sigma)$ le nombre d'orbites de $\sigma$. + - Soient $i,k\in\N^*$. Déterminer l'ordre de $i$ dans $\big(\Z/k\Z,+\big)$. + - Soient $n\in\N^*$ et $\sigma,\tau\in S_n$. On dit que $\sigma$ et $\tau$ sont conjuguées s'il existe $\phi\in S_n$ tel que $\sigma=\phi\tau\phi^{-1}$. + +Montrer que $\sigma$ et $\tau$ sont conjuguées si et seulement si $\colon\forall k\in\db{1,n},n_k(\sigma)=n_k(\tau)$. + - Soit $n\in\N^*$. Calculer $\det\big(i\wedge j\big)_{1\leq i,j\leq n^*}$. + +Ind. Considérer les matrices $A=(1\!1_{i|j})$ et $B=(\phi(j)1\!1_{j|i})$. + - Montrer que $\sigma$ et $\tau$ sont conjuguées si et seulement si : $\forall i\in\db{1,n},m(\sigma^i)=m(\tau^i)$. + - Montrer que $\sigma$ et $\tau$ sont conjuguées si et seulement si les matrices de permutation $P_{\sigma}$ et $P_{\tau}$ sont semblables. +#+end_exercice + + +# ID:nil # Cf année précédente. +#+begin_exercice [ENS MP 2024 # 13] +Soient $G$ un groupe, $A$ une partie finie non vide de $G$. Montrer que $|A|=|AA|$ si et seulement si $A=xH$ avec $x\in G$ et $H$ sous-groupe de $G$ tel que $x^{-1}Hx=H$. +#+end_exercice + +# ID:nil # Cf année précédente. +#+begin_exercice [ENS MP 2024 # 14] +Soient $G$ un groupe et $A\subset G$ fini non vide tel que $|AA|\lt \frac{3}{2}|A|$. Montrer que $A^{-1}A$ est un sous-groupe de $G$. +#+end_exercice + +# ID:7653 +#+begin_exercice Groupe dihédral [ENS MP 2024 # 15] + - Soient $n\geq 3$ et $\mc Q$ un polygone régulier à $n$ côtés. Montrer que l'ensemble des isométries affines du plan préservant $\mc Q$ est un groupe à $2n$ éléments. + - s On note maintenant $n=q$, nombre premier impair, et $D_{2q}$ le groupe précédent. Montrer que tout groupe de cardinal $2q$ est isomorphe à $\Z/2q\Z$ ou à $D_{2q}$. +#+end_exercice +#+BEGIN_proof + - Action sur les sommets, nice. + - Il ne peut pas avoir que des éléments d'ordre $2$. Donc il a un élément d'ordre $q$, et un autre qui agit sur $\Z/q\Z$ par conjugaison. Si l'action est trivial, le groupe est commutatif. Sinon, on est isomorphe à $D_{2q}$. +#+END_proof + + + +# ID:7654 +#+begin_exercice [ENS MP 2024 # 16] + - Trouver tous les groupes d'ordre $8$ dont l'ordre maximal des éléments est $4$. + - Trouver tous les groupes d'ordre $8$ à isomorphisme pres. +#+end_exercice +#+BEGIN_proof + - Tu as un $\Z/4\Z$, tu prend un autre élément. Soit il commute, alors c'est $\Z/4\Z\times \Z/2\Z$. + + Si le $\Z/4\Z$ est normal, la conjugaison a une action, et on est un groupe dihédral. + + Sinon, ça veut dire qu'on a d'autres copies du $\Z/4\Z$, on est le groupe des quaternions. + - Si l'ordre maximal est $2$, c'est $\left(\Z/2\Z\right)^3$. +#+END_proof + + +# ID:7675 +#+begin_exercice [ENS MP 2024 # 17] + - Donner des exemples de groupes d'ordre $12$ commutatifs ainsi qu'un exemple non commutatif. + - Montrer que tout groupe d'ordre $12$ admet un élément d'ordre $2$. + - Trouver à isomorphisme prêt les groupes commutatifs d'ordre $12$. + - Montrer que tout groupe d'ordre $12$ admet un élément d'ordre $3$. + - Trouver tous les groupes d'ordre $12$ à isomorphisme pres. +#+end_exercice +#+BEGIN_proof + - + - Sinon, tous les ordres sont impairs, donc tous les éléments sont d'ordre $3$, impossible pour des raisons de cardinal. + - Regarder l'ordre maximal : soit cyclique, soit un élément d'ordre $6$, donc $\Z/6/\times \Z/2\Z$, soit un élément d'ordre $4$, auquel tout autre élément doit être d'ordre $3$ (dans le quotient), sinon on obtiendrait un sous-groupe d'ordre $8$. Soit un élément d'ordre $3$. + - On découpe $G$ en classes de conjugaisons. Chaque classe est de cardinal un diviseur de $|G|$. Si un élément qui n'est pas dans le centre a un centralisateur de cardinal divisible par $3$, alors on trouve un élément d'ordre $3$ dedans. Sinon, on a découpé $|G| = |Z(G)| + 3k$, donc on trouve un élément d'ordre $3$ dans le centre. + - Si $G$ non commutatif. Si $G$ admet un élément d'ordre $6$, c'est $D_{12}$ (tout sous-groupe d'indice deux est normal). + + Sinon, $G$ admet un élément $e$ d'ordre $3$. + + + S'il engendre un sous-groupe normal, il faut trouver les actions de $\Z/4/Z$ et $(\Z/2\Z)^2$ sur $\Z/3\Z$. + + La seconde donne un élément qui agit trivialement, donc qui commute, donc $\Z/2/Z\times \Z/3\Z = \Z/6/Z$ + + Pour la première, l'élément d'ordre $2$ agit trivialement. C'est $(x,y) . (x', y') = (x + x', y + y'^{\pm 1})$. Ce n'est pas un groupe usuel… + + Sinon, on considère les groupes de cardinal $3$. + + On fait agir $e$ par conjugaison dessus. $e$ préserve son groupe, et c'est le seul qu'il préserve, car si $e$ normalise $H_1$, le groupe engendré par $e$ et $H_1$ admet $H_1$ comme sous-groupe normal, et trop d'éléments d'ordre $3$. + + On en déduit qu'on trouve exactement $4$ sous-groupes d'ordre $3$ ($7$ ferait trop) + + On a donc trouvé $8$ éléments d'ordre $3$ + le neutre. + + Il en reste $3$. Prenons-en un, $e_2$. S'il commutait avec un élément d'ordre $3$, on aurait un $\Z/6\Z$. Donc on trouve exactement trois éléments d'ordre $2$, tous conjugués. + + $e_1e_2$ conjugue $e_2$ comme $e_1$, donc $e_1e_2$ ne peut pas être d'ordre $3$. Donc les éléments d'ordre $2$ forme un groupe, qui est normal. + + On trouve un produit semi-direct de $\Z/2\Z\times \Z/2\Z$ par $\Z/3\Z$, qui est isomorphe à $\mc A_4$. +#+END_proof + + +# ID:nil # Bof, quel intérêt +#+begin_exercice [ENS MP 2024 # 18] +Soit $A=\begin{pmatrix}0&0&-1\\ 1&0&1\\ 0&1&0\end{pmatrix}\in\M_3(\mathbb{F}_3)$. On admet que $A^{13}=-I_3$. + - Quels calculs auriez-vous fait pour justifier que $A^{13}=-I_3$? + - Montrer que $A\in\op{GL}_3(\mathbb{F}_3)$ et que $A$ est d'ordre $26$ dans ce groupe. + - On note $G$ le sous-groupe de $\op{GL}_3(\mathbb{F}_3)$ engendré par $A$, et on pose $V=G\cup\{0\}$. Montrer que $V=\text{Vect}(I_3,A,A^2)$. + - On pose $W=\text{Vect}(I_3,A)$. Montrer que, pour tout $M\in G$, il existe $N,P\in W\setminus\{0\}$ telles que $M=P^{-1}N$. + - On note $H$ le sous-groupe de $\op{GL}_3(\mathbb{F}_3)$ engendré par $A^2$. Montrer que $H$ est isomorphe à $\Z/13\Z$, puis que $|H\cap W|=4$. +#+end_exercice +#+BEGIN_proof + - Polynôme caractéristique, ou exponentiation rapide. + - Déterminant, et $A$ n'est pas d'ordre $2$. + - Polynôme caractéristique. + - +#+END_proof + + +# ID:7658 +#+begin_exercice [ENS MP 2024 # 19] + - Montrer que toute rotation du plan complexe est composée de deux symétries orthogonales par rapport à des droites. + - Montrer que toute permutation d'un ensemble fini non vide $X$ est produit de deux éléments d'ordre au plus $2$ du groupe des permutations de $X$. + - Le résultat de la question précédente subsiste-t-il si $X$ est infini? +#+end_exercice +#+BEGIN_proof + - + - Suffit de le faire pour des cycles. + - Prendre une translation sur $\Z$, si elle s'écrit $t = \tau_1 \circ \tau_2$, ils sont d'ordre $2$, donc correspondent à des ensembles $\op{Fix}_1$, et $S_1, T_1$, idem $S_2,T_2$. On a $\tau_1 t \tau_1 t = \op{id}$, ou $\tau_1(x+1) = \tau_1(x)-1$. Si $\tau_1$ a un pont fixe, c'est la symétrie, autour de celui-ci. Ça marche… + + Donc ça persiste, en découpant selon les orbites finies et infinies. +#+END_proof + + +# ID:nil # Trivial +#+begin_exercice [ENS MP 2024 # 20] +Soit $P\in\C[X]$ non constant à coefficients dans $\{-1,1\}$. Soit $z\in\C$ une racine de $P$. Montrer que $|z|\lt 2$. +#+end_exercice + + +# ID:7667 +#+begin_exercice [ENS MP 2024 # 21] +Soient $m\in\N^*$ et $(a_0,...,a_m)\in\R^{m+1}$. + +On pose $f(X,Y)=a_0X^m+a_1X^{m-1}Y+a_2X^{m-2}Y^2+...+a_mY^m$ et on suppose que le polynôme $f(X,1)\in\R[X]$ est scindé. + +Montrer que, pour tout $(n,p)\in\N^2$, le polynôme $\frac{\partial^{n+p}f}{\partial X^n\partial Y^p}(X,1)$ est nul ou scindé sur $\R$. +#+end_exercice +#+BEGIN_proof +Si on dérive par rapport à $X$ c'est clair. Il suffit de le justifier quand on dérive par rapport à $Y$ : Si $a_0 X^m + a_1 X^{m-1} + \dots + a_m$ est scindé, alors $a_1X^{m-1} + 2 a_2 X^{m-2} + \dots + ma_m$ est scindé ? Passer par le polynôme réciproque. +#+END_proof + +# ID:7436 +#+begin_exercice [ENS MP 2024 # 22] +Soit $(a_n)\in(\R^*)^{\N}$. On suppose qu'il existe $C\gt 0$ tel que $\forall n\in\N$, $|a_n|\in[1/C,C]$. Pour $n\in\N$, on pose $P_n=\sum_{k=0}^na_kX^k=a_n\prod_{k=1}^n(X-x_{k,n})$, ou l'on a note $x_{k,n}$ les racines complexes de $P_n$. + - Montrer que $\{x_{k,n}\,;\,n\in\N^*,k\in\db{1,n}\}$ est borné. + - Montrer que $\sum_{k=1}^nx_{k,n}^2=\frac{a_{n-1}^2-2a_{n-2}a_n}{a_n^2}$ pour tout $n\geq 2$. + - Montrer que, pour $n$ suffisamment grand, $P_n$ n'est pas scindé sur $\R$. +#+end_exercice + + +# ID:417 # classique +#+begin_exercice [ENS MP 2024 # 23] + - Soit $P=X^n+\sum_{k=0}^{n-1}a_kX^k$ unitaire de degre $n\geq 2$ à coefficients dans $\C$, avec $a_{n-1}\in\R_+$. Montrer, pour $M=\max(|a_0|,\ldots,|a_{n-2}|)$, que toute racine $z$ de $P$ vérifie $\mathfrak{Re}(z)\leq 0$ ou $|z|\leq\dfrac{1+\sqrt{1+4M}}{2}$. + - Soit $p$ un nombre premier et $b\geq 3$ un entier. On écrit $p=\overline{c_nc_{n-1}\cdots c_0}^b$ en base $b$. Montrer que $\sum_{k=0}^nc_kX^k$ est irreductible dans $\Z[X]$. +#+end_exercice + + +# ID:7676 +#+begin_exercice [ENS MP 2024 # 24] +Soit $P$ un polynôme à $n$ indéterminées $X_1,X_2,\ldots,X_n$. On dit que $P$ est symétrique si, pour toute permutation $\sigma$ de $\{1,2,\ldots,n\}$, on a $P(X_{\sigma(1)},X_{\sigma(2)},\ldots,X_{\sigma(n)})=P(X_1,X_2,\ldots,X_n)$. On dit que $P$ est homogène de degre $k\in\N$ s'il est somme de mo- nomes de la forme $cX_1^{k_1}X_2^{k_2}\cdots X_n^{k_n}$ avec $k_1+k_2+\cdots+k_n=k$. + - Montrer qu'il existe une famille presque nulle $(e_i(X_1,X_2,\ldots,X_n))_{i\geq 0}$ de polynômes à $n$ indéterminées symétriques et homogènes tels que, pour tout $t\in\R$, + $(1+tX_1)(1+tX_2)\cdots(1+tX_n)=\sum_{i\geq 0}e_i(X_1,X_2, \ldots,X_n)t^i$. + - Montrer qu'il existe une famille $(h_i(X_1,X_2,\ldots,X_n))_{i\geq 0}$ de polynômes à $n$ indéterminées symétriques et homogènes tels que, pour tous $x_1,x_2,\ldots,x_n\in\R$ et tout $t\in\R$ au voisinage de $0$, + + $\dfrac{1}{(1-tx_1)(1-tx_2)\cdots(1-tx_n)}=\sum_{i=0}^{+\i}h_i(x_1,x_2,\ldots,x_n)t^i$. + +On pose $\mc{P}_n=\{\ \lambda=(\lambda_1,\ldots,\lambda_n)\in\N^n,\ \lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n\ \}$ et, si $\alpha\in\N^n$, on pose $\Lambda(\alpha)$ le $n$-uplet obtenu en ordonnant les entiers de $\alpha$ par ordre decroissant, puis pour tout $\lambda\in\mc{P}_n$, $m_{\lambda}=\sum_{\alpha\in\N^n,\,\Lambda(\alpha)=\lambda}X_1^{ \alpha_1}X_2^{\alpha_2}\cdots X_n^{\alpha_n}$. + - Calculer $m_{\lambda}$ avec $\lambda=(2,1,0,0)$ et $\lambda$ le $n$-uplet contenant $r$ fois 1 et $n-r$ fois 0. + - Pour $\lambda,\mu\in\mc{P}_n$, on note $M_{\lambda,\mu}$ le nombre de matrices dont les coefficients valent $0$ ou $1$ et telles que la somme des coefficients de la $i$-ieme ligne vaut toujours $\lambda_i$ et celle des coefficients de la $j$-ieme colonne vaut toujours $\mu_j$. Montrer que $\prod_{i=1}^ne_{\lambda_i}(X_1,\ldots,X_n)=\sum_{\mu\in\mc{P}_n}M_{\lambda,\mu}m_{\mu}$. +#+end_exercice +#+BEGIN_proof + - Sans difficulté, unicité des coefficients polynomiaux. + - DSE, on a une expression de $h_i$. + - $e_{\la_i}(X_1,\dots, X_n)$ est le polynôme homogène symétrique total de degré $\la_i$. + + Si on fixe un $\mu$, $M_{\la, \mu}$ est le nombre de façons de répartir $\mu_1,\dots \mu_n$ sur les lignes de sorte à respecter le $\la$ : pour faire $\la_1$ on pioche dans $\mu_1,\dots, \mu_n$, puis pour faire $\la_2$, on pioche dans le reste, etc. + + On peut l'écrire comme la somme + + Le coefficient en $\mu$ de $\prod_{i=1}^n e_{\la_i}(X_1,\dots,X_n)$ est la même chose : à chaque $\la_i$, on choisit sur quelles variables (= colonnes) mettre les $\la_i$ coefficients qui valent $1$, de sorte que les choix totaux respectent $\mu$. +#+END_proof + + +# ID:7677 +#+begin_exercice Théorème de Liouville [ENS MP 2024 # 25] +Soient $A,B,C\in\C[X]$ non tous constants et premiers entre eux deux à deux. + - On veut montrer que si $A+B=C$ alors $\max{(\deg(A),\deg(B),\deg(C))}\leq M(ABC)-1$ ou $M(P)$ est le nombre de racines distinctes du polynôme $P$. + + Si $P,Q\in\C[X]$, on note $W_{P,Q}=PQ'-P'Q$. + - Montrer que $W_{A,B}=W_{C,B}=W_{A,C}\neq 0$. + - Montrer que $\deg(A\wedge A')+\deg(B\wedge B')+\deg(C\wedge C') \leq\deg(W_{A,B})$. + - Conclure. + - Soit $d\in\N^*$. Donner un exemple de $(A,B,C)\in\C[X]^3$ avec $\deg(A)=d$ et pour lequel $\max(\deg(A),\deg(B),\deg(C))=M(ABC)-1$. + - Soient $A,B,C\in\C[X]$ premiers entre eux dans leur ensemble et tels que $A^n+B^n=C^n$ avec $n\in\N^*$. Montrer que $n\leq 2$. Montrer qu'il existe des solutions pour $n=2$. +#+end_exercice +#+BEGIN_proof + - + - Si ces quantité sont nulles, $\frac{P}{Q}$ est constante. + - Comme les polynômes sont premiers entre eux, ces quantités sont premières entre elles, et divisent $W$. + - + - + - +#+END_proof + + +# ID:7678 +#+begin_exercice [ENS MP 2024 # 26] +Soit $\R\left[X,X^{-1}\right]$ l'ensemble des fractions rationnelles dont le dénominateur est une puissance de $X$. + - Montrer que $\R[X,X^{-1}]$ est un sous-anneau de $\R(X)$. En est-ce un sous-corps? Quels sont ses éléments inversibles? + - Déterminer les automorphismes de l'anneau $\R$. + - Déterminer les automorphismes de la $\R$-algèbre $\R[X,X^{-1}]$. + - Déterminer les automorphismes de l'anneau $\R[X,X^{-1}]$. +#+end_exercice +#+BEGIN_proof + - + - classique + - $\op{Id}$ et $X\mapsto X^{-1}$, clairement. + - $X$ est envoyé sur un élément de degré $1$ ou $-1$, qui doit être inversible. Cette fois, on peut envoyer $X$ sur $\a X$. +#+END_proof + + + +# ID:7681 +#+begin_exercice [ENS MP 2024 # 27] +Soient $P,Q\in\R[X]$ unitaires. On dit que $P$ et $Q$ sont entrelacés lorsqu'entre deux racines consécutives de l'un (en tenant compte des multiplicités) il y a exactement une racine de l'autre. On suppose que $\deg(Q)=\deg(P)-1$, que $Q$ est scindé à racines simples sur $\R$, et que $P$ et $Q$ n'ont aucune racine commune. On pose enfin $F=\dfrac{P}{Q}$, $\mathbb{H}=\{z\in\C,\ \text{Im}(z)\gt 0\}$. Montrer l'équivalence entre : + + $P$ est scindé sur $\R$ et $P$ et $Q$ sont entrelacés, + + $F(\mathbb{H})\subset\mathbb{H}$ +#+end_exercice +#+BEGIN_proof +Si $Q(x_1) = Q(x_2) = 0$ avec $x_1\lt x_2$, sans racines de $P$ entre les deux, alors en suivant le chemin $z_t = i\eps + t$ de $x_1^-$ à $x_2^+$ l'argument de $Q(z)$ est modifié de $\simeq 2\pi$, alors que celui de $P(z)$ ne change pas. + +Réciproquement, c'est clair. +#+END_proof + + +# ID:7682 +#+begin_exercice [ENS MP 2024 # 28] +Soient $n\in\N^*$, $A\in\op{GL}_n(\R)$ et $u,v\in\R^n\setminus\{0\}$. Exprimer $\det(A+uv^T)$. Dans le cas ou celui-ci est non-nul, exprimer $(A+uv^T)^{-1}$. +#+end_exercice +#+BEGIN_proof +Pour $u,v = \vec e_1$, le déterminant vaut $\det A + \det A_{11}$, et l'inverse ?? + +Plutôt : si $A$ inversible vaut l'identité, c'est $\det (I_n + uv^T) = 1 - \langle u, v\rangle$, qui admet un inverse de la forme $I_n + c uv^T$. +#+END_proof + + +# ID:nil # Trivial +#+begin_exercice [ENS MP 2024 # 29] +Soit $\mathbb{K}$ un sous-corps de $\C$. + - Soient $E$ un $\mathbb{K}$-espace vectoriel et $f\in\mc{L}(E)$. Que dire de $f$ si, pour tout $x\in E$, la famille $(x,f(x))$ est liée? + - Soit $A\in\M_n(\mathbb{K})$ telle que $\op{tr}A=0$. Montrer que $A$ est semblable à une matrice dont la diagonale est nulle. +#+end_exercice + + +# ID:7679 +#+begin_exercice [ENS MP 2024 # 30] + - Calculer $\det\left(i^j\right)_{1\leq i,j\leq n}$. + - Soient $a_1,\dots,a_n$ des réels distincts. Pour $i\in\db{1,n}$, on pose: + $$P_i=\prod_{j \in\db{ 1,n}\setminus\{i\}}(X-a_j)=\sum_{k=1}^n\alpha_{i,k}X^{k-1}$$ + Calculer $\det(\alpha_{i,k})_{1\leq i,k\leq n}$. +#+end_exercice +#+BEGIN_proof + - + - Pour $n = 3$, on trouve le déterminant de Van der Monde. En général, on retire la première colonne aux autres, on obtient une ligne de $0$, une ligne de $a_1 - a_i$, et on peut factoriser par ces $a_1 - a_i$. On obtient exactement le déterminant de taille $n-1$ : en général, les termes sans $a_1$ ni $a_i$ partent, et il reste la différence des termes en $a_i$ et ceux en $a_1$. +#+END_proof + + +# ID:nil # Trivial +#+begin_exercice [ENS MP 2024 # 31] +Soient $n,r,k\in\N$ avec $1\leq r\leq n$ et $r+k\leq n$. Soit $M=\begin{pmatrix}A&B\\ C&D\end{pmatrix}\in\M_n(\C)$, ou $A\in\op{GL}_r(\C)$. Montrer que $M$ est de rang $r+k$ si et seulement si $D-CA^{-1}B$ est de rang $k$. +#+end_exercice + + +# ID:7680 +#+begin_exercice [ENS MP 2024 # 32] +Soient $n\in\N^*$, $m$ un entier supérieur ou egal à $2$. Montrer que la reduction modulo $m$ définit un morphisme de groupes de $\text{SL}_n(\Z)$ dans $\text{SL}_n(\Z/m\Z)$, puis que ce morphisme est surjectif. +#+end_exercice +#+BEGIN_proof +Surjectivité : Dans $SL_n(\Z/m\Z)$, on est produit de transvections : le pgcd des coefficients de la première colonne et de $m$ doit être $1$, donc on peut trouver un coefficient premier avec $m$, et le mettre en haut à gauche. Etc. +#+END_proof + + +#+begin_exercice Sous-algèbre transitive [ENS MP 2024 # 33] :todo: +Soient $n\in\N^*$, $A$ une sous-algèbre de $\M_n(\C)$. On suppose que, pour tout $v\in\C^n$ non nul, on a $\{Mv\ ;\ M\in A\}=\C^n$. Montrer que $A=\M_n(\C)$. +#+end_exercice +#+BEGIN_proof +Si $\dim A\leq n$, chaque matrice est entièrement déterminée par sa première colonne, donc il existe $A_i$ tel que $\forall M\in A,\, M_i = A_i M_1$. En utilisant que $A$ est une algèbre, on obtient que $M$ doit commuter avec les $A_i$, donc $M$ est diagonale par blocs, ce qui est contradictoire. + +Donc $\dim A\gt n$, donc il existe dans $A$ une matrice $B$ dont la première colonne est nulle, en prenant l'image de $M\mapsto MB$ on en obtient d'autres, dont la projection sur chaque colonne non nulle de $B$ est $\R^n$. +#+END_proof + + +#+begin_exercice [ENS MP 2024 # 34] +Soient $A,B\in{\cal M}_n({\R})$ telles que $A^2+B^2=AB$ et $AB-BA\in{\rm GL}_n({\R})$. Montr er que $n$ est divisible par $3$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 35] +Soient $\chi:({\Z}/n{\Z})^{\times}\ra{\C}^*$ un morphisme de groupes non constant. Soit ${\cal A}$ l'ensemble des matrices de la forme $(a+b\chi(r)+c\overline{\chi(s)}+d\chi(r)\overline{\chi(s)})_{r,s\in({\mathbb{ Z}}/n{\Z})^{\times}}$ avec $a$, $b$, $c$ et $d\in{\R}$. - Montr er que ${\cal A}$ est un ${\R}$-espace vectoriel. - Pour $\xi:({\Z}/n{\Z})^{\times}\ra{\C}^*$ un morphisme de groupes, calculer $\sum_{r\in({\Z}/n{\Z})^{\times}}\xi(r)$. + - Montr er que ${\cal A}$ est stable par produit matriciel et que la ${\R}$-algèbre $({\cal A},+,\times,.)$ est isomorphe à ${\cal M}_2({\R})$ (on exhibera un isomorphisme). +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 36] +On s'interesse aux parties de ${\cal M}_n({\R})$ qui sont des groupes pour le produit matriciel. - Donner des exemples de tels groupes, dont certains ne soient pas des sous-groupes de ${\rm GL}_n({\R})$. + - Soit $A\in{\cal M}_n({\R})$. Montr er que $A$ est semblable à une matrice de la forme $\begin{pmatrix}B&0\\ 0&N\end{pmatrix}$, ou $B$ est inversible et $N$ nilpotente. + - Caractériser ces groupes. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 37] +Pour tout $A\in{\cal A}_4({\R})$, soit ${\rm Pf}(A)=a_{1,2}a_{3,4}-a_{1,3}a_{2,4}+a_{1,4}a_{2,3}$. + - Montr er que, pour tout $A\in{\cal A}_4({\R})$, ${\rm Pf}(A)^2=\det(A)$. + - On admet que ${\rm GL}_n^+({\R})$ est connexe par arcs. Montr er que, pour tout $A\in{\cal A}_4({\R})$ et tout $B\in{\cal M}_4({\R})$, ${\rm Pf}(BAB^T)=\det(B){\rm Pf}(A)$. + +_Ind._ Pour le cas $\det B\lt 0$, considérer la matrice $J={\rm diag}(-1,1,1,1)$. + - Soit $R\in{\rm SO}_4({\R})$. On pose $A=R-R^T$. Montr er l'équivalence entre : + +(i) $R$ n'a pas de valeur propre réelle, (ii) ${\rm Pf}(A)\neq 0$, (iii) $A$ est inversible. + - Soient $R_1,R_2\in{\rm SO}_4({\R})$, $A_1=R_1^T-R_1$ et $A_2=R_2^T-R_2$. On suppose $\chi_{R_1}=\chi_{R_2}$ et ${\rm Pf}(A_1)={\rm Pf}(A_2)\neq 0$. Montr er qu'il existe $P\in{\rm SO}_4({\R})$ telle que $R_1=PR_2P^T$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 38] +Déterminer l'image de $\phi:M\in{\cal M}_2({\C})\mapsto\sum_{n\in{\N}}\frac{(- 1)^n}{(2n+1)!}M^{2n+1}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 39] +A quelle condition sur la matrice $A$, la comatrice de $A$ est-elle diagonalisable? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 40] +Pour $i\in{\N}$ et $A\in{\cal M}_n({\C})$, on note $c_i(A)$ le coefficient numero $i$ du polynôme caractéristique $\chi_A(X)$ de la matrice $A$. + - Montr er que $c_i(AB)=c_i(BA)$ pour toutes matrices $A,B\in{\cal M}_n({\C})$ et $i\in{\N}$. + - Le résultat reste-t-il valable pour des matrices à coefficients dans un corps ${\mathbb{K}}$ quelconque? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 41] +Soient $n\in{\N}$ avec $n\geq 2$, $\zeta=e^{2i\pi/n}$ et $S=\Big(\zeta^{(r-1)(s-1)}\Big)_{1\leq r,s\leq n}$. + - Donner une expression simple de $\det(S)$. Ind. On pourra calculer $S^2$. + - On pose $G_n=\sum_{k=0}^{n-1}e^{\frac{2ik^2\pi}{n}}$. Donner une expression simple de ${\left|G_n\right|^2}$ par un calcul direct. - On suppose que $n$ est impair. Déterminer le spectre de $S$ et la multiplicité de chacune de ses valeurs propres. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 42] + - Rappeler l'ordre d'un élément $k$ de $\Z/n\Z$. + - Montrer que deux permutations de $\mc{S}_n$ sont conjuguées si et seulement si elles ont pour tout $k$, le même nombre de cycles de longueur $k$ dans leurs décompositions en produit de cycles à supports disjoints. + - Soit $c$ un cycle de longueur $k$. Déterminer le nombre de cycles dans la décomposition de $c^i$ en produit de cycles à supports disjoints. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 43] +Soient $E$ un $\C$-espace vectoriel de dimension finie et $v,w\in\mc{L}(E)$. On note $u=vw-wv$. Pour $\lambda\in\op{Sp}(u)$, on note $F_u(\lambda)=\bigcup_{m\geq 1}\op{Ker}(u-\lambda\op{id })^m$ + - Montrer que $F_u(\lambda)$ est un sous-espace vectoriel stable par $u$ et qu'il admet un supplemen- taire stable par $u$. + - On écrit $\pi_u=(X-\lambda)^pQ$ avec $(X-\lambda)\wedge Q=1$. Montrer que $E=F_u(\lambda)\oplus\op{Ker}Q(u)$. On suppose de plus que $u$ commute avec $v$. On note $p_{\lambda}$ le projecteur sur $F_u(\lambda)$ parallélément à $\op{Ker}Q(u)$. + - Montrer que $p_{\lambda}$ commute avec $v$. + - Montrer que $\op{tr}(up_{\lambda})=\lambda\op{rg}(p_{\lambda})=0$. + - En déduire que $u$ est nilpotent. + - On suppose desormais que $vw^2-w^2v=w$. Montrer qu'il existe un entier $d$ impair tel que $\pi_w=X^d$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 44] +Soit $A,B,M$ dans $\M_n(\C)$. On munit $\M_n(\C)$ d'une norme arbitraire $\|\ \|$. + - Montrer que $M$ est nilpotente si et seulement si $\forall k\in\N^*,\ \op{tr}(M^k)=0$. + - On suppose que $A(AB-BA)=0$. Montrer que $AB-BA$ est nilpotente. + - On suppose que $A(AB-BA)=(AB-BA)A$. Montrer que $AB-BA$ est nilpotente. + - Soit $(M_k)_{k\in\N}$ une suite de matrices de $\M_n(\C)$, toutes semblables. On suppose que $\|M_k\|\ra+\i$. Montrer qu'il existe une extraction $\phi$ et une matrice nilpotente $N$ telles que $\frac{M_{\phi(k)}}{\|M_{\phi(k)}\|}\underset{k\ra+\i}{ \longrightarrow}N$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 45] +Soit $A\in\M_n(\C)$ de polynôme caractéristique $\chi_A=\prod_{i=1}^r\underbrace{(X-\lambda_i)^{\alpha_i}}_{=P_i}$. + - Montrer que $P_i$ est le polynôme caractéristique de l'endomorphisme induit par $A$ sur $\op{Ker}P_i(A)$. + - Montrer qu'il existe $D$ diagonalisable et $N$ nilpotente telles que $A=D+N$ et $ND=DN$. + - Si $X\in\M_n(\C)$, on note $\op{Comm}_X:M\mapsto MX-XM$. On reprend les notations précédentes. Montrer que $\op{Comm}_A=\op{Comm}_D+\op{Comm}_N$, que $\op{Comm}_D$ et $\op{Comm}_N$ commutent et sont respectivement diagonalisable et nilpotente. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 46] +Soient $n\in\N^*$ et $\mathbb{K}$ un sous-corps de $\C$. Une matrice $A\in\M_n(\mathbb{K})$ est dite toute puissante (TP $\mathbb{K}$) si, pour tout $p\in\N^*$, il existe $B\in\M_n(\mathbb{K})$ telle que $A=B^p$. - Trouver les matrices TP $\mathbb{K}$ pour $n=1$ et $\mathbb{K}=\R,\Q,\C$. + - Soit $A\in\M_n(\mathbb{K})$. On suppose que $\chi_A=\prod_{i=1}^k(X-\lambda_i)^{\alpha_i}$ ou les $\lambda_i$ sont distincts dans $\mathbb{K}$ et les $\alpha_i$ sont des entiers naturels non nuls. + - Montrer qu'il existe $N_1,\ldots,N_k$ nilpotentes telles que $A$ soit semblable à une matrice diagonale par blocs avec comme blocs diagonaux $\lambda_1I_{\alpha_1}+N_1,\ldots,\lambda_kI_{\alpha_k}+N_k$. + - Montrer que $A$ est TP $\mathbb{K}$ si et seulement si les $\lambda_iI_{\alpha_i}+N_i$ le sont. On dit que $M\in\M_n(\mathbb{K})$ est unipotente si $M-I_n$ est nilpotente et on note $\mc{U}_n(\mathbb{K})$ l'ensemble des matrices unipotentes de $\M_n(\mathbb{K})$. + +Pour $A\in\mc{U}_n(\mathbb{K})$, on pose $\ln(A)=\sum_{p=1}^{+\i}\frac{(-1)^{p-1}}{p}(A-I_n)^p$. + - Justifier la définition de $\ln(A)$ pour $A\in\mc{U}_n(\mathbb{K})$. Montrer que $\exp$ est une bijection de l'ensemble des matrices nilpotentes sur l'ensemble $\mc{U}_n(\mathbb{K})$. + - Montrer que les matrices unipotentes sont TP $\mathbb{K}$. + - Déterminer finalement les matrices toutes-puissantes de $\M_n(\C)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 47] +Soient $A$, $B$, $C\in\M_n(\R)$. On considére l'équation $(E):X-AXB=C$ d'inconnue $X\in\M_n(\R)$. On note $\mathrm{Sp}_{\C}(A)$ et $\mathrm{Sp}_{\C}(B)$ les spectres complexes de $A$ et $B$. + - On suppose que, pour tout $(\alpha,\beta)\in\mathrm{Sp}_{\C}(A)\times\mathrm{Sp}_{\C}(B)$, $\alpha\beta\neq 1$. Montrer que l'équation $(E)$ admet une unique solution. + - Que se passe-t-il dans le cas general? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 48] +Combien y-a-t-il de classes de similitude de $\M_{3n}(\R)$ constituées de matrices $M$ telles que $M^3=0$? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 49] +Déterminer les $M$ de $\M_n(\R)$ telles que $M$ soit semblable à $2M$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 50] +Déterminer les matrices $A\in\mathrm{GL}_n(\R)$ telles que, pour tout $k\geq 2$, on dispose de $M\in\M_n(\Z)$ vérifiant $A=M^k$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 51] +Montrer que toute matrice de $\mathrm{GL}_n(\C)$ admet une racine carrée. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 52] + - Montrer que toute $M\in\mathrm{SL}_n(\C)$ s'écrit de facon unique $UD$ ou $U\in\mathrm{SL}_n(\C)$ est de la forme $I_n+N$ avec $N$ nilpotente, $D\in\mathrm{SL}_n(\C)$ est diagonalisable et $UD=DU$. + - Soit $\rho$ un morphisme de groupes de $\mathrm{SL}_n(\C)$ dans $\mathrm{SL}_m(\C)$ tel que les coefficients de $\rho(M)$ soient des fonctions polynomiales de ceux de $M$. Montrer que $\rho$ respecte la décomposition de la question précédente. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 53] + - Soient $A,B\in\M_n(\C)$ diagonalisables. à quelle condition existe-t-il $P\in\mathrm{GL}_n(\C)$ tel que $PAP^{-1}$ et $PBP^{-1}$ soient diagonales? + - Soit $A\in\M_n(\C)$. Montrer que $A$ s'écrit de maniere unique $A=D+N$ avec $D$ diagonalisable, $N$ nilpotente et $DN=ND$. + - Soient $A\in\M_n(\C)$, $\pi_A=\prod_{i=1}^r(X-\lambda_i)^{\beta_i}$ son polynôme minimal et $P\in\C[X]$.Montrer que $P(A)$ est diagonalisable si et seulement si $P^{(j)}(\lambda_i)=0$ pour tous $i\in\db{1,r\rrbracket$ et $j\in\llbracket 1,\beta_i-1}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 54] + - Soient $u,v$ deux endomorphismes diagonalisables d'un $\mathbb{K}$-espace vectoriel $E$ de dimension finie, tels que $uv=vu$. Montrer que $u$ et $v$ sont codiagonalisables. + - Soit $u$ un endomorphisme d'un $\mathbb{K}$-espace vectoriel $E$ de dimension finie. Montrer que $u$ admet au plus une décomposition de la forme $u=d+n$, ou $(d,n)\in\mathbb{K}[u]^2$, l'endomorphisme $d$ est diagonalisable, l'endomorphisme $n$ est nilpotent et $dn=nd$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 55] +Soient $n\in\N$ et $w$ une fonction continue positive non identiquement nulle de $[0,1]$ dans $\R$. + - Soit $(f - {0\leq k\leq n}$ une suite de fonctions continues de $[0,1]$ dans $\R$ telle que, pour tout $(k,\ell)\in\db{0,n}^2$, $\int_0^1f_kf_{\ell}w=\delta_{k,\ell}$. Montrer que $(f - {0\leq k\leq n}$ est libre. + - Montrer qu'il existe une unique suite $(p_k)_{k\in\N}$ telle que, pour tout $(k,\ell)\in\N^2$, $\int_0^1p_kp_{\ell}w=\delta_{k,\ell}$ et que, pour tout $k\in\N$, $p_k$ soit polynomiale de degre $k$ à coefficient dominant positif. + - Montrer que, si $n\in\N^*$, $p_n$ à $n$ racines simples dans $]0,1[$ que l'on note $x_{1,n}\lt \cdots\lt x_{n,n}$. + - Montrer que, si $n\in\N^*$, il existe un unique $(\lambda_{1,n},\ldots,\lambda_{n,n})\in\R^n$ tel que, pour tout $p\in\R_{2n-1}[X]$, $\int_0^1pw=\sum_{k=1}^n\lambda_{k,n}p(x_{k,n})$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 56] +Soient $e_1,\ldots,e_n$ des vecteurs d'un espace euclidien $E$ tels que $\langle e_i,e_j\rangle\leq 0$ pour tous $i,j$ distincts dans $\db{1,n\rrbracket$. Montrer que $(e_1,\ldots,e_n)$ est libre si et seulement s'il existe une forme lineaire $f$ sur $E$ telle que $\forall i\in\llbracket 1,n},\;f(e_i)\gt 0$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 57] +Soient $n,m\geq 1$ des entiers. On note $\langle\;,\;\rangle$ le produit scalaire canonique sur $\R^n$. Montrer qu'il existe un espace prehilbertien $(E,\langle\;,\;\rangle_E)$ et une application $f\colon\R^n\ra E$ tels que, pour tous $x,x'\in\R^n$, $\langle x,x'\rangle^m=\langle f(x),f(x')\rangle_E$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 58] +Trouver un espace prehilbertien $(E,\langle\;,\;\rangle)$ et $f\colon\R\ra E$ tels que, pour tous $x,y\in\R$, $\exp\left(-\frac{(y-x)^2}{2}\right)=\langle f(x),f(y)\rangle$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 59] +Soient $m,n\in\N^*$ tels que $n\lt m$. On munit $\R^m$ de sa structure euclidienne canonique. Soit $r\in\N^*$, on considére $r$ vecteurs de $\R^m$ notes $x_1,\ldots,x_r$. + - Montrer qu'il existe une matrice $U_0\in\M_{m,n}(\R)$ minimisant $\sum_{i=1}^r\|x_i-UU^Tx_i\|_2^2$ parmi toutes les matrices $U\in\M_{m,n}(\R)$ telles que $U^TU=I_n$. + - Montrer que $\min_{U\in\M_{m,n}(\R)}\sum_{i=1}^r\|x_i-UU^Tx_i\|_2 ^2=\min_{U,V\in\M_{m,n}(\R)}\sum_{i=1}^r\|x_i-UV^Tx_{ i}\|_2^2$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 60] +Soient $(\lambda_n)_{n\geq 0}\in\R^{\N}$ une suite strictement croissante vérifiant $\lambda_0=0$ et $k$ dans $\R\setminus\{\lambda_n,\;n\in\N\}$. + - Calculer $I_{n,k}=\inf_{(a_0,\ldots,a_n)\in\R^{n+1}}\int_0^1\left(t^k- \sum_{i=0}^na_it^{\lambda_i}\right)^2dt$. + +On admettra que le déterminant de la matrice de coefficient general $m_{i,j}=\dfrac{1}{1+x_i+y_j}$ vaut $\dfrac{\prod_{1\leq i\lt j\leq n}(x_j-x_i)(y_j-y_i)}{\prod_{1 \leq i,j\leq n}(1+x_i+y_j)}$. + - En déduire une condition suffisante sur $(\lambda_n)$ pour que $F=\mathrm{Vect}(t\mapsto t^{\lambda_n})_{n\in\N}$ soit dense dans $\mc C^0([0,1],\R)$ pour la norme $f\mapsto\left(\int_0^1f^2\right)^{1/2}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 61] +Soit $V$ un $\R$-espace vectoriel de dimension finie, $\left\langle\;,\;\right\rangle_1$ et $\left\langle\;,\;\right\rangle_2$ deux produits scalaires tels que $\forall(x,y)\in V^2$, $\left\langle x,y\right\rangle_1=0\Longleftrightarrow\left\langle x,y\right\rangle_2=0$ + - Soient $x,y\in V$. Montrer que si $\|x\|_1=\|y\|_1$ alors $\|x\|_2=\|y\|_2$. + - En déduire qu'il existe $C\gt 0$ tel que $\forall x\in E$, $\|x\|_1=C\|x\|_2$. + - Soit $u\in\mc{L}(V)$ qui préserve l'orthogonalite : si $\left\langle x,y\right\rangle=0$ alors $\left\langle u(x),u(y)\right\rangle=0$. Montrer qu'il existe $C\in\R^+$ tel que $u\circ u^*=C\,\mathrm{id}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 62] +Soient $E$ un espace euclidien de dimension $n$ et $(e_1,\ldots,e_n)$ une base de $E$. On pose $\Lambda=\Big{\{}\sum_{i=1}^n\lambda_ie_i,\,(\lambda - {1 \leq i\leq n}\in\Z^n\Big{\}}$. + - Soit $r\in\mc{O}(E)$ tel que $r(\Lambda)\subset\Lambda$. Montrer que $r(\Lambda)=\Lambda$. + - Montrer que $G_{\Lambda}=\{r\in\mc{O}(E),r(\Lambda)=\Lambda\}$ est un sous-groupe fini de $\mc{O}(E)$. + - Ici $n=3$. Montrer que tous les éléments de $G_{\Lambda}$ ont un ordre qui divise $12$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 63] +Soient $E$ un espace euclidien de dimension $n$, $G$ un groupe fini et $\rho$ un morphisme injectif de $G$ dans $\text{GL}(E)$ tel que, pour tout $g\in G$, $\rho(g)\in\mc{S}(E)$. Montrer que les éléments de $G$ sont d'ordre $1$ ou $2$, puis que $|G|$ divise $2^n$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 64] + - Déterminer une condition nécessaire et suffisante sur $a\in\R$ pour que la matrice $\begin{pmatrix}1&a\\ a&1\end{pmatrix}$ soit positive, pu définie positive. + - Soit $(a,b,c)\in[-1,1]^3$. On suppose que $1+2abc\geq a^2+b^2+c^2$. Démontrer que $\forall n\in\N^*,\;1+2(abc)^n\geq a^{2n}+b^{2n}+c^{2n}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 65] + - Soit $A\in\mc{S}_n(\R)$ à coefficients strictement positifs. Montrer qu'il existe un vecteur propre de $A$ dont tous les coefficients sont $\gt 0$. + - Soit $A\in\M_2(\R)$ à coefficients $\gt 0$. Montrer que $A$ possede un vecteur propre à coefficients $\gt 0$. + - Soient $a_1,\ldots,a_n\in\N^*$, $M_i=\begin{pmatrix}a_i&1\\ 1&0\end{pmatrix}$ pour $1\leq i\leq n$. Montrer que $M_1\times\cdots\times M_n$ est à spectre inclus dans $\R\setminus\Q$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 66] + - Rappeler la définition de l'adjoint d'un endomorphisme d'un espace euclidien. - Soient $E$ un espace euclidien et $u\in\mc{L}(E)$. Montrer que $u$ et $u^*$ commutent si et seulement s'il existe une base orthonormée de $E$ dans laquelle la matrice de $u$ est diagonale par blocs, les blocs diagonaux etant soit de taille $1$, soit de taille $2$ et de la forme $\left(\begin{array}{cc}a&b\\ -b&a\end{array}\right)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 67] +Montrer que $\text{SO}_3(\Q)$ est dense dans $\text{SO}_3(\R)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 68] +On admet l'existence d'une $\R$-algèbre $\mathbb{H}$ d'unite $1$ admettant une base de la forme $(1,i,j,k)$ avec $i^2=j^2=k^2=-1$ et $ij=k=-ji,jk=i=-kj,ki=j=-ik$. Montrer que le groupe des automorphismes de la $\R$-algèbre $\mathbb{H}$ est isomorphe à $\text{SO}_3(\R)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 69] +On munit $\M_n(\R)$ du produit scalaire défini par $\left\langle A,B\right\rangle=\op{tr}(A^TB)$. + +Soient $A,B\in\mc{S}_n(\R)$. Montrer que $\colon\inf\limits_{\|G\|=1}\|AG-GB\|=\min\limits_{(\lambda_1,\lambda_2)\in\text {Sp}(A)\times\text{Sp}(B)}|\lambda_1-\lambda_2|$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 70] +Soient $X$ un ensemble et $K:X\times X\ra\R$. On suppose que, pour tous $n\geq 1$ et $x_1,\ldots,x_n\in X$, $(K(x_i,x_j))_{1\leq i,j\leq n}\in\mc{S}_n^+(\R)$. Pour $x\in X$, on note $K_x:y\mapsto K(x,y)$. Soit $E$ le sous-espace de $\R^X$ engendre par les fonctions $(K_x)_{x\in X}$. + +Soit $a,b\in E$. Par définition de $E$, il existe $(\lambda_x)_{x\in X}$ et $(\mu_x)_{x\in X}$ dans $\R^X$ n'admettant qu'un nombre fini de coefficients non nuls tels que $a=\sum_{x\in X}\lambda_xK_x$ et $b=\sum_{x\in X}\mu_xK_x,$ et on pose + +$$\left\langle a,b\right\rangle=\sum_{x,y\in X}\lambda_x\mu_yK(x,y).$$ + - Montrer que cela définit bien un produit scalaire sur $E$. - Montrer qu'il existe $f:X\ra E$ telle que $\forall x,y\in X$, $K(x,y)=\left\langle f(x),f(y)\right\rangle$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 71] +Soient $p\geq 1$ et $A,B\in\mc{S}_p^{++}(\R)$. + - Montrer que $\op{tr}\left(I_p-A^{-1}B\right)\leq\ln\left(\frac{\det A}{ \det B}\right)$. + - Soient $n\geq 1$, $u_1,\ldots,u_n\in\R^p$ et $\lambda\gt 0$. Pour $1\leq m\leq n$, on pose $A_m=\sum_{k=1}^mu_k\ u_k^T$ et $B_m=\lambda I_p+A_m$. Montrer que, pour $1\leq m\leq n$, $B_m$ est symétrique définie positive. + - Soient $\lambda_1,\ldots,\lambda_p$ les valeurs propres (avec multiplicité) de $A_n$. + +Montrer que $\sum_{m=1}^n\left\langle u_m,B_m^{-1}u_m\right\rangle \leq\sum_{i=1}^p\ln\left(1+\frac{\lambda_i}{\lambda}\right)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 72] +Si $G$ est un groupe, on note $Z(G)$ son centre. + +On pose $U_n(\C)=\left\{A\in\M_n(\C)\,,\,A^*A=I_n\right\}$ ou $A^*=\overline{A}^T$, l'ensemble des matrices unitaires. + - Montrer que $Z(G)$ est un sous-groupe de $G$ et que $\mc{U}_n(\C)$ est un sous-groupe de $\mathrm{GL}_n(\C)$. + - Soit $A\in\M_n(\C)$ hermitienne, c'est-a-dire telle que $A^*=A$. Démontrer qu'il existe $P\in\mc{U}_n(\C)$ telle que $P^*AP$ soit diagonale. - Démontrer que toute matrice $M\in\M_n(\C)$ s'écrit comme combinaison lineaire d'au plus quatre matrices unitaires. + - Déterminer $Z\left(\mc{U}_n(\C)\right)$. +#+end_exercice + + +** Analyse + +#+begin_exercice [ENS MP 2024 # 73] +Soit $F$ l'application qui à une norme $N$ sur $\R^n$ associe la boule fermée de centre $0$ et de rayon $1$ pour $N$. + - L'application $F$ est-elle injective? + - Quelle est l'image de $F$? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 74] +Soient $E$ un $\R$-espace vectoriel et $\phi:E\ra\R^+$ une application telle que + - pour tout $x\in E$, $\phi(x)=0$ si et seulement si $x=0$, + - pour tout $x\in E$ et tout $\lambda\in\R$, $\phi(\lambda x)=|\lambda|\phi(x)$. + +On note $C=\{x\in E,\ \phi(x)\leq 1\}$. + - Montrer que $\phi$ est une norme si et seulement si $C$ est convexe. + - Soit $K$ un partie de $E$ convexe, compacte, d'interieur non vide et symétrique par rapport à l'origine. Montrer que $K$ est un voisinage de l'origine. + - Soit $x\in E\setminus\{0\}$. Posons $I(x)=\{\lambda\gt 0\,;\ \exists k\in K,\ x=\lambda k\}$. Montrer que $I(x)$ est un convexe ferme, non vide. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 75] +Soit $G$ un sous-groupe de $(\R^n,+)$ dans lequel $0$ est un point isole. Montrer qu'il existe une famille libre $(u_1,\ldots,u_p)$ dans $\R^n$ telle que $G=\left\{\sum_{k=1}^pa_k.u_k,\ (a_1,\ldots,a_p)\in\Z^p\right\}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 76] +Soit $n\in\N^*$. Soit $E$ l'ensemble des paves de $\R^n$, c'est-a-dire des parties de la forme $[a_1,b_1]\times\cdots\times[a_n,b_n]$ avec $a_1\leq b_1$,..., $a_n\leq b_n$. Pour toute partie finie $G\subset\R^n$, on note $f(G)=\{F\cap G,\ F\in E\}$. Déterminer $\sup\{k\in\N\ ;\ \exists G\subset\R^n,\ |G|=k,\ f(G)= \mc{P}(G)\}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 77] + - Soient $E$ un espace vectoriel norme et $K$ un compact convexe non vide de $E$. Soit $(f_i)$ une suite de fonctions affines, continues, qui commutent deux à deux et telles que $f_i(K)\subset K$ pour tout $i\in\N$. Montrer que les fonctions $f_i$ ont un point fixe commun. + - Le résultat précédent reste-t-il valable pour une famille $(f_i)_{i\in I}$ de fonctions indexées par un ensemble non dénombrable? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 78] +Soient $H$ le groupe (pour la composition) des homeomorphismes de $\R$ sur $\R$, $H^+$ le sous-groupe des homeomorphismes croissants. + - Caractériser les groupes finis isomorphes à un sous-groupe de $H$. + - Montrer qu'on peut munir tout sous-groupe dénombrable $G$ de $H^+$ d'une relation d'ordre totale telle que $\forall f,g,h\in G$, $f\leq g\ \Longrightarrow\ h\circ f\leq h\circ g$. + - Réciproquement, montrer que tout groupe dénombrable pouvant être muni d'un tel ordre est isomorphe à un sous groupe de $H^+$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 79] +Soient $m$ et $n$ dans $\N^*$, $F$ une partie finie de $\R^n$, $x\in\R^n\setminus F$, $f$ une application $1$-lipschitzienne (pour les normes euclidiennes canoniques) de $F$ dans $\R^m$. Montrer que l'on peut prolonger $f$ en une application $1$-lipschitzienne de $F\cup\{x\}$ dans $\R^m$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 80] +Soient $\gamma,\tau\in\R^{+*}$. On pose, pour $N\in\N^*$, + + $D_N=\left\{x\in\R^d\;;\;\forall k\in\Z^d\setminus\{0\}, \|k\|\leq N\Rightarrow|\left\langle x,k\right\rangle|\geq\frac{ \gamma}{\|k\|^{\tau}}\right\}$ et $D=\bigcap_{N\geq 1}D_N$. + +Montrer que $D$ est ferme et d'interieur vide. Qu'en est-il de $D_N$? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 81] +Soient $E$ et $F$ deux espaces vectoriels normes. Soit $f:E\ra F$ telle que : $\forall r\in]0,1],$ $\forall x\in E,\ B\left(f(x),\frac{r}{2}\right)\subset f(B(x,r))\subset B(f(x),2r)$. + - Montrer que $f$ est continue et surjective. + - Que peut-on dire de l'image par $f$ d'un ouvert? D'un ferme? + - Soit $\gamma$ un chemin continu de $[0,1]$ dans $F$. Montrer qu'il existe un chemin $c$ continu de $[0,1]$ dans $E$ tel que $f\circ c=\gamma$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 82] + - Soit $X\subset\R^n$ un ferme non vide. Soit $f:X\ra X$. On suppose qu'il existe $\theta\in[0,1[$ tel que $\forall x,y\in X$, $\|f(x)-f(y)\|\leq\theta\|x-y\|$. Montrer que $f$ possede un unique point fixe $c$ et que, pour tout $x\in X$, $f^m(x)\underset{m\ra+\i}{\longrightarrow}c$. + - Soit $X\subset\R^n$ un compact non vide. Soit $f:X\ra X$. + +On suppose que $\forall x,y\in X$, $x\neq y\Rightarrow\|f(x)-f(y)\|\lt \|x-y\|$. + - Soient $Y,Z$ deux compacts non vides tels que $f(Y)\subset Y$ et $f(Z)\subset Z$. Montrer que $Y\cap Z$ est non vide. + - En déduire que $f$ possede un unique point fixe. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 83] +On se place dans $\R^n$ et $\R^m$ munis d'une norme. + - Montrer qu'il existe $C\gt 0$ et $R_0\geq 0$ tels que, pour tout $r\geq R_0,\ \mathrm{card}\{x\in\Z^n\,;\,\|X\|\leq r\} \leq Cr^n$. + +On appelle plongement grossier $f\colon\Z^n\ra\Z^m$ une fonction qui vérifie : + + $\forall a\geq 0,\ \exists b\geq 0,\ \forall x,y\in\Z^n,\ \|x-y\|\leq a\Rightarrow\|f(x)-f(y)\|\leq b$, + + $\forall b\geq 0,\ \exists a\geq 0,\ \forall x,y\in\Z^n,\ \|f(x)-f(y)\|\leq b\Rightarrow\|x-y\|\leq a$. + +Soit $f\colon\Z^n\ra\Z^m$ un prolongement grossier. + - Montrer qu'il existe $\rho\colon\R^+\ra\R^+$ et $\mu\gt 0$ tels que $\lim_{x\ra+\i}\rho(x)=+\i$ et + + $\forall x,y\in\Z^n,\ \rho(\|x-y\|)\leq\|f(x)-f(y)\|\leq\mu\|x-y\|$. + - Montrer que $m\geq n$. + - Adapter pour $f\colon\R^n\ra\R^m$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 84] +On munit $\R^2$ de sa structure euclidienne canonique. Soit $f\colon\R^2\ra\R^2$ un homeomorphisme. Pour $x\in\R^2$ et $r\gt 0$, on pose : + + $L_f(x,r)=\sup\big{\{}\|f(x)-f(y)\|\;;\;y\in\R^2,\ \|x-y\|\leq r \big{\}}$, + + $\ell_f(x,r)=\inf\big{\{}\|f(x)-f(y)\|\;;\;y\in\R^2,\ \|x-y\| \geq r\big{\}}$. + - Montrer que : + + $L_f(x,r)=\sup\big{\{}\|f(x)-f(y)\|\;;\;y\in\R^2,\ \|x-y\|=r\big{\}}$, + + $\ell_f(x,r)=\inf\big{\{}\|f(x)-f(y)\|\;;\;y\in\R^2,\ \|x-y\|=r\big{\}}$. - Pour $x$ fixe, montrer que $r\mapsto L_f(x,r)$ et $r\mapsto\ell_f(x,r)$ sont croissantes. + +On dit que $f$ est quasi-conforme s'il existe $K_f\gt 0$ tel que : + + $\forall(x,r)\in\R^2\times\R^{+*}$, $L_f(x,r)\leq K_f\ell_f(x,r)$. + - On suppose $f$ quasi-conforme. Montrer qu'alors $L_f(x,2r)\leq(1+K_f)L_f(x,r)$. + - Montrer que $f$ est quasi-conforme si et seulement si $f^{-1}$ est quasi-conforme. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 85] +Soient $n\geq 2$ et $e_1$ le premier vecteur de la base canonique de $\R^n$. Soit $\mc{A}$ l'ensemble des matrices $M$ de $\M_n(\R)$ telles que, pour tout $v\in\R^n$, il existe $a_{v,M}\in\R$, tel que la suite $(M^kv)_{k\geq 1}$ tende vers $a_{v,M}e_1$, avec de plus $v\mapsto a_{v,M}$ non identiquement nulle. + +Soit $v\in\R^n$. Montrer que l'application $f_v:M\in\mc{A}\mapsto a_{v,M}$ est continue. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 86] +Soit $(E,\|\ \|)$ un espace vectoriel norme. Pour $X\subset E$ et $x\in E$, on note $d(x,X)=\inf_{y\in X}\|y-x\|$ et $\Pi_X(x)=\{y\in X\;;\;\forall z\in X,\;\|y-x\|\leq\|z-x\|\}$. + - Pour quels ensembles $Y\subset E$ existe-t-il $X\subset E$ et $x\in E$ tels que $Y=\Pi_X(x)$? + - Soient $X\subset E$ et $x\in E\setminus X$ tels que $d(x,X)=0$. Montrer que $\Pi_X(x)=\emptyset$. + - Existe-t-il $X\subset E$ et $x\in E\setminus X$ tels que $d(x,X)\gt 0$ et $\Pi_X(x)=\emptyset$? + - On suppose qu'il existe un produit scalaire $\langle\,\ \rangle$ tel que $\|x\|=\sqrt{\langle x,x\rangle}$ pour tout $x\in E$, que $E$ est de dimension finie et que $X\subset E$ est un ensemble convexe ferme et borne. Montrer que $\Pi_x(X)$ est un singleton. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 87] +Soit $n\geq 1$ un entier, $L\in\left]0,1\right[$, $F\colon\R^n\ra\R^n$ une application $L$-lipschitzienne pour $\left\|\ \right\|_{\i}$, et $x_*\in\R^n$ tel que $F(x_*)=x_*$. + - Soit $(x_k)_{k\geq 1}$ définie par $x_1\in\R^n$ et $\forall k\geq 1$, $x_{k+1}=F(x_k)$. Montrer que $x_k\xrightarrow[k\ra+\i]{}x_*$. + - Pour $I\subset\{1,\ldots,n\}$, on note $F^{|I}\colon\R^n\ra\R^n$ l'application définie, pour tout $x\in\R^n$ et $1\leq i\leq n$, par $F^{|I}(x)_i=\begin{cases}F(x)_i&\text{si }i\in I\\ x_i&\text{si }i\not\in I\end{cases}$. + +Montrer que $F^{|I}$ est $1$-lipschitzienne pour $\left\|\ \right\|_{\i}$. + - Soit $(I_k)_{k\geq 1}$ une suite de sous-ensembles de $\{1,\ldots,n\}$ telle que chaque indice $i\in\{1,\ldots,n\}$ appartienne à une infinite de ces ensembles. Soient $x_1\in\R^n$ et, pour $k\geq 1$, $x_{k+1}=F^{|I_k}(x_k)$. Montrer que cette suite converge vers $x_*$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 88] +On munit l'espace $\ell^{\i}$ des suites réelles bornées de la norme $\|\ \|_{\i}$. + - Soit $(a_n)$ une suite réelle sommable. Montrer que l'application $f:x\mapsto\sum_{n=0}^{+\i}a_nx_n$ définit une forme lineaire continue sur l'espace $\ell^{\i}$. + - On suppose l'existence d'une partie $F\subset\mc{P}(\N)$ telle que : (i) pour tous $A,B\in F$, $A\cap B\in F$, (ii) pour $A\in F$, $F$ contient toute partie $B$ de $\N$ qui contient $A$, (iii) $F$ ne contient que des ensembles infinis, (iv) si $A\in\mc{P}(\N)$, alors $A\in F$ ou $\N\setminus A\in F$. + - Soit $x\in\ell^{\i}$. + +Montrer qu'il existe un unique réel $x^{\i}$ tel que $\forall\eps\gt 0$, $\exists A\in F$, $\forall n\in A$, $|x_n-x^{\i}|\leq\eps$. + - En déduire l'existence d'une forme lineaire continue sur $\ell^{\i}$ qui n'est pas de la forme donnée en question -. + - On note $c_0$ le sous-espace de $\ell^{\i}$ des suites réelles de limite nulle. Montrer que toute forme lineaire continue sur $c_0$ est de la forme donnée en question -. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 89] +Soient $r\in\R^{+*}$, $E$ une partie de $\R^2$ couplant toute boule de rayon $r$ (pour la norme euclidienne canonique), $P\in\R[X,Y]$ s'annulant sur $E$. Montrer que $P=0$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 90] +Soient $E$ un espace vectoriel réel de dimension finie $n\geq 2$, $C$ un convexe ouvert de $E$ ne contenant pas $0$. Montrer qu'il existe une droite vectorielle ne couplant pas $C$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 91] +Soit $n\in\N^*$. On munit $\R^n$ de sa structure euclidienne canonique. + +Soit $\Delta=\left\{x\in(\R^+)^n,\,\sum_{i=1}^nx_i=1\right\}$. On admet que pour tout $x\in\R^n$, il existe un unique point $\pi(x)\in\Delta$ tel que $\forall z\in\Delta$, $\left\langle z-\pi(x),x-\pi(x)\right\rangle\leq 0$. + - Soient $x,u\in\R^n$ et $x'=\pi(x+u)$. + +Montrer que, pour tout $z\in\Delta$, $2\left\langle u,z-x\right\rangle\leq\left\|z-x\right\|_2^2-\left\|z -x'\right\|_2^2+\left\|u\right\|_2^2$. + +Soit $A\in\M_n(\R)$. Soient $x_1,y_1\in\Delta$ et $(\gamma_k)_{k\geq 1}$ une suite strictement positive. Pour $k\geq 2$, on définit par récurrence $x_{k+1}=\pi(x_k+\gamma_kAy_k)$ et $y_{k+1}=\pi(y_k-\gamma_kA^Tx_k)$. + - Montrer qu'on peut choisir la suite $(\gamma_k)_{k\geq 1}$ de sorte que + +$$\max_{x\in\Delta}\sum_{k=1}^N\left\langle x,Ay_k\right\rangle-\min_{y\in \Delta}\sum_{k=1}^N\left\langle x_k,Ay\right\rangle\leq o(N)\text{.}$$ + - En déduire que $\max_{x\in\Delta}\min_{y\in\Delta}\left\langle x,Ay\right\rangle=\min_{y\in \Delta}\max_{x\in\Delta}\left\langle x,Ay\right\rangle$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 92] +Soient $E$ euclidien et $T:E\ra E$. On suppose qu'il existe $C\in\R^+$ tel que : + + $\forall(x,y)\in E^2,\,\left\|\left|T(x)-T(y)\right|-\left\|x-y\right\|\right| \leq C$. + +L'objectif est de montrer qu'il existe $h\in\R^+$ et un unique $u\in\mc{O}(E)$ tels que + + $\forall x\in E,\,\left\|T(x)-u(x)\right\|\leq h$. + - Conclure dans le cas ou $C=0$. + - Prouver l'unicité de $u$. + - Pour tout $x$ de $E$, on pose $u_0(x)=\lim_{n\ra+\i}\dfrac{T(2^nx)}{2^n}$. Montrer que $u_0$ est bien définie, lineaire et conserve la norme. + - Conclure. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 93] +Soient $(E,\left\langle\,\ \right\rangle)$ un espace prehilbertien, $F:E\ra E$ et $G=\dfrac{1}{2}(\op{id}-F)$. + - Montrer que, $F$ est $1$-lipschitzienne pour $\parallel$ si et seulement si + + $\forall x,x'\in E,\,\left\langle G(x')-G(x),x'-x \right\rangle\geq\left\|G(x')-G(x)\right\|^2$. + - On suppose que $F$ est $1$-lipschitzienne pour $\parallel$ et qu'il existe $x_*\in E$ tel que $F(x_*)=x_*$ (autrement dit $x_*$ est un point fixe de $F$). Soit $(x_n)_{n\geq 1}$ la suite définie par $x_1\in E$ et, pour $n\geq 1$, $x_{n+1}=\dfrac{x_n+F(x_n)}{2}$. Montrer que, pour tout $n\geq 1$, $\left\|F(x_n)-x_n\right\|\leq\dfrac{2\left\|x_1-x_*\right\|}{ \sqrt{n}}$. + - En déduire que, si $E$ est un espace euclidien, $(x_n)_{n\geq 1}$ converge vers un point fixe de $F$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 94] +Soient $n\geq 2$ et $I_n(\R)=\{A\in\M_n(\R)\,;\,\exists\lambda\in \op{Sp}(A),\,\,\op{Im}(A)\subset E_{\lambda}(A)\}$, ou $E_{\lambda}(A)$ est le sous-espace propre de $A$ associe à la valeur propre $\lambda$. + - Montrer que $I_n(\R)$ est stable par similitude. + - Soient $A,B\in I_n(\R)$.Montrer que $A$ et $B$ sont semblables si et seulement si $\op{rg}A=\op{rg}B$ et $\op{tr}(A)=\op{tr}(B)$. + - On note $I_n^*(\R)=\{A\in\M_n(\R)\;;\;\exists\lambda\in \op{Sp}(A),\;\op{Im}(A)=E_{\lambda}(A)\}$. Étudier la connexite par arcs de $I_n(\R)$ et de $I_n^*(\R)$. + - Déterminer les classes de similitude incluses dans $I_2(\R)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 95] +Soit $G$ un sous-groupe compact de $\op{GL}_n(\R)$. + - Montrer qu'il existe une norme stricte sur $\R^n$ pour laquelle les éléments de $G$ sont des isométries. + - On suppose que les éléments de $G$ stabilisent un convexe compact non vide de $\R^n$ note $K$. Montrer que les éléments de $G$ ont un point commun dans $K$. + - Montrer qu'il existe un produit scalaire sur $\R^n$ pour lequel les éléments de $G$ sont des isométries. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 96] +Soit $n\in\N^*$. Soit $G$ un sous-groupe compact de $\op{GL}_n(\R)$. Pour tous $g\in G$ et $A\in\M_n(\R)$, on pose $g\cdot A=gAg^T$. + - Donner un exemple de produit scalaire sur $\M_n(\R)$ et la norme $N_0$ euclidienne associée. + - Soit $N:A\mapsto\sup\limits_{g\in G}N_0(g\cdot A)$. Montrer que $N$ est une norme sur $\mc{S}_n(\R)$. + - Soit $K=\{gg^T,g\in G\}$. Montrer qu'il existe un compact convexe $C$ vérifiant : $K\subseteq C$, $\{g\cdot A,\;(g,A)\in G\times C\}\subseteq C$ et $C\subseteq\mc{S}_n^{++}(\R)$. + - Montrer qu'il existe un produit scalaire $G$ invariant pour $\cdot$. + - La borne supérieure $\sup\limits_{A\in C}\sup\limits_{B\in C}\|A-B\|$ est-elle atteinte? Si oui, est-elle atteinte en un unique $A_0$? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 97] +Déterminer les valeurs d'adherence des suites $(\cos n)$ et $(\cos^nn)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 98] +[PSLR] Soit $S$ une partie de $\N^*$ infinie et stable par produit. On range les éléments de $S$ en une suite strictement croissante $(s_n)_{n\geq 1}$. Montrer que la suite $\left(\dfrac{s_{n+1}}{s_n}\right)_{n\geq 1}$ admet une limite dans $[1,+\i[$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 99] +Soit $(z_n)_{n\geq 0}$ une suite complexe telle que $\forall n\in\N,z_{n+1}=z_ne^{-i\op{Im}(z_n)}$. Pour quelles valeurs de $z_0$ cette suite est-elle convergente? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 100] +Trouver un équivalent de $S_n=\sum\limits_{k=1}^{+\i}\dfrac{k^n}{2^k}$ quand $n\ra+\i$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 101] +On fixe un entiers $n\geq 2$ et $(t_i)_{i\in\Z/n\Z}$ une famille d'éléments de $]0,1[$. Soit pour $i\in\Z/n\Z$, $(x_k^i)_{k\geq 0}$ une suite réelle. On suppose que, pour tout $i\in\Z/n\Z$ et tout $k\in\N$, $x_{k+1}^i=(1-t_i)x_k^i+t_ix_k^{i+1}$. Montrer que les $n$ suites $(x_k^i)_{k\geq 0}$ pour $i\in\Z/n\Z$ convergent vers une même limite. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 102] +Soient $m\in\N^*$, $z_1,\ldots,z_m\in\mathbb{U}$ distincts et $a_1,\ldots,a_m\in\C$. On suppose que $\sum\limits_{k=1}^ma_kz_k^n\underset{n\ra+\i}{ \longrightarrow}\;0$. Montrer que $a_1=\cdots=a_m=0$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 103] +Soit $(a_n)_{n\geq 0}\in\C^{\N}$ bornée telle que $\forall h\in\N^*$, $\frac{1}{n}\sum_{k=1}^na_ka_{k+h} \underset{n\ra+\i}{\longrightarrow}0$. Montrer que $\frac{1}{n}\sum_{k=1}^na_k\underset{n\ra+\i}{ \longrightarrow}0$. +#+end_exercice + + +# ID:7657 +#+begin_exercice [ENS MP 2024 # 104] +Pour $x_0\gt 0$, on définit par récurrence $x_{n+1}=x_n+\int_{x_n}^{+\i}e^{-t^2}\dt$. Étudier la suite $(x_n)_{n\geq 0}$. Donner un équivalent de $x_n$ puis un développement asymptotique à deux termes. +#+end_exercice +#+BEGIN_proof +On a $(x_n)$ croissante, donc tend vers $+\i$. Par ailleurs $\int_{x_n}^{+\i}e^{-t^2}\dt = \left[\frac{e^{-t^2}}{2t}\right] + \int_{x_n}^{+\i} \frac{e^{-t^2}}{2t^2}\dt$. Donc $x_{n+1} = x_n + \frac{e^{-x_n^2}}{2x_n}$, puis $e^{x_{n+1}^2} = e^{x_n^2 + x_n \frac{e^{-x_n^2}}{x_n}} = e^{x_n^2} + x_n e^{-x_n^2} \ra 1$. Donc $e^{x_n^2}\sim n$, c'est-à-dire $x_n \sim \sqrt{\ln n}$. En poussant le terme plus loin, on a un terme, en $\frac{1}{n\sqrt{\ln n}}$ et $\frac{1}{n^2 \sqrt{\ln n}}$ le coup d'après (dont la série converge), alors que $x_{n+1} - x_n$ a un terme suivant en $\frac{e^{-x_n^2}}{x_n^3} = \frac{1}{n (\ln n)^{3/2}}$ qui domine, et dont la série converge. La conclusion est que l'on peut s'écrire $u_n + C + o(1)$. + +Si on considère $x_{n+1} - (u_{n+1}) = (x_n - u_n) - \frac{1}{n\sqrt{\ln n}} + \frac{e^{-x_n^2}}{x_n}$. Si $x_n$ prend un $+c$, on devient $\frac{e^{-x_n^2} e^{-x_n c}}{x_n + c} = \frac{1}{n (\sqrt{\ln n} + c)} e^{\pm \sqrt{\ln n}}$, cela permet de justifier que la constante est nulle. + +Puis, sommation des équivalents des restes. +#+END_proof + + +#+begin_exercice [ENS MP 2024 # 105] +Soit $\alpha\in\R^{+*}$. + - Montrer qu'il existe une unique suite $(n_i)_{i\geq 1}\in(\N^*)^{\N^*}$ telle que, pour tout $i\in\N^*$, $n_{i+1}\geq{n_i}^2$ et que $\alpha=\sum_{i=1}^{+\i}\ln\bigg(1+\frac{1}{n_i}\bigg)$. + - Generaliser ce résultat. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 106] +Soit $(a_n)_{n\in\N}$ une suite réelle positive. + +On note, pour $\alpha\geq 0$, $\mc{R}_{\alpha}=\left\{(u_n)_{n\in\N}\in[0,1]^{\N},\ \sum_{n\in\N}u_na_n\leq\alpha\right\}$. + +Soit $(b_n)_{n\geq 1}$ une suite réelle positive sommable. Pour tout $\alpha\gt 0$, construire une suite $(v_n)_{n\in\N}\in\mc{R}_{\alpha}$ telle que $\sum_{n\in\N}v_nb_n=\max_{(u_n)\in\mc{R}_{\alpha}} \left\{\sum_{n\in\N}u_nb_n\right\}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 107] +Soient $p\in]1,+\i[$ et $q\in\R$ tels que $\frac{1}{p}+\frac{1}{q}=1$. + - Soient $(a_n)_{n\geq 0}$ et $(b_n)_{n\geq 0}$ des suites d'éléments de $\R^+$ telles que $\sum{a_n}^p$ et $\sum{b_n}^q$ convergent. Montrer que $\sum a_nb_n$ converge. + - Soit $(a_n)_{n\geq 0}$ une suite de réels positifs telle que $\sum a_n$ converge et $\alpha\in\R^{+*}$. Pour $n\in\N$, soit $R_n=\sum_{k=n}^{+\i}a_k$. Déterminer la nature de $\sum{\frac{a_n}{{R_n}^{\alpha}}}$. + - Soit $(a_n)_{n\geq 0}$ une suite d'éléments de $\R^+$. On suppose que, pour toute suite $(b_n)_{n\geq 0}$ d'ellements de $\R^+$ telle que $\sum{b_n}^q$ converge, $\sum a_nb_n$ converge. Montrer que $\sum{a_n}^p$ converge. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 108] +On admet l'irrationalite de $\pi$. Pour $n\in\N$, on pose $u_n=\frac{(-1)^n}{n^{\alpha}+\cos(n)}$. + - Montrer que $\sum u_n$ converge si $\alpha\gt \frac{1}{2}$. + - Donner une condition nécessaire et suffisante sur $\alpha$ pour que $\sum u_n$ converge. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 109] + - Montrer que, pour $n\in\N^*$ et $(a_1,\ldots,a_n)\in{\R^+}^n$, $\sqrt[n]{\prod_{i=1}^na_i}\leq\frac{1}{n}\sum_{i=1}^na_i$. + - Soit $(a_n)_{n\geq 1}$ une suite d'éléments de ${\R^+}^*$. Montrer que $\sum_{n=1}^{+\i}\sqrt[n]{\prod_{i=1}^na_i}\lt e\sum_{n=1}^{+ \i}a_n$. + - Montrer que la constante $e$ est optimale. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 110] +Soit $(a_n)_{n\geq 0}\in\C^{\N}$. On pose, pour $n\in\N$, $H_{0,n}=a_0+\cdots+a_n$ et, pour $\alpha\in\N^*$, + + $H_{\alpha,n}=\frac{1}{n+1}\sum_{k=0}^nH_{\alpha-1,k}$. Si $(H_{\alpha,n})_{n\geq 0}$ converge, on dit que $(a_n)$ est $H_{\alpha}$-sommable. + - Soit $\alpha\in\N$. Si $(a_n)_{n\geq 0}$ est $H_{\alpha}$-sommable, montrer qu'elle est $H_{\alpha+1}$ sommable. + - On suppose $(H_{0,n})_{n\geq 0}$ periodique. Montrer que $(a_n)_{n\geq 0}$ est $H_{\alpha}$-sommable pour tout $\alpha\in\N^*$. + - Soit $(a_n)_{n\geq 0}$ une suite de réels positifs. On suppose que $\sum a_n$ diverge. Montrer que, pour tout $\alpha\in\N$, $(a_n)_{n\geq 0}$ n'est pas $H_{\alpha}$-sommable. + - Soit $\alpha\in\N$. Si $(a_n)_{n\geq 0}$ est $H_{\alpha}$-sommable, montrer que $a_n=o(n^{\alpha})$. + - Donner un exemple de suite $(a_n)_{n\geq 0}$ qui n'est pas $H_{\alpha}$-sommable mais qui est $H_{\alpha+1}$-sommable. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 111] + - Montrer que : $\cos(k\theta),\frac{\sin((k+1)\theta)}{\sin\theta},\frac{\cos((k+1/2)\theta)}{ \cos(\theta/2)}$ et $\frac{\sin((k+1/2)\theta)}{\sin(\theta/2)}$ + +sont des polynômes en $\cos\theta$. + - Soient $a_0,\ldots,a_n,b_1,\ldots,b_n$ des réels. + +On suppose que : $\forall\theta\in\R,\,g(\theta)=a_0+\sum_{k=1}^n(a_k\cos(k\theta) +b_k\sin(k\theta))\geq 0$. Montrer qu'il existe un polynôme complexe $P$ tel que : $\forall\theta\in\R,\,g(\theta)=|P(e^{i\theta})|^2$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 112] + - Soit $(u_n)$ une suite réelle telle que $\forall n,p,u_{n+p}\leq u_n+u_p+C$, ou $C$ est une constante réelle. Montrer que $\left(\frac{u_n}{n}\right)$ converge ou tend vers $-\i$. + - Soit $f\in\mc C(\R,\R)$ continue et croissante, telle que $\forall x\in\R,\,f(x+1)=f(x)+1$. On note $f^n$ la composée iterée de $f$ ( $n$ fois). + +Montrer que, pour tout $x\in\R$, $\left(\frac{f^n(x)-x}{n}\right)_{n\geq 1}$ converge vers une limite qui ne depend pas de $x$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 113] +Soient $(a_1,\ldots,a_n)$ et $(b_1,\ldots,b_n)$ dans $(\R^{+*})^n$. + +On note $a\geq b$ si : $\forall k\in\db{1,n-1}$, $\sum_{i=1}^ka_i\geq\sum_{i=1}^kb_i$ et $\sum_{i=1}^na_i=\sum_{i=1}^nb_i$. Montrer que $a\geq b$ si + +et seulement si, pour tout $(x_1,\ldots,x_n)\in(\R^{+*})^n$, $\sum_{i=1}^nx_i^{a_{\sigma(i)}}\geq\sum_{\sigma\in\mc{S}_n} \prod_{i=1}^nx_i^{b_{\sigma(i)}}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 114] + - Soit $f:[0,2\pi]\ra\R$ une fonction continue. Montrer qu'il existe $x\in[0,2\pi]$ tel que $f(x)\geq\dfrac{1}{2\pi}\int_0^{2\pi}f(t)dt$. + - Soient $z_1,\ldots,z_n\in\C$. + +Montrer qu'il existe une partie $I$ de $\db{1,n}$ telle que $\left|\sum_{j\in I}z_j\right|\geq\dfrac{1}{\pi}\sum_{j=1}^n|z_j|$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 115] +Soient $a\lt b$. Une dissection du segment $[a,b]$ est une suite finie $(t - {0\leq k\leq n}$ strictement croissante telle que $t_0=a$ et $t_n=b$. Pour $f:[a,b]\ra\R$, on définit la variation de $f$ sur $[a,b]$ par $V(f,[a,b])=\sup_{t\,\text{\tiny{\rm dissection}}\atop\text{\tiny{\rm def}}\,[a,b ]}\sum_{i=0}^{n-1}|f(t_{i+1})-f(t_i)|$. + - Calculer $V(f,[a,b])$ dans le cas ou $f$ est de classe $\mc C^1$ sur $[a,b]$. + - Soit $f:[0,1]\ra\R$. Montrer que $V(f,[0,1])\lt +\i$ si et seulement s'il existe $g$ et $h$ croissantes telles que $f=g-h$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 116] +Soit $f\colon\R\ra\R$ une fonction dérivable. On pose $S_-=\{x\in\R,\ f'(x)\lt 0\}$. + - L'ensemble $S_-$ peut-il être fini non vide? + - On suppose que, pour tout $\eps\gt 0$, il existe une suite $(I_n)_{n\in\N}$ d'intervalles ouverts tels que $S_-\subset\bigcup_{n\in\N}I_n$ et $\sum_{n=0}^{+\i}\ell(I_n)\leq\eps$ (ou $\ell(I_n)$ designe la longueur de $I_n$). Montrer que $f$ est croissante (donc $S_-=\emptyset$). +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 117] +Soient $E$ un espace vectoriel, $C\subset E$ un ensemble convexe non vide, $a\lt b$ deux réels, et $F$ l'ensemble des fonctions $f:C\ra[a,b]$ convexes. Soit $x,y\in C$ fixes. Déterminer $\sup_{f\in F}\left(f(y)-f(x)\right)$. Déterminer les cas ou la borne supérieure est atteinte. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 118] +Pour toute fonction $f\colon\R\ra\R\cup\{+\i\}$, on note $\mathrm{dom}(f)=\{x\in\R,\ f(x)\neq+\i\}$. Si $\mathrm{dom}(f)\neq\emptyset$, on définit $f^*\colon\R\ra\R\cup\{+\i\}$ par $f^*(y)=\sup_{x\in\R}\left\{xy-f(x)\right\},$ pour tout $y\in\R$. + - Soit $f\colon\R\ra\R\cup\{+\i\}$ telle que $\mathrm{dom}(f)\neq\emptyset$. Montrer que $\mathrm{dom}(f^*)$ est un ensemble convexe et que $f^*$ est convexe sur $\mathrm{dom}(f^*)$. + - Soit $g\colon\R\ra\R$ une fonction convexe dérivable. + +On pose $E=\left\{(y,a)\in\R^2\,;\ \forall x\in\R,\ xy-a\leq g(x) \right\}$. + - Montrer que, pour tout $x\in\R$, $g(x)=\sup_{(y,a)\in E}\left(xy-a\right)$. + - En déduire que $(g^*)^*=g$. + - Etendre au cas ou $g$ n'est pas dérivable. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 119] +Soient $I$ un intervalle réel contenant $0$ et $f:I\ra\R$ de classe $\mc C^1$. + +On suppose qu'il existe $A,C\gt 0$ telles que $\forall x\in I,\ |f'(x)|\leq C|f(x)|+A$. + +Montrer que $\forall x\in I,\ |f(x)|\leq|f(0)|e^{C|x|}+\dfrac{A}{C}\left(e^{C|x|}-1 \right)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 120] +Soit $f\colon\R^+\ra\R$ uniformément continue et dont une primitive est bornée. On suppose que, pour tout $x\gt 0$, $|f(x)|\leq\dfrac{2}{x^2}\int_0^x(x-y)\,|f(y)|dy$. Montrer que $f(x)\underset{x\ra+\i}{\longrightarrow}0$. Quelles generalisations peut-on étudier? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 121] +On note $[a,b]$ un segment de $\R$. Une application $\delta:[a,b]\ra{\R^+}^*$ est appelée une jauge. Soit $D=((a - {0\leq i\leq n},(x - {0\leq i\leq n-1})$ une subdivision pointée de $[a,b]$, c'est-a-dire $a_0=a\lt a_1\lt \cdots\lt a_n=b$ et $\forall i\in\db{0,n-1},\ x_i\in[a_i,a_{i+1}]$. On dit que $D$ est $\delta$-fine lorsque pour tout $i$, $|a_{i+1}-a_i|\leq\delta(x_i)$. + - Soit $f\in\mc C^0([a,b],\R)$. Montrer que, pour tout $\eps\gt 0$, il existe une jauge $\delta$ telles que $\forall x,y\in[a,b],y\in[x-\delta(x),x+\delta(x)]\Rightarrow|f(x)-f(y)|\leq\eps$. + - Si $\delta$ est une jauge, montrer qu'il existe une subdivision pointée $\delta$-fine. + - Redémontrer le theoreme de Heine pour $f$ continue. + - Soient $f:[a,b]\ra\R$ une fonction continue par morceaux et $I$ un réel. On dit que $f$ est HK-intégrable, d'intégrale $I$ si, pour tout $\eps\gt 0$, il existe une jauge $\delta$ telle que, pour toute subdivision pointée $((a - {0\leq i\leq n},(x - {0\leq i\leq n-1})$ $\delta$-fine, on a $\left|\sum_{i=0}^{n-1}(a_{i+1}-a_i)f(x_i)-I\right|\leq\eps$. + +Montrer que $I$ est unique. On la note $\int_{HK}f$. + - Montrer que, si $f$ est dérivable, $f'$ est HK-intégrable et $\int_{HK}f'=f(b)-f(a)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 122] +Soient $P\in\C[X]$ non constant tel que $P(0)\neq 0$, $r\in\R^{+*}$, $z_1,\ldots,z_p$ les racines de module strictement inférieur à $r$ de $P$ comptées avec multiplicité. Montrer que $\dfrac{1}{2\pi}\int_{-\pi}^{\pi}\ln(|P(re^{it})|)dt=\ln(|P(0))|+\sum_ {k=1}^p\ln\left(\dfrac{r}{|z_k|}\right)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 123] +Soit $E=\mc C^0([0,1],\R)$. On dit qu'un endomorphisme $u$ de $E$ est positif si, pour tout $f\in E$, $f\geq 0$ implique $u(f)\geq 0$. On pose, pour $i\in\N$, $e_i:x\in[0,1]\mapsto x^i$. + - Soit $u$ un endomorphisme positif de $E$. Montrer que pour tout $f\in E$, $|u(f)|\leq u(|f|)$. + - Soit $f\in E$. Montrer que, pour tout $\eps\gt 0$, il existe $\delta\gt 0$ tel que : $\forall x,y\in[0,1]$, $|f(x)-f(y)|\leq\eps+\dfrac{2\,\|f\|_{\i}}{\delta^2}\,(x-y)^2$. + - Soit $(T_n)_{n\geq 0}$ une suite d'endomorphismes positifs de $E$. On suppose que, pour $i\in\{0,1,2\}$, la suite $(T_n(e_i))$ converge uniformément vers $e_i$ sur $[0,1]$. Montrer que, pour tout $f\in E$, la suite $(T_n(f))$ converge uniformément vers $f$ sur $[0,1]$. + - Démontrer le theoreme de Weierstrass. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 124] +Soit $s\gt 1$. On dit que $f\in\mc C^{\i}(\R,\R)$ est $s$-Gevrey s'il existe $R,C\gt 0$ tels que : $\forall k\in\N$, $\forall x\in\R$, $\left|f^{(k)}(x)\right|\leq CR^k(k!)^s$. + - Soit $f:x\in\R\mapsto\sum_{n=0}^{+\i}e^{-n+in^sx}$. Justifier que $f$ est bien définie et $s$-Gevrey. + - Soit $f:x\in\R\mapsto\mathbf{1}_{\R^+}(x)\,e^{-1/x}$. Montrer que $f$ est de classe $\mc C^{\i}$ et 2-Gevrey. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 125] +Pour $x\gt 0$ et $\alpha,\beta\in\C$, on pose : $F_{\alpha,\beta}(x)=\int_0^{+\i}e^{-xt}t^{\alpha}(1+t)^{\beta}\dt$. + - Pour quels $(\alpha,\beta)$ l'intégrale $F_{\alpha,\beta}(x)$ converge-t-elle absolument? + - Pour un tel couple $(\alpha,\beta)$, étudier la régularite de $F_{\alpha,\beta}$. + - On pose $f:x\in\R^{+*}\mapsto\int_x^{+\i}e^{-t^2}\dt$ et $g:x\in\,]0,1[\mapsto\int_0^x\frac{\dt}{\ln t}$. Exprimer $f$ et $g$ en fonction des $F_{\alpha,\beta}$. + - Déterminer un développement asymptotique de $F_{\alpha,\beta}$ en $+\i$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 126] +Soit $\Gamma:x\mapsto\int_0^{+\i}t^{x-1}e^{-t}\dt$. + - Soit $n\in\N$. Montrer que $\Gamma(n+1/2)=\frac{(2n)!}{2^{2n}n!}\,\Gamma(1/2)$. + - Montrer que, pour $x,y\gt 0$ et $\lambda\in[0,1]$, $\Gamma\left((1-\lambda)x+\lambda y\right)\leq\Gamma(x)^{1-\lambda}\Gamma( y)^{\lambda}$. + - En déduire : + + $\forall n\in\N^*$, $\Gamma(n+1/2)^2\leq\Gamma(n)\,\Gamma(n+1)$; $\forall n\in\N$, $\Gamma(n+1)^2\leq\Gamma(n+1/2)\Gamma(n+3/2)$. + - Montrer que $\Gamma(1/2)=\sqrt{\pi}$. + - On note, pour $n\in\N^*$, $\Gamma_n(x)=\int_0^nt^{x-1}\left(1-\frac{t}{n}\right)^n dt$. Démontrer que la suite $(\Gamma_n)_{n\geq 1}$ converge simplement vers $\Gamma$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 127] +Soient $x,y\in\mc C^{\i}(\R,\R)$ vérifiant $x'(t)=\sin(y(t))$ et $y'(t)=\cos(x(t))$. + - Montrer que $f:t\mapsto\sin(x(t))+\cos(y(t))$ est constante. + - Soit $\phi:t\mapsto\frac{1}{2}\left(x(t)+y(t)-\frac{\pi}{2}\right)$. Montrer que les points $(\sin(\phi(t)),\phi'(t))$ sont situes sur un même cercle dont on déterminera le rayon. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 128] +Soient $A\in\M_n(\R)$, $B\in\M_{n,1}(\R)$, $E$ l'espace des applications continues de $[0,1]$ dans $\R$, $x\in\R^n$. Pour $u\in E$, soit $X_u$ l'unique application de classe $\mc C^1$ de $[0,1]$ dans $\R$ telle que $X_u(0)=x$ et $\forall t\in[0,1],X_u'(t)=AX_u(t)+Bu(t)$. + +Montrer que $\{X_u(1)\;;\;u\in E\}=\R^n$ si et seulement si la matrice $(A|AB|\ldots|AB^{n-1})$ de $\M_{n,n^2}(\R)$ est de rang $n$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 129] + - Que dire du spectre complexe d'une matrice symétrique réelle? d'une matrice antisymétrique réelle? + - Soient $A\in\mc C^1(\R,\M_n(\R))$ et $B\in\mc C^0(\R,\M_n(\R))$ vérifiant : $A'=AB-BA$. On suppose que : $\forall t\in\R,A(t)\in\mc{S}_n(\R)$ et $B(t)\in\mc{A}_n(\R)$. Montrer qu'il existe $P\in\mc C^1(\R,\M_n(\R))$ à valeurs dans $\mc{O}_n(\R)$ telle que : $\forall t\in\R,A(t)=P(t)^{-1}A(0)P(t)$. + - On se place dans le cas $n=2$ avec : $A=\begin{pmatrix}b_1&a_1\\ a_1&b_2\end{pmatrix},B=\begin{pmatrix}0&a_1\\ -a_1&0\end{pmatrix}$ et $(S):a_1'=a_1(b_2-b_1)$, $b_1'=2a_1'$, $b_2'=-2a_1'$, $b_1(0)+b_2(0)=0$ et $a_1(0),b_1(0)\geq 0$. + - Calculer $AB-BA$. + - Trouver une solution particuliere de $(S)$ au voisinage de $0$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 130] +Pour $k\geq 3$, on note $G_k:z\mapsto\sum_{(n,m)\in\Z^2\setminus\{(0,0)\}}\frac{1}{(m+nz)^{ k}}$. - Montrer que $G_k(z)$ est bien défini pour tout complexe $z$ tel que $\op{Im}z\gt 0$ et que la fonction $(x,y)\mapsto G_k(x+iy)$ est de classe $\mc C^{\i}$ sur $\R\times\R^{+*}$. + - Montrre que $G_k(iy)$ admet une limite quand $y\ra+\i$. + - Étudier l'existence des limites suivantes : + +$$\lim_{N\ra\i}\sum_{n=-N}^N\sum_{m\in\Z}\frac{1}{(m+in)^2}\ \text{et}\ \lim_{M\ra\i}\sum_{m=-M}^M\sum_{n\in\Z}\frac{1}{(m+in)^2},\ \text{oi dans les deux cas la somme}$$ + +exclut $(n,m)=(0,0)$. Ces limites sont-elles egales? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 131] +Soit $(x,y,z)\in(\R^+)^3$. + +Démontrer que $(x+y+z)^3+9xyz\geq 4(x+y+z)(xy+yz+zx)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 132] +Soit $F\colon\R^2\ra\R,\ (t,x)\mapsto F(t,x)$ continue et decroissante par rapport à $x$. + +Soient $u$ et $v$ appartenant à $\mc C^2(\R^+\times\R)$ 1-periodiques par rapport à $x$. + - On suppose que $\frac{\partial u}{\partial t}+F\left(\frac{\partial u}{\partial x},\frac{ \partial^2u}{\partial x^2}\right)\leq 0\leq\frac{\partial v}{ \partial t}+F\left(\frac{\partial v}{\partial x},\frac{\partial^2v}{\partial x ^2}\right)$. + +Démontrer que $\sup_{\R^+\times\R}(u-v)=\sup_{\{0\}\times\R}(u-v)$. + - On suppose que $\frac{\partial u}{\partial t}+F\left(\frac{\partial u}{\partial x},\frac{ \partial^2u}{\partial x^2}\right)=0$. Montrer que $u$ est uniformément continue sur $\R^+\times\R$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 133] +Soient $a\gt 0$, $n\geq 1$ et $x_1,\ldots,x_n\gt 0$. Calculer $\inf_{\begin{subarray}{c}y_1,\ldots,y_n\gt 0\\ y_1+\cdots+y_n\leq 1\end{subarray}}\sum_{i=1}^n\frac{x_i}{y_i^a}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 134] +Soit $n\in\N^*$. On munit $\R^n$ de sa structure euclidienne canonique. + +On considére $n+1$ vecteurs $v_1,\ldots,v_{n+1}$ engendrant positivement $\R^n$, c'est à dire tels que $\left\{\sum_{i=1}^{n+1}\lambda_iv_i,\ (\lambda - {1\leq i \leq n+1}\in(\R^+)^{n+1}\right\}=\R^n$. + +Soit $f\colon\R\ra\R^+$ une fonction continue croissante telle que $f(x)\underset{x\ra+\i}{\longrightarrow}+\i$. Pour $x\in\R^n$, + +on définit $g(x)=\sum_{i=1}^{n+1}f(\langle v_i,x\rangle)$. + - Montrer qu'il existe bien $n+1$ vecteurs $v_1,\ldots,v_{n+1}$ engendrant positivement $\R^n$. + - Montrer que $g$ atteint son minimum sur $\R^n$. + - On suppose que $f$ est intégrable en $-\i$. + +Montrer qu'il existe $x\in\R^n$ tel que $\sum_{i=1}^{n+1}f(\langle v_i,x\rangle)v_i=0$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 135] +On munit $\M_n(\R)$ de sa structure euclidienne canonique. + - Montrer que $\op{GL}_n(\R)$ est ouvert. + - Pour $A\in\M_n(\R)$, que vaut $d(A,\op{GL}_n(\R))$? + - On note $S=\M_n(\R)\setminus\op{GL}_n(\R)$. Montrer que, pour tout $A\in\op{GL}_n(\R)$, il existe $M_0\in S$ telle que $d(A,S)=\|A-M_0\|$. - Rappeler le résultat sur les extrema sous contrainte. Que peut-on en déduire sur la matrice $M_0$ définie ci-dessus? +#+end_exercice + + +** Geometrie + +#+begin_exercice [ENS MP 2024 # 136] + - Montrer que, si $n\geq 2$, le groupe des isométries vectorielles de $\R^2$ préservant les points dont les affixes sont les racines $n$-iemes de l'unite est un groupe d'ordre $2n$ que l'on note $\mc{D}_{2n}$. + - Soient $p$ un nombre premier, $G$ un groupe fini d'ordre $2p$. Montrer que $G$ est isomorphe à $\Z/2p\Z$ ou à $\mc{D}_{2p}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 137] + - On note $G$ le groupe (pour la composition) des deplacements du plan, i.e. des applications de $\C$ dans $\C$ de la forme $z\mapsto az+b$ avec $a\in\mathbb{U}$ et $b\in\C$. Montrer que, si $H$ est un sous-groupe de $G$, $H$ est discret si et seulement si l'orbite de tout $z\in\C$ sous l'action de $H$ n'a pas de point d'accumulation. + - Le résultat subsiste-t-il si on remplace $G$ par le groupes des similitudes directes du plan, i.e. des applications de $\C$ dans $\C$ de la forme $z\mapsto az+b$ avec $a\in\C^*$ et $b\in\C$? +#+end_exercice + + +** Probabilités + +#+begin_exercice [ENS MP 2024 # 138] +Soit $E$ un espace vectoriel norme et soit $(u_1,\ldots,u_n)\in E^n$. On considére des variables aléatoires $\eps_1,\ldots,\eps_n$ i.i.d telles que $\mathbf{P}(\eps_i=1)=\mathbf{P}(\eps_i=-1)=\frac{1}{2}$. Si $(v_1,\ldots,v_n)\in E^n$, on pose $N(v_1,\ldots,v_n)=\mathbf{E}\left(\left\|\sum_{k=1}^n \eps_kv_k\right\|\right)$. Démontrer que, pour tout $(\lambda_1,\ldots,\lambda_n)\in[-1,1]^n,\;N(\lambda_1u_1,\ldots, \lambda_nu_n)\leq N(u_1,\ldots,u_n)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 139] +On considére une piece equilibrée et $\eps_n$ la valeur du $n$-ieme lancer que l'on considére à valeurs dans $\{-1,1\}$. Soient $X_n=\sum_{k=1}^n\eps_k$ et $\tau=\min\{n\in\N^*,\;X_n=0\}$. Déterminer $\mathbf{P}(\tau=n)$ ainsi qu'un équivalent de cette quantite lorsque $n$ tend vers $+\i$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 140] +Soient $X$ et $Y$ deux variables aléatoires indépendantes, $X$ suivant la loi de Poisson de paramêtre $\lambda\gt 0$, et $Y$ la loi geometrique de paramêtre $p\in]0,1]$. + - Montrer que $\mathbf{P}(X=Y)=\sum_{k=0}^{+\i}\mathbf{P}(X=k)\mathbf{P}(Y=k)$. On pose $A=\begin{pmatrix}X&X+Y\\ 0&Y\end{pmatrix}$. + - Calculer $\mathbf{E}(\op{rg}(A))$. + - Calculer $\mathbf{P}(A\in\op{GL}_2(\R))$ puis $\mathbf{P}(A\in\op{GL}_2(\Z))$. + - Déterminer la probabilité pour que $A$ soit diagonalisable sur $\R$. + - On pose $B=\begin{pmatrix}X&X+Y\\ X-Y&Y\end{pmatrix}$. Calculer $\mathbf{P}(B\in\mc{O}_2(\R))$. + - Soient $Z$ une variable aléatoire réelle et $C=\begin{pmatrix}X&X+Y\\ Z&Y\end{pmatrix}$. Calculer $\mathbf{P}(C\in\mc{O}_2(\R))$. - Soit $M$ une matrice aléatoire dans ${\cal M}_n(\R)$ dont la famille des coefficients est i.i.d., chaque coefficient suivant la loi uniforme sur $\{0,-1,1\}$. Déterminer $P(M\in{\cal O}_n(\R))$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 141] +On note $E=\db{1,n}$ et $\Delta$ la différence symétrique. Soit $p\in[0,1]$ et $X$ et $Y$ deux variables aléatoires i.i.d de $\Omega$ dans ${\cal P}(E)$ telles que, pour tout $i\in E,\ {\bf P}(i\in X)=p$. + - Calculer ${\bf E}({\rm card}(X\Delta Y))$. + - On note $D(n)$ le cardinal maximal d'une partie ${\cal A}$ de ${\cal P}(E)$ telle que, pour toutes parties $A$ et $B$ distinctes de ${\cal A}$, $|A\Delta B|\geq n/3$. Calculer $D(n)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 142] +Soient $(X_n)_{n\in\Z}$ une suite de variables aléatoires indépendantes suivant la loi uniforme sur $\{-1,1\}$. Si $N$ est une variable aléatoire à valeurs dans $\Z$, on pose $X_{N+n}(\omega)=X_{N(\omega)+n}(\omega)$. + - Existe-t-il $N$ tel que ${\bf P}(X_N=1)=1$ et, pour tout $n\in\N^*$, ${\bf P}(X_{N+n}=1)=1/2$? + - Existe-t-il $N$ tel que ${\bf P}(X_N=1)=1$ et, pour tout $n\in\Z^*$, ${\bf P}(X_{N+n}=1)=1/2$? +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 143] +Soient $E=\db{1,n}$ et $p\in]0,1[$. Soit $X$ une variable aléatoire à valeurs dans ${\cal P}(E)$ telle que $\forall i\in E,\ {\bf P}(i\in X)=p$ et, pour $i\neq j\in E$, $(i\in X)$ et $(j\in X)$ sont indépendants. + +Pour $Y$ variable aléatoire de même loi que $X$ et indépendante de $X$, calculer ${\bf E}(|X\Delta Y|)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 144] +Soient $G$ un groupe fini de cardinal $N$, et $A$ une partie de $G$ aléatoire, ou l'on prend chaque élément de $G$ indépendamment avec probabilité $p\gt 0$. + +On note ${\rm AA}=\{xy,\ (x,y)\in A^2\}$. + - Montr per que ${\bf P}(1\in{\rm AA})$ tend vers $1$ quand $N$ tend vers l'infini. + - Montr per que ${\bf P}({\rm AA}=G)$ tend vers $1$ quand $N$ tend vers l'infini. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 145] + - Soit $X$ une variable aléatoire réelle positive $L^2$. + +Montr per que, pour $\lambda\in]0,1[$, ${\bf P}(X\geq\lambda{\bf E}(X))\geq(1-\lambda)^2\frac{{\bf E}(X)^{ 2}}{{\bf E}(X^2)}$. + - Soit $(u_n)$ une suite de variables aléatoires positives indépendantes. Montr per que la série $\sum u_n$ converge presque surement si et seulement si $\sum_{n=0}^{+\i}{\bf E}(\min(u_n,1))\lt +\i$. + - Soit $\alpha\gt 0$. On suppose que ${\bf P}(X_n\geq r)\underset{r\ra+\i}{\sim}r^{-\alpha}$. Trouver une condition nécessaire et suffisante sur $(x_n)_{n\in\N}\in(\R^+)^{\N}$ pour que $\sum x_nX_n$ converge presque surement. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 146] +Soient $\lambda\gt 0$ et $N_{\lambda}$ une variable de Poisson de paramêtre $\lambda$. + +Pour $f\colon\N\ra\R$ bornée, on pose $Tf:n\in\N\mapsto\lambda f(n+1)-nf(n)$. + - Montr per que $Tf(N_{\lambda})$ est d'esperance finie, nulle. + - Pour $\mu$ et $\nu$ deux distributions de probabilités sur $\N$, et $X$ et $Y$ variables aléatoires à valeurs dans $\N$ de lois respectivement données par $\mu$ et $\nu$, on note + + $d(\mu,\nu)=d(X,Y)=\frac{1}{2}\sup_{\|g\|_{\i}\leq 1}{\bf E}(g(X)-g(Y))$. + +Montr per l'existence de $C_{\lambda}\gt 0$ tel que, pour toute variable aléatoire à valeurs dans $\N$, $d(N,N_{\lambda})\leq C_{\lambda}\sup_{\|f\|_{\i}\leq 1}{\bf E}(Tf(N))$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 147] +Soit $X$ une variable aléatoire à valeurs dans $\{x_1,\ldots,x_n\}$. L'entropie de $X$ est définie par ${\cal H}(X)=-\sum_{k=1}^np_i\ln(p_i)$ avec $p_i={\bf P}(X=x_i)$. + - Montrere que ${\cal H}(X)\geq 0$ avec egalite si et seulement si $X$ est constante. + +Soit $(p - {1\leq i\leq n}$ une suite positive telle que $p_1+\cdots+p_n=1$ et $(q_i)$ une autre suite positive de somme $1$. + - Montrere que $\sum_{i=1}^np_i\ln(p_i)\geq\sum_{i=1}^np_i\ln(q_i)$. Expliciter le cas d'egalite. + - Montrere que ${\cal H}(X)\leq\ln(n)$ avec egalite si et seulement si $X$ suit une loi uniforme. + +Soit $(X,Y)$ un couple de variables aléatoires à valeurs dans $\{x_1,\ldots,x_n\}^2$. + +On note $p_{i,j}={\bf P}(X=x_i,Y=x_j)$ pour $1\leq i,j\leq n$. + +L'entropie de $(X,Y)$ est ${\cal H}(X,Y)=-\sum_{i,j=1}^np_{i,j}\ln(p_{i,j})$. + - Montrere que ${\cal H}(X,Y)\leq{\cal H}(X)+{\cal H}(Y)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 148] +Soit $(E,\langle\,\ \rangle)$ un espace euclidien. Soient $v_1,\ldots,v_n\in E$ tels que, pour tout $i\in\db{1,n}$, $\|v_i\|\leq 1$. Soient $\alpha_1,\ldots,\alpha_n\in[-1,1]$ et $w=\sum_{i=1}^n\alpha_iv_i$. Montrere qu'il existe des $\eps_1,\ldots,\eps_n\in\{-1,1\}$ tels que $v=\sum_{i=1}^n\eps_iv_i$ satisfait $\|v-w\|\leq\sqrt{n}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 149] +Soit $(X_n)_{n\geq 1}$ une suite de variables aléatoires i.i.d sur $\Z$ à support fini suivant la loi $\mu$. On pose $\nu(k)=\frac{e^{\lambda k}\mu(k)}{{\bf E}(e^{\lambda X_1})}$ et on considére une suite $(Y_n)_{n\geq 1}$ i.i.d suivant la loi $\nu$. On pose $S_n=\sum_{k=1}^nX_k$ et $T_n=\sum_{k=1}^nY_k$. On prend $\lambda\geq 0,a\in\R,\eps\gt 0,n\geq 1$. + - Montrere que ${\bf P}(na\leq T_n\leq(a+\eps)n)\leq\frac{e^{ \lambda n(a+\eps)}}{({\bf E}(e^{\lambda X}))^n}{\bf P}(S_n\geq na)$. + - On suppose $X\sim-X$ et $\exists k\gt a,\ (a\gt 0),\ \mu(k)\gt 0$. + +Démontrer que $\frac{1}{n}\ln{\bf P}(S_n\geq na)\xrightarrow[n\ra+\i]{}\inf_{s \geq 0}(-sa+\log{\bf E}(e^{sX}))$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 150] +Soient $\sigma\gt 0$, $n\geq 1$ un entier et $X_1,\ldots,X_n$ des variables aléatoires réelles discretes telles que pour tout $1\leq i\leq n$ et $s\gt 0$, ${\bf E}\left(\exp(sX_i)\right)\leq\exp\left(\sigma^2s^2\right)$. Montrere que ${\bf E}\left(\max_{1\leq i\leq n}X_i\right)\leq 2\sigma\sqrt{\ln n}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 151] +Soient $n\geq 1$, $a\gt 0$, et $X_1,\ldots,X_n$ des variables aléatoires discretes, indépendantes, d'esperance nulle, et à valeurs dans $[-a,a]$. + - Montrere que, pour tout $1\leq i\leq n$ et $s\gt 0$, ${\bf E}\left[e^{sX_i}\right]\leq\exp\left(\frac{{\bf V}(X_i)}{a^2} \left(e^{as}-1-as\right)\right)$. - On note $\sigma^2=\dfrac{1}{n}\sum_{i=1}^n\mathbf{V}(X_i)$. Montrer que, pour tout $t\geq 0$, + +$$\mathbf{P}\left(\left|\dfrac{1}{n}\sum_{i=1}^nX_i\right|\geq t\right) \leq 2\exp\left(-\dfrac{nt^2}{2\sigma^2+2at/3}\right).$$ +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 152] +Pour $x\gt 0$, on pose $\Gamma(x)=\int_0^{+\i}t^{x-1}e^{-t}\dt$. On pourra utiliser sans demonstration le fait que $\Gamma(1/2)=\sqrt{\pi}$ et $\Gamma(1)=1$. + - Montrer que, pour tout $k\geq 1$ entier, $\Gamma(k)=(k-1)!$ et $\Gamma(k+1/2)\leq k!$. + - Soient $\sigma\gt 0$ et $X$ une variable aléatoire réelle discrete à valeurs dans un ensemble discret, telle que, pour tout $t\geq 0$, $\mathbf{P}\left(\left|X\right|\gt t\right)\leq 2\exp\left(-\dfrac{t^2}{2 \sigma^2}\right)$. Montrer que, pour tout entier $k\geq 1$, $\mathbf{E}\left(\left|X\right|^k\right)\leq(2\sigma^2)^{k/2}k\Gamma(k /2)$. + - On suppose de plus que $\mathbf{E}\left(X\right)=0$. Montrer que $\forall s\gt 0$, $\mathbf{E}\left[\exp(sX)\right]\leq\exp\left(4\sigma^2s^2\right)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 153] +Soient $n\geq 3$ un entier. Si $\sigma\in\mc{S}_n$, une suite alternante pour $\sigma$ est une suite strictement croissante $(i_1)_{1\leq m}$ d'éléments de $\db{1,n}$ telle que : + + - soit pour tout $k\in\db{2,\ell-1}$, $\sigma(i_k)\gt \max\{\sigma(i_{k-1}),\sigma(i_{k+1})\}$; + + - soit pour tout $k\in\db{2,\ell-1}$, $\sigma(i_k)\lt \max\{\sigma(i_{k-1}),\sigma(i_{k+1})\}$. + +On note $\Delta(\sigma)$ la longueur maximale d'une suite alternante pour $\sigma$ et on considére $\sigma_n$ une variable aléatoire suivant la loi uniforme sur $\mc{S}_n$. Calculer $\mathbf{E}(\Delta(\sigma_n))$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 154] +Soit $(X_k)_{k\geq 1}$ une suite de variables aléatoires discretes réelles i.i.d. Pour $n\geq 1$, on note $M_n=\max\limits_{1\leq k\leq n}X_k$. Soit $\alpha\gt 0$. Montrer que les conditions suivantes sont équivalentes : + +(i) $\exists(a_n)_{n\geq 1}\in(\R^{+*})^{\N^*},\quad \forall x\geq 0,\quad\mathbf{P}\left(\dfrac{M_n}{a_n}\leq x\right)\ \xrightarrow[n\ra\i]{}\exp(-x^{-\alpha})$, + +(ii) $\forall x\gt 0,\quad\dfrac{\mathbf{P}(X_1\gt xt)}{\mathbf{P}(X_1\gt t)} \xrightarrow[t\ra\i]{}x^{-\alpha}\qquad(\text{et }\forall t\gt 0,\mathbf{P}(X_1\gt t)\gt 0)$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 155] +Soit $(X_k)_{k\in\N^*}$ une suite de variables aléatoires indépendantes telle que, pour tout $n\in\N^*$, $X_n\sim\mc{B}\left(1/n\right)$. Pour $n\in\N^*$, on pose $S_n=X_1+\cdots+X_n$. + - Montrer que, pour une indexation de sous-suite $(\phi(n))_{n\geq 1}$ bien choisie, + + $\mathbf{P}\left(\bigcap_{N\geq 1}\bigcup_{k\geq N}\left(\left| \dfrac{S_{\phi(k)}}{H_{\phi(k)}}-1\right|\gt \dfrac{1}{k}\right)\right)=0$. + - Montrer que l'evenement $\propto\left(\dfrac{S_n}{\ln(n)}\right)_{n\geq 1}$ converge $\flat$ est presque s $\hat{\text{\rm{\small ar}}}$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 156] + - Soient $n\in\N$, $(p_0,\ldots,p_n)\in\{-1,1\}^{n+1}$. Montrer que les racines de $\sum_{i=0}^np_iX^i$ dans $\C$ sont de module inférieur ou egal à 1. - Soit $(a_k)_{k\geq 0}$ une suite réelle non identiquement nulle telle que $\sum a_kx^k$ ait pour rayon de convergence $R\gt 0$. Si $j\in\N$, on dit que la suite $(a_i)_{i\geq 0}$ change de signe en $j$ s'il existe $k\in\N^*$ tel que $a_ja_{j+k}\lt 0$ et que $a_i=0$ pour $i\in\db{j+1,j+k-1}$. Montrer que l'ensemble des $x\in]0,R[$ tels que $\sum_{k=0}^{+\i}a_kx^k=0$ est fini de cardinal majore par le nombre de changements de signes de $(a_i)_{i\geq 0}$. + - Soit $(A_k)_{k\geq 0}$ une suite i.i.d. de variables de Rademacher. Pour $n\in\N$, soient $S_n=\sum_{k=0}^nA_k$ et $N_n$ le nombre de $x\in]0,1[$ tels que $\sum_{i=0}^nA_ix^i=0$. Montrer que $N_n\leq\sum_{0\leq k\leq\frac{n-1}{2}}1_{S_{2k+1}=0}$ et en déduire que $\mathbf{E}(N_n)\underset{n\ra+\i}{=}O(\sqrt{n})$. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 157] +Soit $(X_n)_{n\in\N}$ une suite de variables aléatoires à valeurs dans $\N$ telle que, pour tous $n,k\in\N$, $\mathbf{P}\left(X_n=k\right)\gt 0$. Soit $N\in L^2$ une variable aléatoire à valeurs dans $\N$, indépendante de $(X_k)_{k\in\N}$. On pose $X=X_N$. + - Montrer qu'il existe une unique fonction $f_0\colon\N\ra\R$, que l'on déterminera, telle que $\mathbf{E}\left((f_0(X)-N)^2\right)=\min\limits_{g\colon\N\ra\R} \mathbf{E}\left((g(X)-N)^2\right)$. + - On pose, pour tout $n\in\N$ et tout $g\colon\N\ra\R$, $R(g,n)=\mathbf{E}\left((g(X_n)-n)^2\right)$. Montrer que, si la suite $(R(f_0,n))_{n\in\N}$ est constante egale à un certain $R_0$, alors $R_0=\min\limits_{g\colon\N\ra\R}\sup\limits_{n\in\N}R(g,n)$ et $f_0$ est l'unique fonction vérifiant cette condition. +#+end_exercice + + +#+begin_exercice [ENS MP 2024 # 158] +Soient $a\in]0,1[$ et $m\in\N^*$. à l'aide d'une interpretation probabiliste, calculer la borne supérieure, pour $(u_n)_{n\geq 1}$ parcourant l'ensemble des suites à valeurs dans $[0,1]$, de + +$$\sum_{1\leq n_1\lt n_2\lt \cdots\lt n_m}\prod_{\ell=1}^mu_{n_{\ell}}\prod_ {n_{\ell-1}\lt k\lt n_{\ell}}(1-au_k).$$ +#+end_exercice + + +* ENS PSI 2024 :autre: + +** Algèbre + +#+begin_exercice [ENS PSI 2024 # 159] +$\!\!$Résoudre $X^2+X=\begin{pmatrix}1&1\\ 1&1\end{pmatrix}$ dans $\M_2(\R)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 160] +$\!\!$Soient $N\in\N^*$ et $x_0\lt x_1\lt ...\lt x_N$ des réels. On définit $S_3^N$ l'ensemble des fonctions $s$ de classe $\mc C^2$ sur $[x_0,x_N]$ tel que $\forall i\in\db{0,N}$, $s_i=s_{||x_i,x_{i+1}[}$ soit un polynôme de degre au plus 3. + - Montrer que $S_2^3$ est de dimension 5. + - Montrer que $S_3^N$ est de dimension $N+3$. + +Soit $f$ de classe $\mc C^2$ sur $[x_0,x_N]$. - Montrter qu'il existe une unique fonction $s$ de $S^N_3$ telle que $\forall i\in\db{0,N},s(x_i)=f(x_i)$, $s'(x_0)=f'(x_0)$, $s''(x_0)=f''(x_0)$. + - Montrter qu'il existe une unique fonction $s$ de $S^N_3$ telle que : + + $\forall i\in\db{0,N},s(x_i)=f(x_i)$, $s'(x_0)=f'(x_0)$, $s''(x_N)=f''(x_N)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 161] + - Soit $f\in\mc{L}(\C^n)$. Montrter que $f$ est diagonalisable si et seulement si $f^2$ est diagonalisable et $\mathrm{Ker}(f)=\mathrm{Ker}(f^2)$. + - Soit $f\in\mc{L}(\R^n)$. On suppose $f^2$ diagonalisable. Trouver une condition nécessaire et suffisante pour que $f$ soit diagonalisable. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 162] +Soit $E=\R^{\N}$. On définit $F\,\colon\,E\ra E$ par : $\forall u\in E,\;\forall n\in\N,\;(F(u))_n=u_{n+1}$. + - Montrter que $F$ est lineaire. Est-elle injective? Surjective? + - Trouver $G\in\mc{L}(E)$ telle que $F\circ G=\mathrm{id}_E$. Que vaut $G\circ F$? + +Dans la suite de l'exercice, on pose $E=\R^{\Z}$ et on définit $F\,\colon\,E\ra E$ par : + + $\forall u\in E,\;\forall n\in\Z,\;(F(u))_n=u_{n+1}+u_{n-1}$. + - Montrter que $F$ est lineaire. Est-elle injective? + - Soit $\lambda\in\R$. Montrter qu'il existe une matrice $M_{\lambda}\in\M_2(\R)$ dependante de $\lambda$ telle que : + + $\forall u\in E$, $u\in\text{Ker}(F-\lambda\text{id})\Leftrightarrow\forall k\in\Z$, $\left(\begin{array}{c}u_k\\ u_{k+1}\end{array}\right)=M_{\lambda}^k\left(\begin{array}{c}u_0\\ u_1\end{array}\right)$. + +En déduire la dimension de $\text{Ker}(F-\lambda\text{id})$. + - Montrter que si $|\lambda|\neq 2$ alors $M_{\lambda}$ est diagonalisable dans $\C$. Donner ses valeurs propres et une base de vecteurs propres. + - Si $|\lambda|\neq 2$, l'espace $\text{Ker}(F-\lambda\text{id})$ contient-il des suites periodiques non nulles? + - Traiter le cas $|\lambda|=2$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 163] +Pour toute matrice $A\in\M_n(\C),$ on note $\rho(A)=\max_{\lambda\in\mathrm{Sp}(A)}|\lambda|$. On admet que, pour tout $A\in\M_n(\C)$, il existe $D,N$ dans $\M_n(\C)$ respectivement diagonale et nilpotente telles que $DN=ND$, et $P\in\text{GL}_n(\C)$ vérifiant $A=P(D+N)P^{-1}$. + - Pour cette question seulement on pose $A=\left(\begin{array}{cc}a&c\\ 0&b\end{array}\right)$. + - Déterminer $\rho(A)$ et donner une condition nécessaire et suffisante pour que $A$ soit diagonalisable. + - Calculer $A^k$ pour $k\in\N$ et trouver une condition nécessaire et suffisante pour que $\sum A^k$ converge. + - Pour $A\in\M_n(\C)$ montrer que : $\sum A^k$ converge si et seulement si $\rho(A)\lt 1$. + - Soient $A,B,C\in\M_n(\C)$ telles que $\rho(A)\rho(B)\lt 1$. Montrer qu'il existe $D\in\M_n(\C)$ telle que $ADB-D=C$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 164] +On dit que $U\in{\cal M}_n({\C})$ est unipotente si $U-I_n$ est nilpotente. Soit $A\in{\cal M}_n({\C})$. On admet qu'il existe un unique couple $(D_0,N_0)$ avec $D_0$ diagonalisable et $N_0$ nilpotente tel que $A=D_0+N_0$ et $D_0N_0=N_0D_0$. + - Soit $U\in{\cal M}_n({\C})$ unipotente. + - Montrer que ${\rm Sp}(U)=\{1\}$. + - Calculer $U^{-1}$ en fonction de $N=U-I_n$. + - Si $A\in{\cal M}_n({\R})$, montrer que $D_0\in{\cal M}_n({\R})$ et $N_0\in{\cal M}_n({\R})$. + - On suppose $A\in{\rm GL}_n({\C})$. + - Montrer que $D_0\in{\rm GL}_n({\C})$. + - Montrer qu'il existe un unique couple $(D,U)$ tel que $A=DU$, $DU=UD$ et $D$ diagonale, $U$ unipotente. + - Si $A\in{\cal M}_n({\R})$, montrer que $D\in{\cal M}_n({\R})$ et $U\in{\cal M}_n({\R})$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 165] +Pour $A,B\in{\cal M}_n({\C})$, on pose $[A,B]=AB-BA$. + +On note ${\cal S}=\{[A,B]\;,\;(A,B)\in{\cal M}_n({\C})^2\}$ + - Si $M\in{\cal S}$, montrer que ${\rm Tr}(M)=0$. + - Montrer que ${\cal S}$ est stable par multiplication par un scalaire. + - Montrer que ${\cal S}$ est stable par similitude. + - Montrer que toute matrice de trace nulle est semblable à une matrice de diagonale nulle. + - Soient $D={\rm Diag}(1,2,\ldots,n)$ et ${\cal N}$ l'ensemble des matrices de diagonale nulle. Montrer que l'application $M\mapsto[D,M]$ est un automorphisme de ${\cal N}$. + - Montrer que ${\cal N}={\cal S}$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 166] +Soit $B\in{\cal M}_d({\C})$. + - Montrer que si $B$ est diagonalisable alors $e^B$ l'est aussi. + - Soit $A\in{\cal M}_d({\C})$ diagonalisable ayant $n$ valeurs propres distinctes $\mu_1,\ldots,\mu_n$. + - Montrer qu'il existe $Q\in{\C}[X]$ tel que $\forall 1\leq j\leq n$, $Q(\mu_j)=e^{\mu_j}$. + - Montrer que $Q(A)=e^A$. + - On considére $\exp:M\in{\cal M}_d({\R})\mapsto e^M$. + - Soit $C={\rm Diag}(-1,-2,...,-d)$. Pourquoi est-elle inversible? + - Montrer que, si $\lambda\in{\rm Sp}(M)$, alors $e^{\lambda}\in{\rm Sp}(e^M)$. + - Montrer qu'il n'existe pas de matrice $M\in{\cal M}_n({\R})$ telle que $C=e^M$. + - Que dire de $\exp$? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 167] + - On considére la fonction $f$ définie sur ${\cal M}_{n,1}({\R})$ par $:f(X)=\frac{1}{2}X^TAX-B^TX$ ou $A\in{\cal S}_n({\R})$ et $B\in{\cal M}_{n,1}({\R})$. + +Montrer que $f$ est minorée si et seulement si ${\rm Sp}(A)\subset{\R}^+$ et $B\in{\rm Im}(A)$ + - Soient $A_1,A_2\in{\cal S}_n({\R})$ et $B_1,B_2\in{\cal M}_{n,1}({\R})$. Pour $i=1,2$, on note $f_i:X\mapsto\frac{1}{2}X^TA_iX-B_i^TX$. On suppose que $f_1$ et $f_2$ sont minorées et que, pour tout $X$, $\|\nabla_Xf_1\|=\|\nabla_Xf_2\|$. Montrer que $f_1=f_2$. + - Soient $A_1,A_2$ dans ${\cal S}_n^+({\R})$. Montrer que ${\rm Im}(A_1+A_2)={\rm Im}(A_1)+{\rm Im}(A_2)$. En déduire que ${\rm Ker}(A_1+A_2)={\rm Ker}(A_1)\cap{\rm Ker}(A_2)$. +#+end_exercice + + +** Analyse + +#+begin_exercice [ENS PSI 2024 # 168] +Soit $M\in{\cal M}_n({\R})$. On pose $\|M\|_{\i}=\sup_{X\neq 0,X\in{\R}^n}\frac{\|MX\|_{\i}}{\|X\|_{ \i}}$. + +Montrer que $\|M\|_{\i}=\sup_{i\in\{1,\ldots,n\}}\sum_{j=1}^n|m_{i,j}|$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 169] +Soit $d\in{\N}^*$. On se place dans ${\cal M}_d({\R})$ muni du produit scalaire canonique. On note ${\mathbb{B}}_d({\R})$ l'ensemble des matrices bistochastiques, c'est-a-dire des matrices $P=(p_{i,j})_{1\leq i,j\leq d}$ à coefficients dans $[0,1]$ telles que $\colon\forall i\in\db{1,d]\!]\,,\,\sum_{k=1}^dp_{i,k}=1$ et $\forall j\in[\![1,d}\,,\,\sum_{k=1}^dp_{k,j}=1$ On note ${\mathbb{P}}_d({\R})$ l'ensemble des matrices de permutation, c'est-a-dire des matrices de ${\cal M}_d({\R})$ de la forme $\big(\delta_{\sigma(i),j}\big)_{1\leq i,j\leq n}$ ou $\sigma\in{\cal S}_n$. + - Montrer que ${\mathbb{B}}_d({\R})$ est convexe. Est-ce un sous-espace vectoriel de ${\cal M}_d({\R})$? + - Montrer que ${\mathbb{B}}_d({\R})$ est borne et ferme. + - Montrer que ${\mathbb{P}}_d({\R})\subset{\mathbb{B}}_d({\R})$. + - Montrer que ${\mathbb{P}}_d({\R})$ est ferme. + +On admet le_theoreme de Birkhoff_ : La matrice $P$ appartient à ${\mathbb{B}}_d({\R})$ si et seulement s'il existe un entier naturel $m\leq{(d-1)}^2+1$, des matrices $P_1,\ldots,P_m$ dans ${\mathbb{P}}_d({\R})$ et des réels $\lambda_1,\ldots,\lambda_m$ positifs de somme $1$ tels que $P=\sum_{i=1}^m\lambda_iP_i$. + - Soit $\phi$ une forme lineaire sur ${\cal M}_d({\R})$. Montrer que $\phi$ admet un minimum sur ${\mathbb{B}}_d({\R})$, et que celui-ci est atteint sur ${\mathbb{P}}_d({\R})$. + - Soient $M\in{\cal M}_d({\R})$ et $P,Q\in{\cal O}_d({\R})$. Montrer que $\|QMP\|=\|M\|$. + - Soient $A,B\in{\cal S}_d({\R})$ orthosemblables aux matrices diagonales $D_A$ et $D_B$. Montrer l'existence de $P\in{\cal O}_d({\R})$ telle que $\|A-B\|=\|D_AP-PD_B\|$. + +On note $R=\big(p_{i,j}^2\big)_{1\leq i,j\leq d}$. + - Montrer que $R\in{\mathbb{B}}_d({\R})$. + - Montrer que $\|A-B\|^2=\sum_{1\leq i,j\leq d}r_{i,j}|\lambda_i(A)- \lambda_j(B)|^2$ ou les $\lambda_i(A)$ (resp. $\lambda_i(B)$) sont les valeurs propres de $A$ (resp. $B$). +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 170] +On dit qu'une suite $(u_n)$ à valeurs dans un espace vectoriel norme $(E,\|\ \|)$ est de Cauchy si $\forall\eps\gt 0\;,\;\exists N\in{\N}\;,\;\forall(m,n)\in{\mathbb{ N}}^2,\;m,n\geq N\Rightarrow\|u_n-u_m\|\leq\eps$. + +On admet qu'une suite réelle est de Cauchy si et seulement si elle est convergente. + - Montrer qu'une suite complexe est de Cauchy si et seulement si elle converge. + - On se place dans l'espace $\ell^2({\C})$ des suites complexes $(u_n)$ telles que $\sum|u_n|^2$ converge. + +Si $u=(u_n)_{n\in{\N}}\in\ell^2({\C})$ on pose $\|u\|_2=\left(\sum_{n=0}^{+\i}|u_n|^2\right)^{1/2}$. + +Montrer qu'une suite à valeurs dans $\ell^2({\C})$ est de Cauchy (au sens de $\|\ \|_2$) si et seulement si elle converge. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 171] +Pour $n\in{\N}^*$, on pose $\omega_n=e^{2i\pi/n}$.On définit une application $\mc{F}$ sur $\M_{n,1}(\C)$ en posant, pour $v=\left(v_1\,\cdots\,v_n\right)^T$, $\mc{F}(v)=\left(\zeta_1\,\cdots\zeta_n\right)^T$ ou, pour $k\in\db{1,n}$, $\zeta_k=\sum_{j=1}^nv_j\omega_n^{(k-1)(j-1)}$. + - Montrter que $\mc{F}$ est lineaire et donner sa matrice $A$ dans la base canonique. + - Calculer $\overline{A}^TA$ et déterminer $\mc{F}^{-1}$. + - Pour $v=\left(v_1\,\cdots\,v_n\right)^T\in\M_{n,1}(\C)$, on pose $\|v\|_2=\left(\sum_{k=1}^n\left|v_k\right|^2\right)^{1/2}$. + +Montrer que $\|\mc{F}(v)\|_2=\sqrt{n}\,\|v\|_2$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 172] +Soit $M\in\M_n(\R)$. On pose $\|M\|_2=\sup_{X\neq 0,X\in\M_{n,1}(\R)}\frac{|MX|_2}{|X|_{ 2}}$ et $k(A)=\|A\|_2\|A^{-1}\|_2$ si $A\in\mathrm{GL}_n(\R)$. + - Rappeler la définition de la norme euclidienne $|\ |_2$ et montrer que $\|M\|_2=\sup_{|X|_2=1}|MX|_2$. + - Montrer que $\|\ \|_2$ est une norme et que $\forall M_1,M_2$, $\|M_1M_2\|_2\leq\|M_1\|_2\|M_2\|_2$. + - Montrer qu'il existe un vecteur non nul $X\in\M_{n,1}(\R)$ tel que $|MX|_2=\|M\|_2|X|_2$. + - Soit $A\in\mathrm{GL}_n(\R)$. Montrer que $A^TA\in\mc{S}_n^{++}(\R)$. + - Soient $\sigma_1\leq\cdots\leq\sigma_n$, les valeurs propres de $A^TA$. Montrer que $k(A)=\sqrt{\frac{\sigma_n}{\sigma_1}}$. + - On suppose $A\in\mc{S}_n^{++}(\R)$. Calculer $\|A\|_2$ et en déduire $k(A)$. + - Montrer que $k(A)=1$ si et seulement s'il existe $\alpha\in\R^*$ et $Q\in\mc{O}_n(\R)$ tels que $A=\alpha Q$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 173] +On se place dans $\R^n$ muni de sa norme euclidienne canonique. Pour toute partie $A$ non vide bornée on définit le diamêtre de $A$ par $d(A)=\sup\{\|x-y\|,\ (x,y)\in A^2\}$. + +Soit $X$ une partie bornée. Pour $\rho\gt 0$ on définit un $\rho$-recouvrement de $X$ comme une famille $(A_k)_{k\in\N}$ dénombrable de parties bornées telle que $X\subset\bigcup_{k\in\N}A_k$ et $\forall k\in\N\colon\ d(A_k)\leq\rho$. + +On définit, pour $s\geq 0$ : + +$$H_s^{\rho}(X)=\inf\left\{\sum_{k\geq 0}d(A_k)^s\,\ (A_k)_{k\in \N}\ \rho\text{-recouvrement de}\ X\right\}.$$ + - Montrer que $H_s^{\rho}(X)$ est fini et qu'il est decroissant en $\rho$. + - Montrer que $H_s(X)=\sup_{\rho\gt 0}(H_s^{\rho}(X))=\lim_{\rho\ra 0}(H_s^{\rho}(X))$ est decroissante par rapport à $s$. + - Calculer $H_0(X)$ et $H_s(X)$ pour $s\gt n$. + - Pour $X$ partie bornée et $v$ vecteur de $\R^n$, comparer $H_s(X+v)$ et $H_s(X)$. + - Pour $\lambda\gt 0$, comparer $H_s(\lambda X)$ et $H_s(X)$. + - Soient $X$ et $Y$ deux parties bornées telles que $\inf_{x\in X,y\in Y}\|x-y\|\gt 0$. + +Montrer que $H_s(X\cup Y)=H_s(X)+H_s(Y)$. + - Soit $s\geq 0$. Montrer que si $H_s(X)\lt +\i$ alors $H_t(X)=0$ pour tout $t\gt s$. + - Soit $s\geq 0$. Montrer que si $H_s(X)\gt 0$ alors $H_t(X)=+\i$ pour tout $t\lt s$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 174] +Pour $f\in\mc C^3([-1,1],\R)$ et $w_1,w_2,w_3\in\R$, on note $I_{app}(f)=w_1f(-2/3)+w_2f(0)+w_3f(2/3)$. + - Déterminer $w_1,w_2,w_3$ de sorte que $\forall P\in\R_2[X]$, $I_{app}(P)=\int_{-1}^1P$. + +On prendra ces valeurs de $w_1,w_2,w_3$ dans toute la suite. + - A-t-on toujours $I_{app}(P)=\int_{-1}^1P$ pour $\deg(P)\geq 3$? + - Soient $f\in\mc C^3([-1,1],\R)$ et $P\in\R_2[X]$ tel que $f(-2/3)=P(-2/3)$, $f(0)=P(0)$ et $f(2/3)=P(2/3)$. + +Montr per que $\|f-P\|_{\i,[-1,1]}\leq C\|f^{(3)}\|_{\i,[-1,1]}$ ou $C=\frac{1}{6}\sup_{x\in[-1,1]}|x(x+2/3)(x-2/3)|$. + +_Ind._ Considérer l'application $t\mapsto f(t)-P(t)-(f(x)-P(x))\frac{(t+2/3)t(t-2/3)}{(x+2/3)x(x-2/3)}$ pour $x\notin\{-2/3,0,2/3\}$. + - En déduire une majoration de $\left|I_{app}(f)-\int_{-1}^1f\right|$ en fonction de $\|f^{(3)}\|_{\i,[-1,1]}$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 175] +Soit $f\in\mc C^0(\R^+,\R)$ carre intégrable. Pour $x\in\R^{+*}$, on pose $g(x)=\frac{1}{x}\int_0^xf(t)\dt$. + - Montr per que $g$ est prolongeable en une fonction continue sur $\R^+$. + - Montr per que $g^2$ est intégrable et que $\int_0^{+\i}g^2(t)\dt\leq 4\int_0^{+\i}f^2(t) \dt$ + - Rappeler l'inegalite de Cauchy-Schwarz et la condition nécessaire et suffisante d'egalite. Discuter de l'optimalite de la constante $4$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 176] + - Soient $a,b\in\R$ avec $0\lt a\lt b\lt 1$, $I=[a,b]$ et $\phi:x\in I\mapsto 2x(1-x)$. Soit $(\phi_n)_{n\geq 0}$ définie par $\phi_0=\phi$ et $\forall n\in\N,\,\phi_{n+1}=\phi\circ\phi_n$. Étudier la convergence sur $I$ de la suite de fonctions $(\phi_n)_{n\geq 0}$ + - Soit $P:I\ra\R$ une fonction polynomiale. Montr per qu'il existe une suite de fonctions polynomiales à coefficients entiers qui converge uniformément vers $P$ sur $I$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 177] + - Existe-t-il une fonction $g\colon\R^+\ra\R^+$ telle que pour toute fonction $f$ polynomiale on ait $f(x)\underset{x\ra+\i}{=}o(g(x))$? + - Donner le rayon de convergence de la série entiere $\sum n!\,z^{n^2}$ + - Existe-t-il une fonction $g\colon\R^+\ra\R^+$ telle que, pour toute fonction $f$ développable en série entiere, $f(x)\underset{x\ra+\i}{=}o(g(x))$? + - Une fonction est dite analytique si elle est développable en série entiere au voisinage de tout point de son domaine de définition. Montr per que, si $f$ est limite simple de polynômes à coefficients positifs sur $\R^+$, alors elle est analytique sur $\R^+$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 178] + - Soit $P\in\R[X]$. Donner une condition nécessaire et suffisante pour que $t\mapsto|P(t)|^{-1/2}$ soit intégrable sur $\R$. + - Soit $F$ une fraction rationnelle complexe. Montr per que $F$ est intégrable sur $\R$ si et seulement si $\deg(F)\leq-2$ et $F$ n'a pas de pole réel.On écrit alors $F(X)=\sum_{x\in\C}\left(\frac{a_{x,n_x}}{(X-x)^{n_x}}+\ldots+\frac{a_{x,1}}{X-x}\right)$, ou les $a_{x,j}$ sont dans $\C$. + +Montrer que $\int_{\R}F(t)\dt=i\pi\sum_{x\in\C}\xi(x)a_{x,1}$ ou $\xi(x)$ designe le signe de $\text{Im}(x)$. + - Soit $P$ un polynôme complexe non constant. En etudiant la fonction $F:r\mapsto\int_0^{2\pi}\frac{dt}{P(re^{it})}$, démontrer le theoreme de d'Alembert-Gauss. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 179] +On munit $\R^n$ du produit scalaire canonique note $\langle\,\ \rangle$. On considére $Q$ ensemble des fonctions $\mc C^1$ de $\R^n$ dans $\R$ telles que : + + $\forall\lambda\in[0,1],\forall x,y\in\R^n,f(\lambda x+(1-\lambda)y) \leq\max(f(x),f(y))$ + - Pour $n=1$, trouver une fonction $f\in Q$ autre qu'une fonction affine. + - On fixe $x,h\in\R^n$. On pose, pour $t\in\R$, $g(t)=f(x+th)-f(x)$. Exprimer $g'(t)$. + - Montrer que $f\in Q$ si et seulement si, pour tous $x,y\in\R^n$ tels que $f(y)\leq f(x)$, on a $\langle\nabla f(x),y-x\rangle\leq 0$. + - Pour $f\in Q$ de classe $\mc C^2$, on pose $\forall t\in\R$, $g(t)=f(x+th)-f(x)$. Calculer $g''(0)$. + - On suppose $\langle\nabla f(x),h\rangle=0$. Que dire du signe de $\sum_{i=1}^n\sum_{j=1}^n\frac{\partial^2f}{\partial x_i\partial x_j}(x)h_ih_j$? +#+end_exercice + + +** Probabilités + +#+begin_exercice [ENS PSI 2024 # 180] +Soit $X$ une variable aléatoire à valeurs dans $\N$. On note $G$ sa série generatrice et $R$ le rayon de convergence de $G$. + - Justifier que $R\geq 1$. + - Soit $(X_n)_{n\geq 1}$ une suite de variables aléatoires à valeurs dans $\N$, telle que $(G_{X_n})_{n\geq 1}$ converge simplement sur $]-R,R[$ vers une fonction notée $G$. La fonction $G$ est-elle nécessairement la série generatrice d'une variable aléatoire? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 181] +On considére un de equilibre cubique. On note $X$ la variable aléatoire qui donne le nombre obtenu à un lancer. Donner sa série generatrice. + - On note $Y$ la variable aléatoire qui correspond à la somme des lancers de deux des cubiques equilibres. Donner sa série generatrice. + - Est-il possible de truquer le de utilise de sorte que $Y$ suive la loi uniforme sur $\db{2,12}$? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 182] +Soit $(Z_k)_{k\in\N}$ une suite de variables aléatoires indépendantes suivant toute la loi de Bernoulli de paramêtre $p\in]0,1[$. + - On note $U=\min\left\{k\in\N^*,\ Z_k=0\right\}\in\N^*\cup\{+\i\}$. + - Déterminer $\mathbf{P}(U\gt k)$ et $\mathbf{P}(U=k)$. + - En déduire $\mathbf{P}(U=+\i)$. + - Donner $\mathbf{E}(U)$ et $\mathbf{V}(U)$. + - On définit $V=\min\{k\in\N\setminus\{0,1\},\ Z_{k-1}=Z_k=1\}\in\N\cup\{+ \i\}$. + - Déterminer $\mathbf{P}(V=k)$ pour $k=1,2,3,4$. + - Montrer que $\mathbf{P}(V\gt n)\leq\mathbf{P}(V\gt n-2)p(2-p)$. + - En déduire $\mathbf{P}(V=+\i)$. - Trouver une relation de récurrence lineaire vérifiée par la suite $(\mathbf{P}(V=k))$. - Montrer que $V$ est d'esperance finie et calculer $\mathbf{E}(V)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 183] +On munit $\Omega=\{\omega_1,\ldots,\omega_n\}$ de la distribution uniforme de probabilité. + +On se donne $X_1,\ldots,X_{n-1}$ des variables aléatoires réelles telles que : + + - pour $i\neq j$, $X_i$ et $X_j$ sont indépendantes. + + - pour tout $i$, $X_i(\Omega)$ est de cardinal au moins $2$. + + - pour tout $i$, $\mathbf{E}(X_i)=0$ et $\mathbf{V}(X_i)=1$. + +Pour tout $i\in\db{1,n-1}$, on pose $x_i=(X_i(\omega_1),\ldots,X_i(\omega_n))$ et on note $x_n=(1,\ldots,1)$. On note $\langle\,\ \rangle$ le produit scalaire canonique sur $\R^n$. + - On se donne une variable aléatoire $Z$ à valeurs discretes et on note $Z(\Omega)=\{\alpha_1,\ldots,\alpha_m\}$, les $\alpha_i$ etant deux à deux distincts. On suppose $m\geq 3$. + - Montrer que $\mathbf{E}(Z)=\frac{1}{\pi}\langle z,x_n\rangle$ ou $z=(Z(\omega_1),\ldots,Z(\omega_n))$. + - Montrer qu'il existe $\beta_1,\ldots,\beta_m\in\R$ non tous nuls tels que : + +$$\sum_{k=1}^m\mathbf{P}(Z=\alpha_k)\beta_k=0\text{ et }\sum_{k=1}^m \mathbf{P}(Z=\alpha_k)\beta_k\alpha_k=0$$ + - En déduire qu'il existe $Q\in\R_{m-1}[X]$ tel que : + + $Q(Z)\neq 0\,\ \mathbf{E}(Q(Z))=0$ et $\mathbf{E}(Q(Z)Z)=0$. + - Montrer que, pour $(i,j)\in\db{1,n-1}^2$, $\langle x_i,x_j\rangle=n\delta_{i,j}$. + - En déduire que $\sum_{k=1}^{n-1}X_k^2=n-1$. + - Montrer que, pour tout $i$ entre $1$ et $n-1,$ on a $\mathbf{E}(X_i^3)=0$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 184] +Soient $X$ et $Y$ deux variables aléatoires réelles discretes telles que $\mathbf{E}(X)=\mathbf{E}(Y)=0$ et $\mathbf{V}(X)=\mathbf{V}(Y)=1$. On pose $\rho=\mathbf{E}(XY)$. + - Enoncer les inegalites de Markov et de Bienayme-Tchebychev. + - Montrer que $\forall t\in[-1,1]$, $\forall(x,y)\in\R^2$, $x^2+y^2-2txy\geq(1-t^2)\max(x^2,y^2)$. + - Soit $\lambda\gt 0$. Montrer que $\mathbf{P}(|X|\geq\lambda$ ou $|Y|\geq\lambda)\leq\frac{2}{\lambda^2}$. + - Montrer que $2(1-t\rho)\geq(1-t^2)\lambda^2\,\mathbf{P}(|X|\geq \lambda$ ou $|Y|\geq\lambda)$. + - Montrer que $\mathbf{P}(|X|\geq\lambda$ ou $|Y|\geq\lambda)\leq\frac{1+\sqrt{1-\rho^2}}{\lambda^2}$. + - Montrer que l'inegalite de Bienayme-Tchebychev en est une consequence. + - Pour $(\alpha,\beta)\in(\R^+)^2$, donner une majoration de $\mathbf{P}(|X|\geq\alpha$ ou $|Y|\geq\beta)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 185] +Soient $X_1,\ldots,X_n$ des variables aléatoires indépendantes suivant la loi de Rademacher. + - On pose $S_n=X_1+\cdots+X_n$. Déterminer la loi, l'esperance et la variance de $S_n$. + - Montrer que, pour tout $a\in\R$, on a $\mathbf{P}(S_n\geq na)\leq\frac{1}{na^2}$. + - Montrer que, pour toute variable aléatoire à valeurs réelles $X$, pour tout $a\in\R$ et pour tout $s\in\R^{+*}\colon\mathbf{P}(X\geq a)\leq\frac{\mathbf{E}\left(e^ {sX}\right)}{e^{sa}}$. + - Montrer que $\forall s\in\R^{+*}$, $\mathbf{P}(S_n\geq na)\leq\left(\frac{\mathrm{ch}(s)}{e^{sa}} \right)^n$. - Montrer que: $\forall s\in\R$, $\mathrm{ch}(s)\leq e^{s^2/2}$_._ +#+end_exercice + + - En déduire que: $\mathbf{P}(S_n\geq na)\leq e^{-na^2/2}$. +#+begin_exercice [ENS PSI 2024 # 186] +Soit $a\lt 0\lt b$ des nombres réels. On pose $f\,\colon\,x\mapsto\frac{ax}{b-a}+\ln\bigg(1+\frac{a(1-e^x)}{b-a} \bigg)$. + - Déterminer l'ensemble de définition $D$ de $f$ et montrer: $\forall x\in D$, $0\leq f''(x)\leq 1/4$. + - En déduire: $\forall x\in D$, $0\leq f(x)\leq x^2/8$. + - Soient $X$ une variable aléatoire telle que $X(\Omega)\subset[a,b]$. + +Montrer: $\forall\lambda\in\R$, $\mathbf{E}(e^{\lambda X})\leq\frac{b-\mathbf{E}(X)}{b-a}e^{ \lambda a}+\frac{\mathbf{E}(X)-a}{b-a}e^{\lambda b}$. + - Dans le cas particulier ou $\mathbf{E}(X)=0$, montrer: $\mathbf{E}(e^{\lambda X})\leq e^{f(\lambda(b-a))}$. + - En déduire dans le cas general: $\mathbf{E}(e^{\lambda X})\leq e^{\lambda\mathbf{E}(X)+\lambda^2(b-a)^{ 2}/8}$. + - Pour tout $\eps\gt 0$, montrer: $\mathbf{P}(|X-\mathbf{E}(X)|\geq\eps)\leq 2\exp\bigg(- \frac{2\eps^2}{(b-a)^2}\bigg)$. + - Soient $X_1,\ldots,X_n$ des variables aléatoires mutuellement indépendantes. On pose $S_n=X_1+\cdots+X_n$. On suppose que, pour tout $k\in\{1,\ldots,n\}$, $X_k(\Omega)\subset[a_k,b_k]$. Montrer pour tout $\eps\gt 0\colon\mathbf{P}(|S_n-\mathbf{E}(S_n)|\geq\eps) \leq 2\exp\bigg(-\frac{2\eps^2}{\sum_{k=1}^n(b_k-a_k)^{2 }}\bigg)$. +#+end_exercice + + + + +* ENS - PC :autre: + +** Algèbre + +#+BEGIN_exercice [ENS PSI 2024 # 187] + - Soit $X\subset\N$ telle que $0$ et $1$ appartiennent à $X$ et $\lim\limits_{n\ra+\i}\frac{1}{n}\mathrm{Card}(X\cap\db{0,n})=0$. + +Montrer que, pour tout $k\in\N^*$, il existe $j\in\N$ telle que $\mathrm{Card}(X\cap\db{j,j+k})=2$. + - Montrer qu'il existe $100$ entiers consécutifs contenant exactement $5$ nombres premiers. +#+END_exercice + +#+begin_exercice [ENS PSI 2024 # 188] +Soient deux réels $a$ et $b$. On pose $P=X^4+aX^3+bX^2+X$. On suppose que les racines de $P$ sont toutes distinctes deux à deux et qu'elles appartiennent à un même cercle du plan complexe. Montrer que $3\lt ab\lt 9$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 189] +Soit $(P_n)_{n\in\N}$ une suite définie par $P_0\in\R[X]$ de degre $\geq 2$ et $\forall n\in\N$, $P_{n+1}=XP_n'$. Montrer qu'il existe une suite de réels positifs $(\lambda_n)_{n\in\N}$ convergeant vers $0$ telle que, pour tout $n\in\N$, les racines complexes de $P_n$ appartiennent au disque de centre $0$ et de rayon $\lambda_n$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 190] +Soit $E$ l'ensemble des matrices $M\in\M_n(\R)$ à coefficients $0$ ou $1$ qui sont inversibles. Quel est le nombre maximal de $1$ d'un élément de $E$? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 191] +On dit qu'une matrice est positive si tous ses coefficients sont positifs. Soit $A\in\M_n(\R)$. Montrer l'équivalence entre: + +(i) $A$ est monotone, c'est-a-dire $A$ est inversible et $A^{-1}$ positive, + +(ii) $\forall X\in\R^n$, $AX\geq 0\Rightarrow X\geq 0$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 192] +Soit $E$ un sous-espace vectoriel de ${\cal M}_n\,(\R)$ tel que $\forall A\in E$, $\mbox{rg}\,(A)\leq 1$. Montrer que $\dim\,(E)\leq n$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 193] +Déterminer les $X\in{\cal M}_2(\R)$ telle que $X^2+X=\begin{pmatrix}1&1\\ 1&1\end{pmatrix}$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 194] +Caractériser les matrices $A\in{\cal M}_n(\R)$ nilpotentes d'indice $n-1$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 195] +Existe-t-il deux matrices $N$ et $P$ de ${\cal M}_n(\R)$ telles que $N^2=0$, $P^2=P$, $NP$ est nilpotente et $(NP)^2\neq 0\,$? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 196] +Soit $M\in{\cal M}_n(\R)$ telle que $M^3=0$. Montrer qu'il existe une unique matrice $X\in{\cal M}_n(\R)$ telle que $X+MX+XM^2=M$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 197] +On pose $A_0=\begin{pmatrix}3/4&1/2\\ 1/2&5/4\end{pmatrix}$ puis, pour tout $n\in\N$, $A_{n+1}=2A_n-A_n^2$. Déterminer la limite de $(\det(A_n))$ quand $n\ra+\i$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 198] +On pose $A_1=\left(\begin{array}{cc}0&1\\ 1&0\end{array}\right)$ et, pour $n\in\N^*$, $A_{n+1}=\left(\begin{array}{cc}A_n&I_{2^n}\\ \hline I_{2^n}&A_n\end{array}\right)$. Montrer que $A_n$ admet $(n+1)$ valeurs propres $\lambda_0\lt \lambda_1\lt \cdots\lt \lambda_n$ d'ordres respectifs $\begin{pmatrix}n\\ k\end{pmatrix}$ pour $0\leq k\leq n$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 199] +On munit $\R_n\,[X]$ du produit scalaire défini par $\langle P,Q\rangle=\int_0^1\!\!P(x)Q(x)\dx$. Montrer que $M=\left(\langle X^i,X^j\rangle\right)_{(i,j)\in\db{0,n}^2}$ est inversible. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 200] +Soient $a_0,\ldots,a_n$ des réels. + +Pour des polynômes $P,Q\in\R_n[X]$, on définit $\langle P,Q\rangle=\sum_{k=0}^nP^{(k)}(a_k)\,Q^{(k)}(a_k)$. + - Montrer que $\langle\,\ \rangle$ est un produit scalaire sur $\R_n[X]$. + - Montrer qu'il existe une base $(P_0,\ldots,P_n)$ de $\R_n[X]$, orthonormée pour ce produit scalaire et telle que, pour chaque $i\in\db{0\,;\,n}$, le polynôme $P_i$ soit de degre $i$ et à coefficient dominant strictement positif. + - Déterminer $P_k^{(k)}(a_k)$ pour tout $k\in\db{0\,;\,n}$. + - On suppose $a_0=\cdots=a_n=a$. Déterminer les polynômes $P_k$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 201] +Soient $n,k\in\N^*$ et $(f_1,\ldots,f_k)$ une famille de vecteurs de $\R^n$. On suppose que $\forall x\in\R^n\setminus\{0\},\exists i\in\{1,\ldots,k\},\ \langle x,f_i\rangle\gt 0$. + - Donner un exemple de famille de $\R^n$ vérifiant cette propriété. + - Montrer que $(f_1,\ldots,f_k)$ est une famille generatrice de $\R^n$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 202] +Soit $n\in\N$ et $M\in\mc{S}_n(\R)$. En notant $(s_1,\ldots,s_n)$ les valeurs propres de $M$, on pose + +$$N_p(M)=\left(\sum_{i=1}^n|s_i|^p\right)^{1/p}.$$ + - Montrer que $(A,B)\mapsto\op{tr}(AB)$ est un produit scalaire sur $\mc{S}_n(\R)$. En déduire que $N_2$ est une norme sur $\mc{S}_n(\R)$. + - Montrer que $N_1(M)=\sup\{|\op{tr}(MO)|,\ O\in\mc{O}_n(\R)\}$. En déduire que $N_1$ est une norme sur $\mc{S}_n(\R)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 203] +Soient $n\in\N^*$ et $A\in\mc{A}_n(\R)$. + - Montrer que les valeurs propres dans $\C$ de $A$ sont imaginaires pures. + - Que dire de $\det(A)$ si $n$ est impair? + - On suppose $n$ pair et on considére la matrice $J\in\M_n(\R)$ dont tous les coefficients valent $1$. Montrer que $\det(A+J)=\det(A)$ +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 204] +Soit $A=(a_{i,j})_{(i,j)\in[1,n]^2}$ vérifiant $\forall(i,j)\in[1,n]^2,\ a_{i,j}\in\{0,1\}$. On note $J\in\M_n(\R)$ la matrice dont tous les coefficients sont egaux à $1$. On suppose qu'il existe $k\in\N^*$ tel que $A^TA=kI_n+J$. Montrer que $A$ est inversible. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 205] +Soit $A\in\M_n(\R)$ telle que $A^2=A^T$. + - Quelles sont les valeurs propres complexes possibles de $A$? + - Donner un exemple de matrice $A$ qui vérifie $A^2=A^T$ et qui possede toutes les valeurs propres possibles trouvées à la question précédente. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 206] + - Soit $S\in\mc{S}_n(\R)$ inversible. Montrer que les assertions sont équivalentes : + +(i) $S$ admet $k$ valeurs propres positives (comptées avec multiplicité), + +(ii) il existe des sous-espaces vectoriels $F$ et $G$ de $E$ tels que $\dim F=k$, $\dim G=n-k$ et $\forall X\in F$, $X^TSX\geq 0$ et $\forall Y\in G$, $Y^TSY\leq 0$. + - Soit $S\in\mc{S}_n(\R)$ inversible. Soit $P\in\text{GL}_n(\R)$. Montrer que $P^TSP$ et $S$ ont le même nombre de valeurs propres positives. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 207] + - Montrer que toute matrice symétrique positive admet une racine carrée. + - Montrer que $A\in\M_n(\R)$ est diagonalisable si et seulement s'il existe $S$ symétrique définie positive telle que $SA=A^TS$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 208] +Soit $E$ un espace euclidien. Soit $a$ un endomorphisme autoadjoint de $E$. Soient $u\in E$ non nul et $V=\text{Vect}\left\{a^k(u)\ ;k\in\N\right\}$. Montrer que l'endomorphisme induit par $a$ sur $V$ n'a que des valeurs propres simples. +#+end_exercice + + +** Analyse + +#+begin_exercice [ENS PSI 2024 # 209] +Soit $A$ un ensemble de $\R^2$. On dit que $x,\ y\ \in A$ sont connectes si et seulement s'il existe $f\in\mc C^0\left(\left[0,1\right],A\right)$ telle que $f\left(0\right)=x$ et $f\left(1\right)=y$. + - Montrer que tous les points de $\R^2$ sont connectes. + - Déterminer les points connectes de $\R^2\setminus\{(0,0)\}$. + - Déterminer les points connectes de $\R^2\setminus\{x,\ \|x\|=1\}$. - Déterminer les points connectes de $\R^2\setminus\underset{i\in\Z^2}{\cup}\mc{B}_o\left(i, \eps\right)$ ou $\eps\in\R^{+*}$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 210] +On considére $f:A\in\M_n(\R)\mapsto\underset{\lambda\in\mathrm{Sp}(A)}{ \sup}\left|\lambda\right|$, ou le spectre est pris sur $\C$. L'application $f$ est-elle lipschitzienne? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 211] +On pose $a_1\geq 0$ puis $a_{n+1}=10^n{a_n}^{n^2}$ pour tout $n\in\N^*$. à quelle condition sur $a_1$ la suite $\left(a_n\right)$ tend-elle vers $0$? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 212] +On pose, pour $n\in\N$, $f\left(n\right)=\sum_{k=0}^n\frac{n^k}{k!}$. Donner un équivalent de $f\left(n\right)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 213] +Étudier les suites $u$ et $v$ telles que $u_0=v_0=0$ et $u_1=v_1=1$ et, pour tout $n\geq 1$, $\left\{\begin{array}{lll}u_{n+1}&=&au_n+bv_n+cu_{n-1}+dv_{n-1}\\ v_{n+1}&=&a'u_n+b'v_n+c'u_{n-1}+d'v_{n-1} \end{array}$. avec toutes les constantes réelles. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 214] +Pour $a\in\R$, soit $\left(u_n\right)$ définie par $u_0\in\left[0,1\right]$ et $\forall n,x_{n+1}=ax_n(1-x_n)$. + - Pour quelles valeurs de $a$ a-t-on $\forall n,u_n\in\left[0,1\right]$? Que peut-on dire alors de la suite $\left(x_n\right)$? + - Montrer que, si $a\in\left[1,2\right]$, alors $x_n$ tend vers $\frac{a-1}{a}$. + - On suppose que $a\in\left[2,3\right]$ et que $\left(x_n\right)$ converge. Quelle est la limite de $\left(x_n\right)$? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 215] +Soit $\left(p_{i,j}\right)_{(i,j)\in\N^2}$ une famille de réels positifs ou nuls telle que $p_{i,j}=0$ si $j\gt i$. On suppose que $\forall n\in\N,\ \sum_{j=0}^np_{n,j}=1$. Montrer l'équivalence des deux assertions suivantes : + - pour chaque $j\in\N$, la suite $\left(p_{n,j}\right)_{n\in\N}$ tend vers $0$, + - pour toute suite convergente $\left(s_n\right)_{n\geq 0}$ de limite $S$, on a $\underset{n\ra\i}{\lim}\sum_{j=0}^np_{n,j}s_j=S$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 216] +Soit $f\colon\R\ra\R$ de classe $\mc C^2$ telle que $f(0)=0$ et $f'(0)\neq 0$. Soit $\left(u_n\right)_{n\in\N}$ une suite réelle vérifiant $u_0\neq u_1$ et $\forall n\in\N$, $u_{n+1}=u_n-\frac{u_n-u_{n-1}}{f(u_n)-f(u_{n-1})}f(u_n)$. + - Montrer que, si $u_0$ et $u_1$ sont assez petits, alors $\underset{n\ra+\i}{\lim}\ u_n=0$. + - Sous les hypotheses de -, déterminer un équivalent de $u_n$ lorsque $n$ tend vers $+\i$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 217] +Pour $c\in\C$, on définit la suite $\left(z_n\right)$ par $z_0=0$ et $z_{n+1}=z_n^2+c$ pour tout $n\in\N$. On pose $\M=\left\{c\in\C,\ (z_n)\text{ est bornée}\right\}$. + - Montrer que, si $\left|c\right|\leq 1/4$, alors $c\in\M$. + - Montrer que, si $\left|c\right|\geq 3$, alors $c\notin\M$. + - Discuter de l'ensemble $\M\cap\R$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 218] +Soient $\left(a_n\right)_{n\geq 0}$ et $\left(b_n\right)_{n\geq 0}$ deux suites réelles. Soit $S\in\R$. On suppose que : + +(i) $\forall n\in\N,\ b_n\gt 0$ ; (ii) la série $\sum b_n$ diverge ; (iii) $\underset{n\ra\i}{\lim}\frac{a_n}{b_n}=S$.Montrer que $\lim_{n\ra\i}\frac{a_0+\cdots+a_n}{b_0+\cdots+b_n}=S$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 219] +Soit $(x_n)_{n\in\N}$ une suite de réels positifs telle que $\sum_{n=0}^{+\i}x_n=A$. Quelles sont les valeurs que peut prendre $\sum_{n=0}^{+\i}x_n^2$? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 220] +Soit $g:[0,+\i[\ra\R$ de classe $\mc C^2$ telle que $g(0)=g'(0)=0$ et $g''(0)\gt 0$. Pour $\lambda\gt 0$, on pose $A_{\lambda}=\{x\gt 0,\ g(x)=\lambda x\}$. Montrer qu'il existe $\mu\gt 0$ tel que $\forall\lambda\in]0,\mu]$, $A_{\lambda}\neq\emptyset$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 221] +Soient $f:[0,1]\ra\R$ continue par morceaux et $g:[0,1]\ra\R$ continue. On suppose que $f+g$ est croissante. Montrer que $f([0,1])$ est un intervalle. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 222] +Trouver toutes les fonctions $f\in\mc C^2(\R,\R)$ telles que : $\forall t\in\R,\ f(t)^2=f\left(t\sqrt{2}\right)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 223] +Soient $f,g:[0,1]\ra[0,1]$ continues. On suppose $f\circ g=g\circ f$ et $g$ croissante. Montrer que $f$ et $g$ admettent un point fixe commun. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 224] +Déterminer les fonctions $f$ de classe $\mc C^1$ sur $[-1,1]$ telles que : $\forall(x,y)\in[-1,1]^2,\ f(x)-f(y)\geq f(x)^2(x-y)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 225] +Déterminer les fonctions $f\in\mc C^2\left(\R,\R\right)$ telles que $\forall x\in\R$, $f\left(7x+1\right)=49f\left(x\right)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 226] +Soit $f:t\mapsto\sum_{k=1}^Na_k\sin(2\pi kt)$ ou les $a_k$ sont des nombres réels avec $a_N\neq 0$. On note $N_j$ le nombre de racines comptées avec multiplicité (notion qu'on admettra) de $f^{(j)}$ sur $[0,1]$. Montrer que $(N_j)_{j\geq 0}$ est une suite croissante qui tend vers $2N$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 227] +Soit $V=\{f\in\mc C^1(\left[\,0\,;1\,\right],\R)\ ;\ f(0)=0$ et $f(1)=1\}$. Trouver tous les réels $\alpha$ tels que : $\forall f\in V,\ \exists x\in\left[\,0,1\,\right],\ f(x)+\alpha=f'(x)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 228] +Soit $g\colon\R\ra\R$ de classe $\mc C^1$ telle que $\lim_{t\ra+\i}g(t)=0$ et $f$ telle que $f'(t)-f(t)=g(t)$. On pose $a=f(0)$. Montrer qu'il existe une unique valeur $a$ pour laquelle $\lim_{t\ra+\i}f(t)=0$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 229] +Soit $f\in\mc C^1(\R^+,\R)$ ne prenant pas les valeurs $0$ et $1$. + +On suppose que $\forall x\geq 0,f'(x)=\frac{1}{f(x)}+\frac{1}{f(x)-1}$. Déterminer la limite de $f$ en $+\i$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 230] +Soit $f\colon\R\ra\R$ $1$-lipschitzienne, $\lambda\in\left]0,1\right[$ et $a\in\R$. Montrer qu'il existe une unique application $F\colon\R\ra\R$ lipschitzienne telle que $\forall x\in\R,F(x)=f(x)+\lambda F(x+a)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 231] +Soit $f\colon\R\ra\R$ de classe $\mc C^{\i}$. On pose $f_a:x\mapsto f(x+a)$ et $F_f=\mathrm{Vect}(f_a)_{a\in\R}$. + - Trouver $f$ telle que $F_f$ est de dimension finie. Preciser la dimension. - Montrter que, si $F_f$ est de dimension finie, alors $F_{f'}$ est aussi de dimension finie. + - Trouver les fonctions $f$ telles que $\dim F_f=1$. +#+end_exercice + + - Trouver les fonctions $f$ telles que $\dim F_f=2$. +#+begin_exercice [ENS PSI 2024 # 232] +Soit $f\colon\R\ra\R$ de classe $\mc C^2$ telle que $\lim\limits_{x\ra\pm\i}f(x)=0$. On pose, pour $t\in\R^+$, $\lambda(t)=\max\limits_{x\in\R}\left(\left|f(x)\right|\exp(-tx^2) \right)$. + - On suppose $f(0)\neq 0$. Déterminer $\lim\limits_{t\ra+\i}\lambda(t)$. + - On suppose maintenant $f(0)=0$ et $f'(0)\neq 0$. Déterminer un équivalent de $\lambda$ en $+\i$. +#+end_exercice + + - Même question en supposant la fonction $f$ de classe $\mc C^{\i}$, $f(0)=\cdots=f^{(k-1)}(0)=0$ et $f^{(k)}(0)\neq 0$. +#+begin_exercice [ENS PSI 2024 # 233] +On dit que $(x_n)$ converge au sens de Cesaro vers $\ell$ lorsque $\dfrac{x_1+\cdots+x_n}{n}\underset{n\ra+\i}{\longrightarrow}\ell$. + +Déterminer toutes les fonctions $f\colon\R\ra\R$ qui vérifient la propriété suivante: pour toute suite réelle $(x_n)$, si la suite $(x_n)$ converge au sens de Cesaro vers $\ell$, alors la suite $(f(x_n))$ converge au sens de Cesaro vers $f(\ell)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 234] +Soit $g$ une fonction $\mc C^2$ de $\R^+$ dans $\R$ telle que $g(0)=g'(0)=0$ et $g''(0)\gt 0$. On pose, pour $\lambda$ dans $\R^{+*}$, $A(\lambda)=\{x\gt 0,\ g(x)=\lambda x\}$. + - Montrez qu'il existe $\lambda_0\gt 0$ tel que, pour tout $\lambda$ dans $]\,0\,;\lambda_0\,[$, $A(\lambda)\neq\emptyset$. + - On pose $\lambda^*=\sup\{\lambda\gt 0,\ A(\lambda)\neq\emptyset\}$ (cela peut être $+\i$). Montrer que, pour tout $\lambda$ dans $]\,0\,;\lambda^*\,[$, $A(\lambda)$ est non vide. + - On pose $X_{\lambda}=\inf\{x\gt 0,\ g(x)=\lambda x\}$ quand c'est défini. Montrer que $X_{\lambda}\neq 0$. + - En toute generalite, la fonction $h\colon\lambda\mapsto X_{\lambda}$ est-elle continue sur $]\,0\,;\lambda^*\,[$? + - Montrer que cette fonction $h$ est continue au voisinage de $0$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 235] +Soit $M\in\mc C^1\left(\R,\M_n(\C)\right)$. On suppose que, pour tout $t$, $M(t)$ est inversible. L'objectif est de montrer que $\dfrac{d}{dt}\left(\det\left(M\left(t\right)\right) \right)=\det\left(M\left(t\right)\right)\mathrm{tr}\left(M\left(t\right)^{-1} \dfrac{d}{dt}\left(M\left(t\right)\right)\right)$. + - Le montrer si $M$ est diagonale. + - Montrer que $\forall U\in\M_n\left(\R\right)$, $\lim\limits_{\eps\ra 0}\dfrac{\det\left(I_n+\eps U\right)-\det \left(I_n\right)}{\eps}=\mathrm{tr}\left(U\right)$. + - On suppose qu'il existe $t_0\in\R$ tel que $M\left(t_0\right)=I_n$. Montrer la relation en $t_0$. + - Traiter le cas general. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 236] +Soit $f\in\mc C^{\i}\left(\R,\R\right)$. On pose $f_a:x\mapsto f\left(a+x\right)$ et $F_f=\mathrm{Vect}\left(f_a,a\in\R\right)$. + - Si $F_f$ est de dimension finie, montrer que $F_{f'}$ l'est aussi. + - Quelle est la dimension de $F_f$ lorsque $f=\exp$? + - Réciproquement, montrer que si $\dim\left(F_f\right)=1$, alors $f=\exp$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 237] +On suppose $e=\dfrac{p}{q}\in\Q$. Montrer que $q\int_0^1x^ne^x\dx\in\N^*$. Conclure. + +Adapter la preuve précédente pour prouver $e^2\notin\Q$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 238] +Soit $f\colon\R\mapsto\R$ une application continue. On suppose que $x\mapsto f(x)+\int_0^xf(t)dt$ tend vers le réel $\ell$ en $+\i$. Montrer que $f$ possede une limite en $+\i$ que l'on déterminera. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 239] +Soit $f:[0,1]\ra\R^+$ et intégrable telle que, pour tout $x\in[0,1]$, $f\left(x\right)f\left(1-x\right)=1$. Montrer que $\int_0^1f\geq 1$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 240] +Soit $f\in\mc C^1(\left[\,0\,;1\,\right],\R)$ telle que $\int_0^1f(t)\dt=0$. Montrer que $\left|\int_0^1f(t)\dt\right|\leq\frac{1}{8}\left\|f^{ '}\right\|_{\i}$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 241] +Soit $f\colon\R^{+*}\ra\R$ une fonction de classe $\mc C^1$, à valeurs dans $\R^{+*}$, decroissante et intégrable sur $\R^{+*}$. + - On suppose que $\frac{f'(x)}{f(x)}\underset{x\ra+\i}{\longrightarrow}0$. Montrer que $\frac{f(x)}{\int_x^{+\i}f(t)\dt}\underset{x\ra+\i}{ \longrightarrow}0$ + - On suppose que $\frac{f'(x)}{f(x)}\underset{x\ra+\i}{\longrightarrow}-\i$. Que dire de $\lim_{x\ra+\i}\frac{f(x)}{\int_x^{+\i}f(t)\dt}$? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 242] +Soit $(P_n)$ une suite de polynômes de $\R[X]$ telle que $\lim_{n\ra+\i}\sup_{x\in[-1,1]}|P_n(x)-e^x|=0$. Montrer que $\deg(P_n)\underset{n\ra+\i}{\longrightarrow}+\i$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 243] +Soit $(f_n)_{n\in\N}$ une suite de fonctions définie sur $[0,+\i[$ par $f_0=1$ et $\forall n\in\N$, $f_n(0)=1$ et $f_{n+1}'(x)=e^x\sqrt{f_n(x)}$. Justifier l'existence de $\lim_{n\ra+\i}f_n(x)$ et déterminer sa valeur. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 244] +Encadrer et donner un équivalent en $+\i$ de $S:x\mapsto\sum_{k=0}^{+\i}\frac{x^k}{\sqrt{k!}}$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 245] + - Montrer que la suite $\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right)_{n\in\N^*}$ converge. On note $\gamma$ sa limite. + - Montrer que la fonction $\Gamma:x\mapsto\int_0^{+\i}t^{x-1}e^{-t}\dt$ est de classe $\mc C^1$ sur $\R^{+*}$. + - Calculer $\Gamma(n)$ pour $n\in\N^*$. Donner un développement asymptotique de $\ln(\Gamma(n+1))$ à la precision $O(\ln(n))$. En considérant la fonction $\Psi:x\mapsto\frac{\Gamma'(x)}{\Gamma(x)}$, montrer que $\Gamma'(1)=-\gamma$. + +Ind. On admet que l'on peut \lt \lt deriver \gt \gt le développement précédent c'est-a-dire que $\Psi(n+1)=\ln(n)+O(1/n)$. + - Montrer que $\Psi$ est croissante et justifier le développement admis precedemment. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 246] +Soient $f,g\colon\R\ra\R$ des fonctions continues. On suppose qu'il existe des constantes $C_1,C_2,a,b\in\R^{+*}$ telles que $\forall x\in\R$, $|f(x)|\leq\frac{C_1}{(1+|x|)^a}$ et $|g(x)|\leq\frac{C_2}{(1+|x|)^b}$.Lorsque c'est possible, on pose $f*g(x)=\int_{-\i}^{+\i}f(x-y)\,g(y)\,dy$. + - à quelle condition sur $C_1,C_2,a,b$ la fonction $f*g$ est-elle définie sur $\R$? + - On suppose maintenant $a$ et $b$ strictement supérieurs à $1$. Montrer qu'il existe $C_3\gt 0$ telle que $\forall x\in\R$, $\ \ |f*g(x)|\leq\dfrac{C_3}{(1+|x|)^{\min(a,b)}}$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 247] +Soit $(a,b)\in\R^2$. Trouver toutes les fonctions $f\in\mc C^1(\R^2,\R)$ bornées sur $\R^2$ et telles que $f=a\dfrac{\partial f}{\partial x}+b\dfrac{\partial f}{\partial y}$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 248] :todo: +Montrer que la fonction $f\colon P\in\R_n[X]\mapsto f(P) = \int_0^1 \big(P(x) - e^x\big)^2 \dx$ admet un unique point critique. +#+end_exercice + +** Géométrie + +#+BEGIN_exercice [ENS PSI 2024 # 249] +Montrer qu'un polygone à $n$ sommets inscrit dans le cercle unité est d'aire maximale si et seulement s'il est régulier +#+END_exercice + +#+BEGIN_exercice [ENS PSI 2024 # 250] +Soit $\eps\in \interval]{0, 1}[$. Soit $n$ le plus grand entier naturel tel que $n\eps \leq 1$. On trace les cercle de rayon $1$ et de centres $(k\eps, -1)$, pour $0\leq k\leq n$. Donner un développement limité de la somme des longueurs des arcs de cercle qui forment une courbe longeant la droite des abscisses. +#+END_exercice + +** Probabilités + +#+BEGIN_exercice [ENS PSI 2024 # 251] +On considère une urne contenant $n\geq 2$ boules : 2 boules sont rouges et les $n-2$ autres sont blanches. On tire les boules une par une sans remise. On s'arrête une fois qu'on a tiré les deux boules rouges. En moyenne, combien reste-t-il de boules dans l'urne ? +#+END_exercice + +#+BEGIN_exercice [ENS PSI 2024 # 252] +On considère une urne vide qu'on remplit successivement d'une boule blanche (avec une probabilité $p$) ou d'une boule rouge (avec une probabilité $1-p$). On arrête de la remplir lorsqu'on obtient la première boule rouge. Puis on la vide jusqu'à tirer la boule rouge. Déterminer le nombre moyen de boules blanches restantes à la fin. +#+END_exercice + +#+BEGIN_exercice [ENS PSI 2024 # 253] +On dispose d'une urne vide. On ajoute des boules une par une et on s'arrête dès qu'on a ajouté une boule rouge. La probabilité d'ajouter une boule rouge à chaque étape est égale à $\frac{1}{m}$. On mélange ensuite les boules et on les retire une à une jusqu'à retirer la boule rouge. Calculer l'espérance du nombre de boules restantes. +#+END_exercice + + +#+begin_exercice [ENS PSI 2024 # 254] +Soit $n\in\N^*$. Pour $A$ partie de $\db{1,n\rrbracket^2$, on note $M(A)$ la matrice carrée de taille $n$ à coefficients dans $\{0,1\}$ caractérisée par $\forall(i,j),M(A)_{i,j}=1\iff(i,j)\in A$. On considére l'ensemble $P$ des parties de $\llbracket 1,n}^2$ de cardinal $n$ et une variable aléatoire $X$ suivant la loi uniforme sur $P$. Quelle est la probabilité que $M(X)$ soit inversible? +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 255] +Soit $(X_n)_{n\in\N^*}$ des variables aléatoires indépendantes suivant la loi uniforme sur $\{-1,1\}$. Pour $n\in\N^*$, soit $M_n=\frac{X_1+\cdots+X_n}{\sqrt{n}}$. Déterminer $\mathbf{E}\left(M_n^k\right)$ pour $k\in\N$._Ind._ Distinguer selon la parite de $k$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 256] +Soient, pour $\lambda\gt 0$, $A_{\lambda}$, $B_{\lambda}$, $C_{\lambda}$, $D_{\lambda}$ quatre variables aléatoires indépendantes suivant la loi de Poisson de paramêtre $\lambda$. + - Calculer $\lim\limits_{\lambda\ra+\i}\mathbf{P}\left(A_{\lambda}X^2+B_{ \lambda}X+C_{\lambda}\ \mathbf{n}\text{'a que des racines réelles}\right)$. + - Même question pour $A_{\lambda}X^3+B_{\lambda}X^2+C_{\lambda}X+D_{\lambda}$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 257] +Soit $n\in\N^*$. On munit l'ensemble $S_n$ des permutations de $\{1,2,\ldots,n\}$ de la probabilité uniforme. Soit $k\in\{1,\ldots,n\}$. Pour $\sigma\in S_n$, on note + + $P_k(\sigma)=\left\{(i_1,\ldots,i_k)\in\{1,\ldots,n\}^k\,\ i_1\lt i_2\lt \cdots\lt i_k\text{ et }\sigma(i_1)\lt \sigma(i_2)\lt \cdots\lt \sigma(i_k)\right\}$ + +l'ensemble des sous-suites croissantes de longueur $k$ de la permutation $\sigma$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 258] +Soient $\lambda\in\left[0,1\right]$, $\left(X_{k,n}\right)_{\genfrac{}{}{0.0pt}{}{1\leq k\leq n}{n\geq 1}}$ des variables aléatoires mutuellement indépendantes, ou $X_{k,n}$ suit la loi $\mc{B}\left(\lambda/n\right)$. On pose $X_n=X_{1,n}+\cdots+X_{n,n}$. + +Soit $t\in\R$. Déterminer $\lim\limits_{n\ra+\i}\mathbf{E}(\exp(tX_n))$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 259] +On dit qu'une variable aléatoire $X$ à valeurs réelles est infiniment divisible si, pour tout $n\in\N^*$, il existe des variables aléatoires $X_{1,n},\ldots,X_{n,n}$ indépendantes et de même loi telles que $X\sim X_{1,n}+\cdots+X_{n,n}$. + - Donner des exemples de variables aléatoires indéfiniment divisibles. + - Soit $X$ une variable aléatoire infiniment divisible non nulle telle que $\mathbf{E}(X)=0$ et $\mathbf{E}(X^2)\lt +\i$. Montrer que, pour tout $A\gt 0,\ \mathbf{P}(X\gt A)\gt 0$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 260] +Pour $x\in\R$, on pose $\gamma(x)=\frac{1}{\sqrt{\pi}}e^{-x^2/2}$. Soit $(X_n)_{n\in\N^*}$ des variables aléatoires indépendantes qui suivent la loi uniforme sur $\{-1,1\}$. Pour $n\in\N^*$, on pose $M_n=\frac{X_1+\cdots+X_n}{\sqrt{n}}$. + +Montrer que pour tout polynôme $P\in\R[X]$, on a $\lim\limits_{n\ra\i}\mathbf{E}(P(M_n))=\int_{-\i}^{+\i} \gamma(x)P(x)\dx$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 261] +On note $D(X)$ le nombre de diviseurs premiers de $X$, ou $X$ suit la loi uniforme sur $\db{1,n}$. + - Calculer $\lim\limits_{n\ra+\i}\mathbf{E}(D(X))$. + - On admet que $\sum\limits_{p\text{ premier, }p\leq n}\frac{1}{p}\sim\ln(\ln n)$. Montrer $\lim\limits_{n\ra+\i}\mathbf{P}\left(\left|\frac{D(X)}{\ln(\ln n)}-1\right|\geq\eps\right)=0$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 262] +Soient $C\in\text{GL}_q(\R)$ et $N\in\M_q(\R)$ nilpotente. Soit $p\in]0,1[$. On définit $(B_n)_{n\in\N}$ par $B_0=I_n$ et $B_{n+1}=A_nB_n$, ou $\mathbf{P}(A_n=C)=p$ et $\mathbf{P}(A_n=N)=1-p$. + +Déterminer $\lim\limits_{n\ra+\i}\mathbf{P}(B_n\neq O)$. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 263] +On munit $\R^2$ de sa structure euclidienne canonique. + +Soient $\theta\in\left]-\pi,\pi\right]$ et $p\in\left]0,1[$. On note $R(\theta)=\begin{pmatrix}\cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta)\end{pmatrix}$ et $M=\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$. + +Soit $(u_n)$ une suite de vecteurs aléatoires de $\R^2$ avec $u_0=(1,0)^T$ et telle que, pour tout $n\in\N$, $\mathbf{P}\left(u_{n+1}=R(\theta)u_n\right)=p$ et $\mathbf{P}\left(u_{n+1}=Mu_n\right)=1-p$. Déterminer la limite de $(\mathbf{E}(\left\|u_n\right\|)$ pour $\theta=\dfrac{2\pi}{3}$ puis pour $\theta$ quelconque. +#+end_exercice + + +#+begin_exercice [ENS PSI 2024 # 264] +Soit $\theta\in\left[\,0\,;2\pi\,\right]$. Soit $p\in\left]\,0\,;1\right[$. On pose $R=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$ et $Q=\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$. + +Les variables aléatoires $(A_n)_{n\in\N^*}$ sont indépendantes et vérifient $\mathbf{P}(A_n=R)=p$ et $\mathbf{P}(A_n=Q)=1-p$. On note $U_0=\begin{pmatrix}1\\ 0\end{pmatrix}$ puis, pour chaque $n\in\N^*$, $U_n=A_nU_{n-1}$. + +On note $t_1\lt t_2\lt t_3\lt \dots$ les instants $n$ successifs ou $A_n=Q$. + - Trouver la loi de $\left\|U_{t_1}\right\|$ ou $\left\|\,\right\|$ designe la norme euclidienne canonique. + - Pour $N\in\N^*$, donner une approximation du nombre d'indices $i$ tels que $t_i\leq N$. + +Dans toute la suite, on suppose que $\theta=\dfrac{2\pi}{3}$. + - Calculer $\mathbf{E}(\ln\left\|U_{t_1}\right\|)$. + - Déterminer la loi de $\left\|U_{t_2}\right\|$. + - Déterminer la loi de $\left\|U_{t_k}\right\|$ pour $k\in\N^*$. + - Déterminer $\mathbf{E}(\ln\left\|U_{t_k}\right\|)$ pour $k\in\N^*$. +#+end_exercice + +* X - MP :xens: + +** Algèbre + +#+begin_exercice [X MP 2024 # 265] +Pour toute partie finie non vide $X$ de $\R$ dont on note $x_1\lt x_2\lt \dots\lt x_n$ les éléments, on pose : $a^+(X)=\prod_{i=1}^{n-1}(x_{i+1}-x_i+1)$ et $a^-(X)=\prod_{i=1}^{n-1}(x_{i+1}-x_i-1)$. L'objectif est d'etablir que : $\sum_{\begin{subarray}{c}B\subset A\\ B\neq\emptyset\end{subarray}}a^-(B)=a^+(A)$ pour n'importe quelle partie finie non vide $A$ de $\R$. + +On se donne donc $A=\{a_1\dots,a_n\}$ une partie finie non vide de $\R$, avec $a_1\lt \dots\lt a_n$. + - On suppose le résultat acquis. Trouver une expression de : $\alpha(A)=\sum_{\begin{subarray}{c}B\subset A\\ a_n\in B\end{subarray}}a^-(B)$. + - Etablir le résultat cherche. + - On suppose $A=\db{1,n}$. Calculer : $\sum_{\begin{subarray}{c}B\subset A\\ B\neq\emptyset\\ B\cap(B+1)=\emptyset\end{subarray}}a^-(B)$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 266] + - Soit $n$ un entier supérieur à l et premier avec 10. Montrer que $n$ possede un multiple dont l'écriture en base 10 n'a que des 9. + - On remarque que $\frac{1}{2}=0,\underline{142857}$ $\underline{142857}\ldots\underline{142857}\ldots$ avec $142+857=999$. + + $\frac{285+714}{7}=0,\underline{285714}$ $\underline{285714}\ldots\underline{285714}\ldots$ $076+923=999$ + + $\frac{1}{13}=0,\underline{076923}$ $\underline{076923}\ldots\underline{076923}\ldots$ + +Expliquer. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 267] +Pour $r$ un rationnel non nul s'écrivant $r=2^ka/b$ avec $k\in\Z$ et $a,b$ deux entiers impairs, on définit la valuation dyadique de $r$ par $v_2(r)=k$. + +On admet que : $\forall x,y\in\Q^*$, $v_2(xy)=v_2(x)+v_2(y)$ et si $x+y\neq 0$, $v_2(x+y)\geq\min(v_2(x),v_2(y))$, avec egalite si $v_2(x)\neq v_2(y)$. + +On note enfin, pour tout $n\in\N^*$, $H_n=\sum_{k=1}^n\frac{1}{k}$. + - Montrer que pour tout $n\gt 1$, $H_n\notin\Z$. + - Montrer que pour tous $m,n\in\N^*$ tels que $m\leq n-2$, on a $v_2(H_n-H_m)\lt 0$. + - Montrer les propriétés admisses plus haut. + - La question - peut-elle s'adapter à la valuation 3-adique? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 268] +Quels sont les $m$ de $\N^*$ tels qu'il existe $m$ éléments consécutifs de $\N^*$ divisibles par des cubes d'éléments de $\N^*\setminus\{1\}$? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 269] +Montrer que tout $n\in\Z$ s'écrit sous la forme $\sum_{k=0}^N\eps_k(-2)^k$ avec $N\geq 0$ et les $\eps_k$ dans $\{0,1\}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 270] +Soit $n\in\N^*$. On note $\mc{F}$ l'ensemble des entiers naturels qui ne sont pas divisibles par le carre d'un entier supérieur ou egal à $2$, et $q(n)=|\mc{F}\cap\db{1,n}|$. +On note $\mc{E}(n,k)=\R^{+*}\cap\left\{\sum_{i=1}^k\sqrt{a_i}-\sum_{i= 1}^k\sqrt{b_i},\ (a_1,\ldots,a_k,b_1,\ldots,b_k)\in\db{0,n}^{2k}\right\}$ et + + $\Delta(n,k)=\min\mc{E}(n,k)$. + - On admet que $(\sqrt{n})_{n\in\mc{F}}$ est libre dans le $\Q$-espace vectoriel $\R$. + +Montrer que $\Delta(n,k)\leq\frac{k(\sqrt{n}-1)}{\left(q(n)+k-1\right)-1}$. + - On démontre dans cette question le résultat admis dans la précédente. + - Soit $\mathbb{K}$ un sous-corps de $\R$, et $x$ un élément de $\mathbb{K}\cap\R^{+*}$. Montrer que $\mathbb{K}[\sqrt{x}]=\mathbb{K}+\mathbb{K}\sqrt{x}$ est un sous-corps de $\R$, et que si $\sqrt{x}\not\in\mathbb{K}$ alors il existe un unique automorphisme $\sigma$ de l'anneau $\mathbb{K}[\sqrt{x}]$ différent de l'identite et fixant tous les éléments de $\mathbb{K}$. + +Dans la suite, on fixe un entier $n\geq 1$ on suppose acquire, pour tout ensemble fini $A$ constitue de $n$ nombres premiers, la libert $\acute{\text{e}}$ de la famille des $\sqrt{m}$, ou $m$ parcourt l'ensemble des éléments de $\mc{F}$ ayant tous leurs diviseurs premiers dans $A$. Soit $A$ un ensemble forme de $n+1$ nombres premiers $p_1,\ldots,p_{n+1}$. + - On construit par récurrence une suite $(\mathbb{K}_0,\ldots,\mathbb{K}_n)$ de corps : $\mathbb{K}_0=\Q$ et $\mathbb{K}_i=\mathbb{K}_{i-1}[\sqrt{p_i}]$ pour tout $i\in\db{1,n}$. Montrer que $\mathbb{K}_n$ est de dimension $2^n$ comme $\mathbb{K}_0$-espace vectoriel, et en preciser une base. Montrer qu'il existe un automorphisme $\sigma$ du corps $\mathbb{K}_n$ qui fixe $\sqrt{p_1},\ldots,\sqrt{p_{n-1}}$ et envoie $\sqrt{p_n}$ sur $-\sqrt{p_n}$. Dans la suite, on raisonne par l'absurde en supposant que $\sqrt{p_{n+1}}\in\mathbb{K}_n$. + - Montrer que $\sqrt{p_{n+1}}=\alpha+\beta\sqrt{p_n}$ pour un $\alpha\in\mathbb{K}_{n-1}$ et un $\beta\in\mathbb{K}_{n-1}$, puis montrer qu'en fait $\sqrt{p_{n+1}}=\beta\sqrt{p_n}$. + - Montrer que $\sqrt{p_{n+1}}=\lambda\prod_{k=1}^n\sqrt{p_k}$ pour un $\lambda\in\Q$, et conclure à une contradiction. + - Conclure. +#+end_exercice + +#+begin_exercice [X MP 2024 # 271] +Soit $p$ un nombre premier congru à $3$ modulo $4$. On note $L$ l'ensemble des carres de $\mathbb{F}_p^*$. + - Montrer que $|L|=\frac{p-1}{2}$. + - Montrer que si $x\in L$, alors $-x\notin L$. + - On fixe $x\in\mathbb{F}_p^*$ et l'on pose $A=\big{\{}(\ell_1,\ell_2)\in L^2\ ;\ x=\ell_1-\ell_2\big{\}}$. Calculer $\op{card}A$. +#+end_exercice + +#+begin_exercice [X MP 2024 # 272] +Soit $p$ un nombre premier impair. + - Dénombrer les $(x,y)\in(\mathbb{F}_p)^2$ tels que $x^2+y^2=1$. + - Soit $z\in\mathbb{F}_p\setminus\{0\}$. Dénombrer $\{\,(x,y)\in\mathbb{F}_p^2,\ x^2+y^2=z\}$. +#+end_exercice + +#+begin_exercice [X MP 2024 # 273] +Soit $p$ un nombre premier impair. On pose $q=2p+1$ et l'on suppose $q$ premier. On considére l'équation : $(E):x^p+y^p+z^p=0$ d'inconnue $(x,y,z)\in\Z^3$. Soit $(x,y,z)\in\Z^3$ une solution de $(E)$ telle que $p$ ne divise aucun des entiers $x,y$ et $z$ et telle que $x,y,z$ soient premiers entre eux deux à deux. + - Montrer que $q$ divise $x,y$ ou $z$. + - Montrer qu'il existe $(a,b,c)\in\Z^3$ tel que : $y+z=a^p$, $x+y=b^p$, $x+z=c^p$. + - Factoriser $y^p+z^p$. + - Conclure à une contradiction. +#+end_exercice + +# ID:6237 # Classique :) +#+begin_exercice [X MP 2024 # 274] +Soient $p$ un nombre premier congru à $1$ modulo $4$ et $S$ l'ensemble $S=\{x,y,z)\in\N^3\ ;\ p=x^2+4yz\}$. Pour $(x,y,z)\in S$ on pose : + + - si $x\lt y-z$, $f(x,y,z)=(x+2z,z,-x+y-z)$ ; + + - si $y-z\lt x\lt 2y$, $f(x,y,z)=(2y-x,y,x-y+z)$ ; + + - si $x\gt 2y$, $f(x,y,z)=(x-2y,x-y+z,y)$. + +Montrer que $f$ définit une involution de $S$. En déduire que $p$ s'écrit $u^2+v^2$ avec $(u,v)\in\N^2$. +#+end_exercice + +#+begin_exercice [X MP 2024 # 275] +Soit $d\in\Z\setminus\{0\}$. On considére l'équation $(*):x^2-dy^2=1$ d'inconnue $(x,y)\in\Z^2$. + - Traiter les cas $d\lt 0$ et $d=k^2$ avec $k\in\N$. + - Dans la suite, on suppose $d\gt 0$ et $\sqrt{d}\not\in\N$. Soit $(x_0,y_0)\in\N^2\setminus\{(\pm 1,0)\}$ solution de $(*)$. + +On pose $z=x_0+\sqrt{d}\,y_0$. Montrer que, pour tout $n\in\N^*$, il existe un unique $(x_n,y_n)\in\N^2$ tel que $z^{n+1}=x_n+\sqrt{d}\,y_n$. - En déduire que, si l'ensemble des solutions de $(*)$ est non trivial, i.e. n'est pas reduit à $\{(\pm 1,0)\}$, il en existe une infinite. + - Soit $x\in\R$. Montrer que, pour tout $n\in\N$, il existe $(p,q)\in\Z^2$ tel que $|p-qx|\lt \dfrac{1}{n}$. + - Montrer qu'il existe une infinite de couples $(p,q)\in\Z\times\N^*$ tels que $|p-qx|\lt \dfrac{1}{q}$. + - Montrer qu'il existe $K\in\R$ pour lequel il existe une infinite de couples d'entiers $(p,q)$ tels que $|p^2-dq^2|\lt K$. + - Conclure que $(*)$ possede des solutions non triviales. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 276] + - Soit $\mathbb{F}$ un corps fini. On admet que le groupe multiplicatif $\mathbb{F}^{\times}$ est cyclique. + +Soient $n\geq 1$ et $u\in\mathbb{F}$. On note $\widehat{\mathbb{F}^{\times}}$ l'ensemble des morphismes de $\mathbb{F}^{\times}$ dans $\C^*$ prolonges par 0 en 0. On note $N(X^n=u)$ le nombre de zeros du polynôme $X^n-u$ dans $\mathbb{F}$. On note $\widehat{\mathbb{F}^{\times}}[n]$ l'ensemble des $\chi\in\widehat{\mathbb{F}^{\times}}$ tels que $\chi^n=1$. Montrer que $N(X^n=u)=1+\sum_{\chi\in\widehat{\mathbb{F}^{\times}}[n],\chi\neq 1}\chi(u)$. + - On suppose $\mathbb{F}=\Z/p\Z$ avec $p\equiv 1\pmod{3}$ et $p$ impair. + +Montrer que $N(X^3+Y^3=1)=p+\sum_{\chi_1,\chi_2\in\widehat{\mathbb{F}^{\times}}[3] \setminus\{1\}}J(\chi_1,\chi_2)=p-2+2\Re\mathfrak{e}\,J(\omega,\omega)$ si + + $\omega\in\widehat{\mathbb{F}^{\times}}[3]\setminus\{1\}$, ou $J(\chi_1,\chi_2)=\sum_{a+b=1}\chi_1(a)\chi_2(b)$. + - On admet que $|J(\omega,\omega)|=\sqrt{p}$ et $pJ(\omega,\omega)=g_{\omega}^3$ ou $g_{\omega}=\sum_{x\in\mathbb{F}}\omega(x)\zeta_p^x$ avec $\zeta_p=e^{\frac{2i\pi}{p}}$. + +Montrer que $N(X^3+Y^3=1)=p-2-a_p$ avec $a_p^2+27b_p^2=4p$ ou $b_p\in\Z$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 277] +Pour $p$ premier impair, on note $\chi\colon\mathbb{F}_p\ra\{1,-1,0\}$ la fonction définie par $\chi(0)=0$, $\chi(x)=1$ pour tout élément $x$ de $\mathbb{F}_p^{\times}$ qui est un carre, et $\chi(x)=-1$ dans toute autre situation. + +Pour $x\in\mathbb{F}_p$, on note $e^{\frac{2i\pi x}{p}}$ la quantite $e^{\frac{2i\pi k}{p}}$, ou $k\in\Z$ est un representant quelconque de $x$. + +Pour $t\in\N$, on pose $g_p(t)=\sum_{x\in\mathbb{F}_p}\chi(tx)e^{\frac{2i\pi x}{p}}$. + - Soit $p$ un nombre premier impair, et des entiers $a$ et $b$ tels que $0\lt a\lt b\lt p$. Montrer que $g_p(1)\sum_{n=a}^{b-1}\chi(n)=\sum_{x\in\mathbb{F}_p}\chi(x)\sum_{t=a}^{b -1}e^{\frac{2i\pi tx}{p}}$. On admettra dans la suite que $|g_p(1)|=\sqrt{p}$. + - Montrer qu'il existe une constante $M\gt 0$ telle que, quels que soient $p$ premier impair, et $a,b$ entiers tels que $0\leq a\lt b\lt p$, on ait $\op{card}\{k\in\db{a,b-1},\ k\ \text{est un carre modulo}\ p\}=\dfrac{b-a}{2}+u_{p,a,b}$ ou $|u_{p,a,b}|\leq M\,\sqrt{p}\,\ln p$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 278] +Soit $G$ un groupe fini de cardinal $2n$ ou $n$ est impair. + - Montrer que $G$ possede un élément d'ordre 2. + - Montrer que $G$ possede un sous-groupe d'ordre $n$. + +_Ind._ Considérer l'application $\Phi$ qui à $g\in G$ associe $\Phi(g):G\ra G$ telle que, pour tout $x\in G$, $\Phi(g)(x)=gx$. + - Trouver un contre-exemple si $n$ est pair. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 279] +Soit $p$ un nombre premier. On dit qu'un groupe $G$ est un $p$-groupe si, pour tout $g\in G$, l'ordre de $g$ est une puissance de $p$. Si $k\in\N^*$, on dit que $G$ est $k$-divisible si, pour tout $g\in G$, il existe $x\in G$ tel que $x^k=g$. + - Montrer qu'un $p$-groupe non trivial et $p$-divisible est infini. + - Donner un exemple de tel groupe. + - Montrer que $G$ est alors $k$-divisible pour tout $k$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 280] +Soit $G$ un groupe d'ordre $n\geq 1$. Pour $g_1$,..., $g_k\in G$, on note $E(g_1,\ldots,g_k)=\{g_{i_1}\cdots g_{i_s}\;;\;s\in\N,\;\;1 \leq i_1\lt \cdots\lt i_s\leq k\}$ (avec la convention que l'élément neutre est le produit vide donc appartient à cet ensemble). + - Soient $g_1$,..., $g_k\in G$ tel que $G=E(g_1,\ldots,g_k)$. Montrer que $k\geq\lfloor\log_2(n)\rfloor$. + - Soit $A\subset G$. Montrer que $\sum\nolimits_{x\in G}\lvert A\cap Ax\rvert=\lvert A\rvert^2$. + - Montrer qu'il existe $g_1,\ldots,g_k\in G$ tels que $G=E(g_1,\ldots,g_k)$ avec $k\leq\lfloor\log_2(2n\ln n)\rfloor$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 281] +On note $\mc{S}(\C)$ le groupe des permutations de $\C$. Soit $G$ un sous-groupe cyclique de $\mc{S}(\C)$ d'ordre $2^n$, ou $n\geq 2$, contenant la conjugaison complexe. + - Montrer que, pour tout $z\in\C\setminus\R$, il existe $\tau\in G$ tel que $\tau(z)\neq\pm z$. + - Soit $H$ un sous-groupe de $G$ d'ordre $2^{n-1}$. Montrer que $H$ contient au moins deux applications $\R$-lineaires. + - Montrer que $G$ contient exactement deux applications $\R$-lineaires. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 282] +Soit $\mc{A}$ une $\C$-algèbre. On suppose que $\mc{A}$ est munie d'une norme $N$ vérifiant : $\forall a,b\in\mc{A},\,N(ab)=N(a)N(b)$. + - Soit $x\in\mc{A}$. En posant $z=z\cdot 1_{\mc{A}}$, on identifie $\C$ à une sous-algèbre de $\mc{A}$. Montrer qu'il existe $z_0\in\C$ tel que $\forall z\in\C,N(x-z_0)\leq N(x-z)$. On pose $a=x-z_0$. + - On suppose que $N(a)=2$. Montrer que $\forall n\in\N^*,\,\forall z\in\mathbb{U}_n,N(a-z)\geq 2$. Montrer ensuite que $N(a-1)=2$ puis $N(a-5)=2$. + - Montrer que $\mc{A}$ est isomorphe à $\C$, i.e. $\dim\mc{A}=1$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 283] + - Soit $f$ l'application qui à $z\in\mathbb{U}\setminus\{i\}$ associe le point d'intersection de $\R$ et de la droite passant par $z$ et $i$. Montrer que $f(z)\in\Q\Leftrightarrow z\in\Q(i)$. + - Montrer qu'il existe une infinite de triplets non proportionnels $(a,b,c)\in\Z^3$ tels que $a^2+b^2=c^2$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 284] +On appelle nombre de coefficients positifs du polynôme $P=\sum_{k=0}^na_kX^k\in\R[X]$ de degre $n\geq 1$ le cardinal de l'ensemble $\{i\in\db{0,n},\;a_i\geq 0\}$. + - Soit $P\in\R[X]$ de degre $n\geq 2$. Montrer que $P^2$ à au moins trois coefficients positifs. + - Montrer que, pour tout entier $n\geq 2$, il existe $P\in\R[X]$ de degre $n$ tel que $P^2$ ait exactement trois coefficients positifs. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 285] +Soient $n\in\N$, $P\in\Z[X]$ de degre majore par $n$, $\Delta$ le pgcd de $P(0),P(1),\ldots,P(n)$. Montrer que, pour tout $k\in\Z$, $\Delta$ divise $P(k)$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 286] +Soit $P=\sum_{k=0}^na_kX^k\in\C[X]$ de degre $n\geq 2$ dont on note $z_0,\ldots,z_{n-1}$ les racines. On note $t_1,\ldots,t_{n-1}$ les racines complexes de $P'$ et l'on suppose que : $\forall k\in\db{0,n-1},|z_k|\leq 1$. + - Montrer que : $\forall k\in\db{1,n-1},|t_k|\leq 1$. + - On suppose que $z_0$ est racine simple de $P$. Calculer $\dfrac{P''(z_0)}{P'(z_0)}$ deux facons : + +(i) en fonction de $z_0$ et des $t_k$ ; (ii) en fonction de $z_0$ et des $z_k$. + - Soit $z\in\C\setminus\{-1\}$ tel que $|z|\leq 1$. Montr er que $\mathfrak{Re}\left(\dfrac{1}{1+z}\right)\geq\dfrac{1}{2}$. + - On suppose que $z_0=1$ et que $z_0$ est racine simple. Montr er qu'il existe $k\in\db{1,n-1}$ tel que $|1-t_k|\leq 1$. + - On suppose que $|z_0|=1$. Montr er qu'il existe $i\in\db{1,n-1}$ tel que $|z_0-t_i|\leq 1$. + - Soient $Q\in\R[X]$ non constant et $\alpha\in\R^*$. On pose $P=Q^2+\alpha^2$. Montr er qu'il existe une racine $z$ de $P$ et une racine $t$ de $P'$ telles que $|z-t|\leq|z|$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 287] +Pour $P = a_0 + a_1 X + \dots + a_n X^n\in\C[X]$, on pose $\lN P\rN = \left(\sum_{i=0}^n |a_i|^2\right)^{1/2}$. + - Montrer que $n\geq 2$, $a_0\gt 0$, et $\theta$ est racine simple de $P$. + - On pose $Q=X^nP(1/X)$ et $f:z\mapsto\frac{P(z)}{Q(z)}$. Montrer que si $\theta^{-1}$ est un pole de $f$ alors $a_0=1$ et $n$ est pair. + - Montrer qu'il existe un réel $r\gt 0$ et une suite $(b_n)_{n\in\N}$ d'entiers telle que, pour tout $z\in D_o(0,r)$, on ait $f$ définie en $z$ et $f(z)=\sum_{n=0}^{+\i}b_nz^n$. +#+end_exercice + +#+BEGIN_exercice [X MP 2024 # 288] + - Soient $P = a_0 + a_1 X + \dots + a_nX^n\in\C[X]$ de degré $n\geq 1$. On pose $r = \min \{|z|,\, z\in\C,\, P(z) = 0\}$, et on suppose $r\gt 0$. Si $a_k\neq 0$, montrer que $r^k\leq {~n~\choose k}\frac{|a_0|}{|a_k|}$. + - Soit $An = \{P\in\C[X]\mid \deg P = n,\, P(-1) = P(1) = 0\}$. Montrer que $\sup_{P\in A_n}\{\min \{|z|,\, P'(z) = 0\}\}\lt +\i$. +#+END_exercice + +#+BEGIN_exercice [X MP 2024 # 289] +Soient $\om,q\in\C^*$ tels que $\om^2$ n'est pas une puissance entière de $q$. On considère l'équation $(*)\colon \om f(z) g(qz) = \om^2 f(qz) g(z) + P(z)$, d'inconnues $(P,f,g)\in\C[X]^3$, avec $g,P$ unitaires. + - Si $(P,f,g)$ vérifie $(*)$, trouver une relation entre les degrés de $P,f,g$. + - On fixe $P$. Montrer l'existence de $(f,g)$ tel que $(P,f,g)$ vérifie $(*)$. + - On fixe $(P,f)$. Y a-t-il unicité de $g$ tel que $(P,f,g)$ vérifie $(*)$ ? +#+END_exercice + +#+BEGIN_exercice [X MP 2024 # 290] +Soit $\theta\in\C$ un nombre algébrique. + - Montrer que l'ensemble des polynômes annulateurs de $\theta$ est l'ensemble des multiples d'un certain polynôme $P\in\Q[X]$ unitaire, déterminé de manière unique. On écrit $P = \sum_{k=0}^n a_k X^k$, avec $a_n = 1$, et on suppose $P$ à coefficients entiers, $\theta$ irrationnel et $a_0\geq 0$. + - Montrer que $n\ge q2$, $a_0\gt 0$ et $\theta$ est racine simple de $P$. + - On pose $Q = X^n P(1/X)$ et $f\colon z\mapsto \frac{P(z)}{Q(z)}$. Montrer que si $\theta^{-1}$ est un pôle de $f$, alors $a_0 = 1$ et $n$ est pair. + - Montrer qu'il existe $r\gt 0$ et une suite $(b_n)$ d'entiers telle que pour tout $z\in D(0, r)$, on ait $f$ définie en $z$ et $f(z) = \sum_{n=0}^{+\i} b_n z^n$. +#+END_exercice + + +#+begin_exercice [X MP 2024 # 291] +Soit $M\in\M_n(\R)$. + - Si $M$ est inversible, combien de coefficients de $M$ faut-il modifier au minimum pour la rendre non-inversible? + - Si $M$ n'est pas inversible, combien de coefficients de $M$ faut-il modifier au minimum pour la rendre inversible? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 292] +Soient $V$ un $\R$-espace vectoriel de dimension finie, et $a,b\in\mc{L}(V)$. Pour $u,v\in\mc{L}(V)$, on pose $[u,v]=uv-vu$. On suppose que $a$ est nilpotent et que $[a,[a,b]]=0$. Montrer que $[a,b]$ et $ab$ sont nilpotents. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 293] +Soit $V_0,\ldots,V_n$ des espaces vectoriels, $(v_0^+,\ldots,v_{n-1}^+)\in\mc{L}(V_0,V_1)\times\cdots\times \mc{L}(V_{n-1},V_n)$ et $(v_1^-,\ldots,v_n^-)\in\mc{L}(V_1,V_0)\times\cdots\times \mc{L}(V_n,V_{n-1})$. On suppose que $v_{i-1}^+\circ v_i^-=-v_{i+1}^-\circ v_i^+$ pour tout $i\in\db{1,n-1}$, et que $v_{n-1}^+\circ v_n^-=0$. Montrer que l'endomorphisme $v_1^-\circ v_0^+$ de $V_0$ est nilpotent. Déterminer l'indice de nilpotence maximal possible de $v_1^-\circ v_0^+$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 294] +Pour tout $\sigma\in\mc{S}_n$, on note $P_{\sigma}\in\M_n(\R)$ la matrice de permutation associée et, pour tout $k$, $n_k(\sigma)$ le nombre de cycles de longueur $k$ dans la décomposition de $\sigma$ en produit de cycles à supports disjoints. + - Soit $\sigma\in S_n$. Calculer, pour tout $k$, $\op{tr}(P_{\sigma}^k)$ en fonction des $n_r(\sigma)$. + - En déduire que deux permutations $\sigma$, $\tau\in\mc{S}_n$ sont conjuguées dans $\mc{S}_n$ si et seulement si les matrices $P_{\sigma}$ et $P_{\tau}$ sont semblables. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 295] +Soient $V=\C^n$ et $T=(\C^*)^n$. Pour tout $v\in V$ et toute partie $H\subset V$, on note $H\cdot v=\{(h_1v_1,\ldots,h_nv_n),\ h\in H\}$. + - Soit $v\in V$. Déterminer la nature topologique de $T\cdot v$. Preciser notamment son adherence. + - Quels sont les sous-espaces $W\subset V$ tels que, pour tout $v\in T$, $W\cdot v=W$? + - Dénombrer les familles $(W - {i\in\db{0,n}}$ de sous-espaces vectoriels satisfaisant la condition de la question précédente et les inclusions strictes $W_0\subsetneq W_1\subsetneq\cdots\subsetneq W_n$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 296] +Soient $V$ un $\C$-espace vectoriel de dimension finie non nulle et $\phi$ un morphisme de groupes de $\mathbb{U}$ dans $\op{GL}(V)$ tel que $\{0\}$ et $V$ soient les seuls sous-espaces vectoriels de $V$ stables par tous les $\phi(g)$ pour $g\in\mathbb{U}$. + - Montrer que $\dim V=1$. + - On suppose $f\colon\theta\in\R\mapsto\phi(e^{i\theta})$ dérivable en $0$. Déterminer $\phi$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 297] +Soit $M=(m_{i,j})_{1\leq i,j\leq n}\in\M_n(\C)$. On dit que $(V,A,B)$ est une realisation de $M$ si :- $V$ est un $\C$-espace vectoriel de dimension $d$, + + - $A=(a_1,\ldots,a_n)$ est une famille libre de formes lineaires sur $V$, + + - $B=(b_1,\ldots,b_n)$ est une famille libre de vecteurs de $V$, + + - pour tous $i,j$, $a_i(b_j)=m_{i,j}$. + +On dit que $d$ est la dimension de la realisation. + - Montrer que si $M$ est realisée par un espace de dimension $d$, elle l'est aussi par un espace de dimension $d'\gt d$. + - Trouver une realisation de la matrice $M_0=\left(\begin{matrix}1&-1\\ -1&1\end{matrix}\right)$ + - Trouver la dimension minimale d'une realisation de $M_0$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 298] +Soient $A,B\in\M_n(\R)$ commutant à $AB-BA$. Calculer $\exp(A+B)$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 299] +Soient $A,B,M\in\M_n(\R)$ telles que $\chi_A=\chi_B$ et $AM=MB$. + - Montrer que, pour tous $r\in\N$ et $X\in\M_n(\R)$, on a $\op{tr}((A-MX)^r)=\op{tr}((B-XM)^r)$. + - En déduire que, pour tout $X\in\M_n(\R)$, on a $\det(A-MX)=\det(B-XM)$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 300] +La matrice $\left(\begin{array}{cc}1&2024\\ 0&1\end{array}\right)$ peut-elle s'écrire $\sum_{n=0}^{+\i}\frac{(-1)^n}{(2n+1)!}A^{2n+1}$ avec $A\in\M_2(\R)$? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 301] +Soit $E$ un $\C$-espace vectoriel de dimension finie. Soient $a,b\in\mc{L}(E)$. On suppose que : $ab-ba=f\circ v$ avec $f\in\mc{L}(\C,E)$ et $v\in\mc{L}(E,\C)$. + - Calculer $\det(ab-ba)$. + - Montrer que $a$ et $b$ sont cotrigonalisables. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 302] +Soit $\mc{A}$ un sous-espace vectoriel de $\M_n(\R)$ stable par crochet de Lie : pour $M,N\in\mc{A}$, $[M,N]=MN-NM\in\mc{A}$. + - On suppose que, pour tout $M\in\mc{A}$, $N\mapsto[M,N]$ induit un endomorphisme diagonalisable de $\mc{A}$. Montrer que $\forall M,N\in\mc{A}$, $[M,N]=0$. + - On suppose que $\dim\mc{A}\leq 3$ et que, pour tout $M\in\mc{A}$, $N\mapsto[M,N]$ induit un endomorphisme nilpotent de $\mc{A}$. On pose $\mc{A}_0=\mc{A}$ et, pour $j\in\N$, $\mc{A}_{j+1}=\{[M,N],\ (M,N)\in\mc{A}_j^2\}$. Montrer que $\mc{A}_3=\{0\}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 303] +Soit $E$ un espace vectoriel de dimension $n$. Un drapeau de $E$ est une famille de sous-espaces $(F - {i\in\db{0,n}}$ telle que $F_0\subsetneq F_1\subsetneq\cdots\subsetneq F_n$. + - Soit $(F - {i\in\db{0,n]\!]}$ un drapeau de $E$. Déterminer $\dim F_k$ pour tout $k\in[\![0,n}$. + +On considére dorenavant deux drapeaux $(F - {i\in\db{0,n]\!]}$ et $(G - {i\in[\![0,n}}$. + - Soient $i\in\db{1,n]\!]$, $j_0\in[\![0,n}$ tels que $F_{i-1}+G_{j_0}=F_i+G_{j_0}$. Montrer que, pour tout $j\geq j_0$, $F_{i-1}+G_j=F_i+G_j$. + - Soit $i\in\db{1,n]\!]$. Montrer qu'il existe $j\in[\![1,n}$ tel que $F_{i-1}+G_j=F_i+G_j$. + - Montrer que l'application $\sigma$ qui à $i$ associe $\min\{j\in\db{1,n]\!],\ F_{i-1}+G_j=F_i+G_j\}$ est une permutation de $[\![1,n}$. + - Montrer qu'il existe une base $(e_1,\ldots,e_n)$ de $E$ telle que $\forall i\in\db{1,n},\ e_i\in F_i\cap G_{\sigma(i)}$. + - Soit $A\in\op{GL}_n(\mathbb{K})$. Montrer qu'il existe une unique permutation $\tau\in\mc{S}_n$ pour laquelle il existe deux matrices $U$ et $V$ triangulaires supérieures dans $\M_n(\mathbb{K})$ vérifiant $A=UP_{\tau}V$ou $P_{\tau}=(\delta_{i,\tau(j)})_{1\leq i,j\leq n}$, et montrer qu'on peut en outre imposer que $1$ soit la seule valeur propre de $U$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 304] +On considére un groupe fini $G$ et un $\C$-espace vectoriel $V$ de dimension finie. Soit $\rho$ un morphisme injectif de $G$ dans $\mathrm{GL}(V)$. + - Calculer $\mathrm{tr}(\rho(e))$ ou $e$ est le neutre de $G$. + - Montrer que, pour tout $g\in G$, $\rho(g)$ est diagonalisable. + - Montrer que, si $\mathrm{tr}(\rho(g))=\mathrm{tr}(\rho(e))$, alors $\rho(g)=\rho(e)$. + - Soit $f:G\ra\C$. Pour $m\in\N$, on note $a_m=\sum_{g\in G}f(g)\left(\mathrm{tr}(\rho(g))\right)^m$. Démontrer qu'il existe $m\in\N$ tel que $a_m\neq 0$ lorsque $f(e)\neq 0$. + - Montrer que $\Phi:z\mapsto\sum_{m=0}^{+\i}a_mz^m$ est une fonction rationnelle. + - On prend $G=\mathfrak{S}_3$ et $\rho\colon\mathfrak{S}_3\ra\mathrm{GL}(V)$. + +Montrer qu'il existe une décomposition de $V$ sous la forme $\bigoplus_iE_i$ telle que : + +(i) $\forall i,\ \forall g\in G,\ E_i$ est stable par $\rho(g)$, (ii) $\forall i,\ \dim E_i\in\{1,2\}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 305] +Soit $d\geq 2$. On munit $\R^d$ de sa structure euclidienne canonique. Soient $\delta_1,\delta_2\gt 0$ avec $\delta_1\neq\delta_2$. Soient $x_1,\ldots,x_n\in\R^d$. On suppose que $\forall i\neq j$, $\|x_i-x_j\|\in\{\delta_1,\delta_2\}$. Montrer que $n\leq\dfrac{(d+1)(d+5)}{2}$. + +Ind. Montrer que les $f_i:y\mapsto\left(\left\|y-x_i\right\|^2-\delta_1^2\right)\left(\left\| y-x_i\right\|^2-\delta_2^2\right)$ sont lineairement indépendantes. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 306] +Pour $n\in\N^*$, soit $H_n=\big{\{}M\in\M_n(\{-1,1\})\;;\;M^TM=nI_n\big{\}}$. + - Déterminer $H_1$, $H_2$ et $H_3$. + - Soit $n\geq 4$ tel que $H_n\neq\emptyset$. Montrer que $4$ divise $n$. + - à l'aide de $A\in H_n$, construire une matrice $B\in H_{2n}$. + - Soit $p$ un nombre premier tel que $p\equiv 3\,[4]$. Montrer que $H_{p+1}$ n'est pas vide. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 307] +On munit $\R^3$ de sa structure euclidienne canonique. Soit $u\in\R^3$ unitaire. + +Soient $\sigma_u:x\mapsto x-2\left\langle x,u\right\rangle u$ et $\Omega_u=\big{\{}x\in\R^3\;;\;\left\langle x,u\right\rangle \geq 0\text{ et }\left\langle x,\sigma_u(x)\right\rangle\leq 0 \big{\}}$. + - Décrire et representer $\Omega_u$. + - Montrer que $\Omega_u$ est auto-dual, c'est-a-dire que $\Omega_u=\big{\{}y\in\R^3\;;\;\forall x\in\Omega_u,\;\left\langle x,y\right\rangle\geq 0\big{\}}$. + - On dit que $x\in\Omega_u$ est extremal si $\colon\forall x_1,x_2\in\Omega_u$, $x=x_1+x_2\Rightarrow x,x_1,x_2$ colineaires. + +Quels sont les points extremaux de $\Omega_u$? + - Si $f\in\mc{L}(\R^3)$, on dit que $f$ est extremal si $f(\Omega_u)\subset\Omega_u$ et, pour tous $g,h\in\mc{L}(\R^3)$ tels que $f=g+h$, $g(\Omega_u)\subset\Omega_u$, $h(\Omega_u)\subset\Omega_u$, on a $f,g,h$ colineaires. + +Déterminer les endomorphismes extremaux de rang 1. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 308] +On munit $\R^n$ de sa structure euclidienne canonique. Soient $(v_1,\ldots,v_n)\in\R^n\setminus\{0\}$ et $r=\mathrm{rg}(v_1,\ldots,v_n)$. On cherche à quelle condition il existe une base orthonormée $(f_1,\ldots,f_n)$ de $\R^n$ et un projecteur orthogonal $p$ tels que $\colon\forall i\in\db{1,n}$, $p(f_i)=v_i$. + - Traiter le cas $r=n$. - On suppose dans cette question que $n=2$ et $r=1$. Donner une condition nécessaire et suffisante dans ce cas. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 309] +Combien y a-t-il de matrices orthogonales de taille $n\in\N^*$ à coefficients dans $\Z$? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 310] +Un produit scalaire hermitien $\Phi$ sur le $\C$-espace vectoriel $E$ est une application $\Phi:E\times E\ra\C$ telle que : $\forall y\in E$, $x\mapsto\Phi(x,y)$ est lineaire ; $\forall(x,y)\in E^2$, $\Phi(y,x)=\overline{\Phi(x,y)}$ ; $\forall x\in E\setminus\{0\}$, $\Phi(x,x)\gt 0$. On note alors $\|x\|=\sqrt{\Phi(x,x)}$ pour $x\in E$. + - On munit $\C^2$ du produit scalaire hermitien tel que $\langle(x_1,x_2),(y_1,y_2)\rangle=x_1\overline{y_1}+x_2\overline {y_2}$. Soit $T$ l'endomorphisme de $\C^2$ dont la matrice dans la base canonique est $\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$. Déterminer $\left\{\langle Tx,x\rangle\ ;\ x\in\C^2,\ \|x\|^2=1\right\}$. + - On munit l'espace $\ell^2(\N,\C)$ des suites complexes $(u_n)_{n\geq 0}$ de carre sommable du produit scalaire défini par : $\langle u,v\rangle=\sum_{n=0}^{+\i}u_n\overline{v_n}$. Soit $T$ l'endomorphisme de $\ell^2(\N,\C)$ qui à $(u_n)_{n\geq 0}$ associe la suite $(u_{n+1})_{n\geq 0}$. Déterminer $\left\{\langle Tu,u\rangle\ ;\ u\in\ell^2(\N,\C),\ \|u\|^2=1\right\}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 311] +Soient $n\in\N^*$, $a_1\leq\cdots\leq a_n$ et $b_1\leq\cdots\leq b_n$ des nombres réels, $A$ et $B$ dans $\mc{S}_n(\R)$ telles que $\chi_A=\prod_{k=1}^n(X-a_k)$ et $\chi_B=\prod_{k=1}^n(X-b_k)$. Montrer que $\op{tr}(AB)\leq\sum_{k=1}^na_kb_k$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 312] +On munit l'espace $E=\R^n$ de sa structure euclidienne canonique. Soit $u$ un endomorphisme autoadjoint de $E$. On note $\lambda_1\leq\cdots\leq\lambda_n$ les valeurs propres de $u$. Soit $(e_1,...,e_n)$ une base orthonormée de $E$ telle que $\forall i\in\db{1,n},\ \langle u(e_i),e_i\rangle=\lambda_i$. + +Montrer que $(e_1,...,e_n)$ est une base de vecteurs propres de $u$. +#+end_exercice + + +** Analyse + +#+begin_exercice [X MP 2024 # 313] +Soit $E$ un espace vectoriel normé. Que dire d'une partie $A$ de $E$ à la fois ouverte et fermée? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 314] +Trouver une partie $A$ de $\R$ telle que $A$, $\check{A}$, $\overline{A}$, $\stackrel{{\circ}}{{A}}$ et $\stackrel{{\circ}}{{A}}$ soient toutes distinctes. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 315] +Soit $N$ une norme sur $\R^d$ (ou $d\geq 1$). + - Montrer que la boule unite fermée pour $N$ est fermée, bornée, d'interieur non vide, convexe et symétrique par rapport à $0$. + - Soit $C$ une partie non vide de $E$, fermée, bornée, d'interieur non vide, convexe et symétrique par rapport à $0$. Montrer qu'il existe une norme sur $\R^d$ dont $C$ est la boule unite fermée. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 316] +Soit $f\colon [0,1]\ra\R$. + - Montrer que si $f$ est continue alors le graphe de $f$ note $\Gamma_f$ est ferme dans $\R^2$. La reci-proque est-elle vraie? + - Montrer que si $\Gamma_f$ est compact alors $f$ est continue. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 317] +Soient $E$ l'ensemble des polynômes à coefficients dans $\{-1,0,1\}$ et $A$ l'ensemble des racines des polynômes non nuls de $E$. + - Trouver des propriétés de base sur $A$ (stabilité ou symétrie). + - Montrer que, pour tout $a\in A$, $|a|\lt 2$. + - Montrer que $\overline{A}=[-2,-1/2]\cup\{0\}\cup[1/2,2]$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 318] +Soit $E=\mc C^0([0,1],\R)$ muni de la norme infinie. + - Soit $h_1:E\ra\R$ définie par $h_1(f)=\sum_{\stackrel{{ p}}{{q}}\in 0\cap[0,1]\atop p\wedge q=1}f \left(\frac{p}{q}\right)\frac{1}{q^3}$. Montrer que $h_1$ est bien définie et continue. + - Soient $g\colon\R\ra\R$ croissante et $h_2:E\ra\R$ définie par $h_2(f)=\sup_{t\in[0,1]}g(f(t))$. + +Déterminer les points de continuité de $h_2$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 319] +Existe-t-il une fonction continue $f\colon\C\ra\C$ telle que $f\circ f=\exp$? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 320] + - Soit $A\in\M_n(\R)$, exprimer la norme subordonnée de $A$ relative à la norme infinie, puis à la norme 1. + - Montrer que si $\|A\|_{\mathrm{op},\i}\leq 1$ et $\|A\|_{\mathrm{op},1}\leq 1$, alors $\|A\|_{\mathrm{op},2}\leq 1$. + - Soit $A\in\mathrm{GL}_n(\R)$, montrer que $\inf_{B\notin\mathrm{GL}_n(\R)}\|B-A\|_{\mathrm{op},2}=\sqrt{\lambda_{ 1}}$, ou $\lambda_1$ est la plus petite valeur propre de $AA^T$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 321] +Soit $(u_n)$ une suite réelle majorée telle que $\forall n\in\N^*,\, u_n = \frac{1}{n}\sum_{k=n+1}^{2n} u_k$. Montrer que $(u_n)$ est constante. +#+end_exercice + +# ID:7668 +#+BEGIN_exercice [X MP 2024 # 322] +On définit la suite $(z_n)$ par $z_0\in\C^*$ et, pour tout $n\in\N$, $z_{n+1} = \frac{1}{2}\left(z_n + \frac{1}{z_n}\right)$. + - Lorsque $z_0\in\R^*$, étudier l'existence de la suite $(z_n)$ et sa convergence. + - Même question lorsque $z_0\in\C^*$. +#+END_exercice +#+BEGIN_proof + - $\R_+^*,\R_-^*$ stables. On a $z_{n+1} - \sqrt{2} = \frac{(z_n - \sqrt{2})^2}{2z_n}$, donc si $z_n$ est proche de $\sqrt{2}$, c'est plié. Pour $z_n$ positif, si $|z_n|\geq 2$, $z_{n+1}\lt z_n$. + - On a $|z_{n+1} - \sqrt{2}| = |z_n - \sqrt{2}| \frac{|z_n - \sqrt{2}|}{2 |z_n|}$, donc si $|z_n|\geq 2$, on se rapproche de $\sqrt{2}$. + + Par ailleurs, si $\Re(z_0)\gt 0$, on est bien défini et l'argument de $(z_n)$ décroît (en valeur absolue). Cela justifie que l'on ne peut pas s'approcher de $0$, donc $(z_n)$ est bornée, et toute valeur d'adhérence est réelle (sinon, on diminue strictement l'argument). + + Idem si $\Re(z)\lt 0$. Si $\re(z) = 0$, on tombe sur un des cas précédents. +#+END_proof + + +# ID:7669 +#+BEGIN_exercice [X MP 2024 # 323] + - Si $n\in\N^*$, montrer que l'équation $\sum_{k=1}^n x^k = 1$ admet une unique solution dans $\R_+$ que l'on note $a_n$. + - Montrer que $(a_n)_{n\geq 1}$ converge vers une limite $\l$ à déterminer. Donner un équivalent de $a_n - \l$. +#+END_exercice + +#+BEGIN_exercice [X MP 2024 # 324] +Soit $(a_n)_{n\geq 0}$ une suite strictement décroissante à termes dans $\interval]{0, 1}[$. Soient $\a\gt 0$ et $(u_n)$ définie par $u_0\geq 0$ et $\forall n\in\N,\, u_{n+1} = u_n (u_n^{\a} + a_n)$. Montrer qu'il existe un unique $u_0\geq 0$ tel que la suite $(u_n)$ converge vers un réel $\gt 0$. Déterminer alors cette limite. +#+END_exercice + + + +#+begin_exercice [X MP 2024 # 325] + - Soient $a,b\in\N^*$ avec $a\wedge b=1$. Montrer l'existence de $N\in\N^*$ tel que, pour tout $n\geq N$, il existe $(u,v)\in\N^2$ vérifiant $n=au+bv$. + - Soit $(s_n)_{n\geq 1}$ une suite strictement croissante d'éléments de $\N\setminus\{0,1\}$. On suppose que l'ensemble $S=\{s_n,\ n\in\N^*\}$ est stable par produit. + +Montrer que $\frac{s_{n+1}}{s_n}\ra 1$ si et seulement s'il existe $p,q\in\N^*$ tels que $\frac{\ln(s_p)}{\ln(s_q)}\not\in\Q$. +#+end_exercice +#+BEGIN_proof + +#+END_proof + + +#+begin_exercice [X MP 2024 # 326] +Soit $f\colon\R\ra\R$ 1-periodique. + +On définit $\colon\forall S\in\R^{\N^*},\ \forall n\in\N^*,\ M_n(f,S)=\frac{1}{n}\sum_{k=1}^nf(S_k)$. + - Montrer que la suite $(M_n(f,S))$ converge pour toute suite $S$ si et seulement si $f$ est constante. + - On dit qu'une suite réelle $(u_n)$ est équirépartie modulo 1 lorsque + + $\forall f\in\mc C(\R,\R)$ 1-periodique, $\frac{1}{n}\sum_{k=1}^nf(u_k)\xrightarrow[n\ra+\i]{}\int_0^1f(x) dx$. + +Montrer que la suite $(\sqrt{n})$ est equiperpartie modulo 1. +#+end_exercice + + +# ID:7670 +#+begin_exercice [X MP 2024 # 327] +Calculer la somme de la série de terme general $n^2 2^{-(n+1)}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 328] +Soit $\phi\colon\N^*\ra\N^*$ injective. Nature de $\sum\frac{\phi(n)}{n^2}$ ? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 329] +Déterminer la nature de la série $\sum\frac{\sin(\pi\sqrt{n})}{n^{\alpha}}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 330] +Soit $(u_n)$ une suite réelle strictement positive telle que la série $\sum u_n$ converge. + +Montr per que la série de terme general $v_n=\frac{1}{1+n^2u_n}$ diverge. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 331] + - Soit $(u_n)_{n\geq 0}\in(\R^{+*})^{\N}$. Pour $n\in\N$, on pose $S_n=u_0+\cdots+u_n$. On suppose que $\frac{S_n}{nu_n}\xrightarrow[n\ra+\i]{}a\gt 0$. Déterminer la nature de $(S_n)$. Donner un équivalent de $\frac{1}{u_n}\sum_{k=0}^nku_k$. + - Soient $(u_n)_{n\geq 0},(v_n)_{n\geq 0}\in(\R^{+*})^{\N}$. Pour $n\in\N$, on pose $S_n=u_0+\cdots+u_n$ et $T_n=v_0+\cdots+v_n$. On suppose que $\frac{S_n}{nu_n}\xrightarrow[n\ra+\i]{}a\in\R^{+*}$ et $\frac{T_n}{nv_n}\xrightarrow[n\ra+\i]{}b\in\R^{+*}$. Donner un équivalent de $\frac{1}{u_nv_n}\sum_{k=0}^nu_kv_k$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 332] +Déterminer les fonctions d rivables $f\colon\R\ra\R$ telles que + + $\forall(x,y)\in\R^2,\ f(x)f(y)=f(x+yf(x))$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 333] +Soit $f\colon\N^*\ra\R$ telle que $f(mn)=f(m)+f(n)$ pour tous $m,n\geq 1$. + - On suppose $f$ croissante. Montr per qu'il existe $c\in\R$ tel que $\colon\forall n\in\N^*$, $f(n)=c\ln n$. - On suppose que $f(n+1)-f(n)\ra 0$ quand $n\ra+\i$. Montrer qu'il existe $c\in\R$ tel que : $\forall n\in\N^*$, $f(n)=c\ln n$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 334] +Soit $E$ l'ensemble des $f\in\mc C^{\i}(\R,\R)$ telles que $f\underset{+\i}{\longrightarrow}0$. + +Si $f\in E$, on pose $\Delta(f):x\mapsto f(x+1)-f(x)$. + - Montrer que $\Delta$ est un endomorphisme de $E$. Est-ce un automorphisme? + - Soient $f\in E$, $x\in\R$ et $n\in\N^*$. + +Montrer qu'il existe $x_n\in\left]x,x+n\right[$ tel que $\Delta^n(f)(x)=f^{(n)}(x_n)$. + +Ind. Étudier $y\mapsto f(x+y)$ et $y\mapsto\sum_{k=0}^n\dfrac{y(y-1)\cdots(y-k+1)}{k!}\,\Delta^k(f)(x)$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 335] +Déterminer les $f\in\mc C^2([0,1],\R)$ telles que $\forall x\in[0,1]$, $f(x)=2(f(x/2)+f(1-x/2))$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 336] +Soient $N$ et $d$ deux entiers supérieurs ou egaux à 1. On pose $D=\left[\!\left[-N,N\right]\!\right]^d$ et on note $(e_1,\ldots,e_d)$ la base canonique de $\R^d$. + +On note $\partial D=\left\{\sum_{i=1}^dx_ie_i\,;\,(x_1,\ldots,x_n)\in D,\; \exists i\in\left[\!\left[1,d\right]\!\right],\;|x_i|=N\right\}$ et $\overset{\circ}{D}=D\setminus\partial D$. + +Pour $i\in\left[\!\left[1,d\right]\!\right]$ et $u:D\ra\R$, on pose $\forall x\in\overset{\circ}{D},\;\Delta_iu(x)=2u(x)-u(x+e_i)-u(x-e_i)$. + +On pose, pour $x\in\overset{\circ}{D}$, $Mu(x)=\prod_{i=1}^d\Delta_iu(x)$. + - Construire une fonction $u:D\ra\R^+$ concave, i.e. vérifiant $\forall i\in\left[\!\left[1,d\right]\!\right]$, $\Delta_iu\geq 0$, telle que $\forall x\in\overset{\circ}{D},Mu(x)\gt 0$ et $u|_{\partial D}=0$. + +Pour $f\colon\overset{\circ}{D}\ra\R^+$ fixée, on note $A$ l'ensemble des $h:D\ra\R^+$ concaves, nulles sur $\partial D$ et telles que $Mh\geq f$. Soit $u:x\mapsto\inf_{h\in A}h(x)$. + - Montrer que $A$ est non vide. + - Montrer que $u\in A$ et que $Mu=f$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 337] +Si $f$ est une fonction de $[0,1]$ dans $\R$, on note $V(f)$ la borne supérieure, dans $[0,+\i]$, de l'ensemble $\left\{\sum_{k=0}^{n-1}|f(a_{k+1}-f(a_k)|\;;\;n\in\N^*,0\leq a_0\leq a_1\cdots\leq a_n\leq 1\right\}$. On note $VB$ l'ensemble des fonctions $f$ de $[0,1]$ dans $\R$ telles que $V(f)\lt +\i$. + - Montrer que $VB$ est un sous-espace de l'espace vectoriel des fonctions $f$ de $[0,1]$ dans $\R$ contenant les fonctions monotones et les fonctions lipschitziennes. + - Donner un exemple de fonction continue de $[0,1]$ dans $\R$ n'appartenant pas à $VB$. + - Montrer qu'une fonction $f$ de $[0,1]$ dans $\R$ est dans $VB$ si et seulement si elle est différence deux fonctions croissantes de $[0,1]$ dans $\R$. + - Soit $f\in\R^{[0,1]}$. Montrer que les deux propriétés suivantes sont équivalentes : + +(i) $\forall g\in[0,1]^{[0,1]},V(g)\lt +\i\implies V(f\circ g)\lt +\i$; + +(ii) $f$ est lipschitzienne. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 338] + - Soit $f\colon\R^{+*}\ra\R$ une fonction convexe. - Montrer qu'il existe $\ell\in\overline{\R}$ tel que $\frac{f(x)}{x}\xrightarrow[x\ra+\i]{}\ell$; déterminer les valeurs possibles de $\ell$. - Si $\ell\in\R$, montrer que $f(x)-\ell x$ possede une limite dans $\overline{\R}$ quand $x$ tend vers $+\i$ et déterminer les limites possibles. + - Soient $f,g$ convexes et continues sur $[0,1]$ vérifiant $\max(f,g)\geq 0$. + +Montrer qu'il existe $\alpha,\beta$ positifs et non tous nuls tels que $\alpha f+\beta g\geq 0$. + - Soient $f_1,\ldots,f_n:[0,1]\ra\R$ convexes et continues vérifiant $\max(f_1,\ldots,f_n)\geq 0$. + +Montrer qu'il existe $\alpha_1,\ldots,\alpha_n$ positifs et non tous nuls vérifiant $\sum_{k=1}^n\alpha_kf_k\geq 0$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 339] +Soient $f\in\mc C^{\i}(\R,\R)$ et $(a,b)\in\R^2$ avec $a\lt b$. Montrer l'équivalence entre : + +(i) $f$ n'est pas polynomiale, + +(ii) Vect $\big(\{x\mapsto f(\alpha x+\beta)\;;(\alpha,\beta)\in\R^2\}\big)$ est dense dans $\mc C^0([a,b],\R)$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 340] +Soient $F$ un ferme de $\R$, $O=\R\setminus F$. + - Montrer que $O$ est reunion dénombrable d'intervalles ouverts bornes. + - Montrer qu'il existe une fonction $f$ de classe $\mc C^{\i}$ de $\R$ dans $\R$ telle que $F=f^{-1}(\{0\})$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 341] +On munit $E=\mc C^0([0,1],\R)$ de la norme $\|\;\|_{\i}$. + +Pour $f\in E$, soit $T(f):t\in[0,1]\mapsto\sup_{[0,t]}(f)-f(t)$. Soit $f\in E$. + - Montrer que $T(f)$ est continue, que $T(f)\geq 0$ et que $T(f)(0)=0$. + - Montrer que la suite $(\|T^n(f)\|)_{n\geq 0}$ est decroissante. + - Si $f$ est $K$-lipschitzienne, montrer que $T(f)$ est lipschitzienne. + - Soit $f\in E$ lipschitzienne. Montrer que $(T^nf)$ converge uniformément vers la fonction nulle. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 342] +Soit $f:[0,1]\ra[-a,b]$ continue, ou $a$ et $b$ sont dans $\R^+$. On suppose que $\int_0^1f(t)dt=0$. + +Montrer que $\int_0^1f(t)^2dt\leq ab$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 343] +Pour $r\in\R$ et $n\in\N$, soit $D_n(r)=\int_{-1}^1(1-x^2)^n\cos(rx)dx$. + - Montrer que, pour tout $n\in\N$, il existe $P_n$ et $Q_n$ des polynômes à coefficients entiers de degre au plus $n$ tels que, pour tout $r\in\R$, $D_n(r)=\frac{n!}{r^{2n+1}}(P_n(r)\cos(r)+Q_n(r)\sin(r))$. + - En déduire que $\pi$ est irrationnel. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 344] +$\;\;\;\;$Soinent $f\colon\R\ra\R$ de classe $\mc C^1$ à support compact et $E$ l'ensemble des fonctions $\phi$ de $\R$ dans $\R$, de classe $\mc C^1$ bornées par $1$. Déterminer $\sup\bigg{\{}\int_{-\i}^{+\i}f\phi'\;;\;\phi\in E \bigg{\}}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 345] +Nature de $\int_0^{+\i}\frac{e^x}{e^{-x}+e^{2x}|\sin x|}dx$? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 346] +Soit $f\colon\R\ra\R$ intégrable et lipschitzienne. Peut-il exister un réel $x$ non nul tel que la série de terme general $f(nx)$ diverge? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 347] + - Soit $(f_n)$ une suite de $\mc C^1([0,1],\R)$ convergeant uniformément vers une fonction $f$ sur $[0,1]$. On suppose que, pour toute fonction $g\in\mc C^1([0,1],\R)$, $\int_0^1f'_ng\longrightarrow 0$ quand $n\ra+\i$. Que dire de $f$? + - Soit $x\in\R$. Calculer $\sum_{n\in\N^*}\frac{\cos(nx)}{n^2}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 348] +Montrer que $\sum_{n=0}^{+\i}\big(1-(1-e^{-n})^x\big)\sim\ln(x)$ quand $x\ra+\i$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 349] +Soit $q\in\R^*$. Soit $a\in\mc C^0(\R,\R^*)$. Soit $m,M$ deux réels vérifiant $:0\lt m\lt M$ et $m\leq|a|\leq M$. On suppose egalement que $m\gt 2$ ou $M\lt \frac{1}{2}$. Montrer qu'il existe une unique fonction $F\colon\R\ra\R^*$ continue et bornée vérifiant $\colon\forall t\in\R,F(t)=1+\frac{F(qt)}{a(t)}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 350] +Soit $\sum a_nz^n$ une série entiere dont le rayon de convergence appartient à $]0,+\i[$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 351] +Soit $x\gt 0$. + - Montrer que $\colon\forall n\in\N,\sum_{k=0}^{2n+1}(-1)^k\frac{x^k}{k!}\lt e^{-x}\lt \sum_{k=0}^{2n}(-1)^k\frac{x^k}{k!}$. + - Montrer que $\colon\forall n\in\N,\sum_{k=0}^{2n+1}(-1)^k\frac{x^{2k+1}}{2k+1}\lt \arctan x\lt \sum_{k=0}^{2n}(-1)^k\frac{x^{2k+1}}{2k+1}$. + - Montrer que $\forall n\in\N,\sum_{k=0}^{2n+1}\frac{(-1)^kx^{2k}}{4^k(k!)^2}\lt \frac{2}{\pi}\int_0^1\frac{\cos(xt)}{\sqrt{1-t^2}}\dt\lt \sum_{k=0 }^{2n}\frac{(-1)^kx^{2k}}{4^k(k!)^2}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 352] +Montrer que, pour tous $r\in$ ] $0,1[$ et $\theta\in\R$, $\ln\left|1-re^{i\theta}\right|=-\sum_{n=1}^{+\i}\frac{r^n}{n}\cos(n\theta)$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 353] +Soit $f\colon\R\ra\R$ de classe $\mc C^{\i}$ telle que $\forall n\in\N,\ \forall x\in\R,\ f^{(n)}(x)\geq 0$. + - On suppose que $f(0)=0$. Montrer que $\forall x\leq 0,\ f(x)=0$. + - On suppose que $f(0)=0$. Montrer que $\forall x\geq 0,\forall n\in\N^*,\ f(x)\leq\frac{x}{n}f^{ '}(x)$. Que peut-on en déduire? + - Démontrer que $\forall x\in\R$, $\sum_{k=0}^n\frac{f^{(k)}(0)}{k!}x^k\xrightarrow[n\ra+\i]{}f(x)$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 354] +Soit $(L_n)_{n\geq 0}$ définie par $L_0=L_1=1$ et, si $n\geq 1$, $L_{n+1}=(n+1)L_n-\binom{n}{2}L_{n-2}$, avec $L_{-1}=0$. On pose $f:x\mapsto\sum_{n=0}^{+\i}\frac{L_n}{n!}\,x^n$. + - Montrer que le rayon de convergence de $f$ est strictement positif. + - Montrer que $\frac{L_n}{n!}\ra 0$. + - Déterminer $f$. Ind. Trouver une équation différentielle vérifiée par $f$. + - En déduire un équivalent de $\frac{L_n}{n!}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 355] +Une série $\sum_{n\geq 0}a_n$ est dite primitive lorsqu'elle est à termes entiers et il n'existe pas d'entier $d\gt 1$ divisant tous les $a_n$. + - Soit $\sum_{n\geq 0}a_n$ et $\sum_{n\geq 0}b_n$ deux séries primitives. Montrer que leur produit de Cauchy est une série primitive. + - Soit $F(z)=\sum_{n\in\N}c_nz^n$, ou $c_n\in\Z$ pour tout $n$, telle qu'il existe $P$ et $Q$ dans $\C[X]$ avec $P\wedge Q=1$ et $Q(0)=1$, tels que, pour $z$ voisin de 0, on ait $F(z)=\frac{P(z)}{Q(z)}$. Montrer que $(P,Q)\in\Q[X]^2$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 356] +Soit $n\geq 2$. On pose $g_n=\sum_{k=0}^n\frac{1}{2^{4k}}\binom{2k}{k}^2$. Soit $K_n$ l'élément de $\R_n[X]$ tel que + + $\frac{1}{\sqrt{1-x}}\underset{x\ra 0}{=}K_n(x)+o(x^n)$. + - Montrer que $\frac{1}{2\pi}\int_0^{2\pi}\left|K_n\left(e^{i\theta} \right)\right|^2d\theta=g_n$. + - Soit $\sum a_kz^k$ une série entiere de rayon de convergence supérieur ou egal à $1$, de somme $f(z)$. On suppose que, pour $|z|\lt 1$, $|f(z)|\leq 1$. Montrer que $\left|\sum_{k=0}^na_k\right|\leq g_n$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 357] +Déterminer la limite de la suite de terme general $u_n=n\int_0^{+\i}\sin(t^n)\dt$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 358] +Soit $r\in$]0, $\pi[$. Déterminer la limite de la suite de terme general $u_n=\int_{-r}^r\frac{\sin(nt)}{\sin t}\dt$. +#+end_exercice + + +# ID:7454 +#+begin_exercice [X MP 2024 # 359] +Déterminer un équivalent en $1^-$ de $x\mapsto\int_0^1\frac{1}{\sqrt{(1-t^2)(1-xt^2)}}dt$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 360] +Calculer $f(x)=\int_{\R}\frac{e^{ixt}}{1+t^2}dt$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 361] +Soit $t\gt 0$. Our $p\in\R$, on pose $F_c(p)=\int_0^{+\i}e^{-tx^2}\cos(px^2)\dx$, + + $F_s(p)=\int_0^{+\i}e^{-tx^2}\sin(px^2)\dx$ et $Z=F_c+iF_s$. + - Montrer que $Z$ est de classe $\mc C^{\i}$ sur $\R$. + - Déterminer une équation différentielle du premier ordre satisfaite par $F$. + - En déduire $F_c$ et $F_s$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 362] +Soit $f\colon\R^+\ra\R^{+*}$ de classe $\mc C^1$ telle que $\frac{xf'(x)}{f(x)}\ra a\in\R$ quand $x\ra+\i$. + - Soit $m\in\R^{+*}$. Montrer que $x\mapsto\frac{f(mx)}{f(x)}$ admet une limite en $+\i$ ; la calculer. + +Soit $I:t\mapsto\int_0^{+\i}e^{-tx}f(x)dx$. + - Montrer que $I$ est définie sur $\R^{+*}$. + - Montrer que $I$ admet une limite finie en $+\i$. + - Supposons $a\lt -1$. Déterminer la limite de $I$ en $0^+$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 363] + - Soient $I$ et $J$ deux segments de $\R$, et $f:I\times J\ra\R$ continue. Montrer l'existence et l'egalite des deux quantites $\int_I\left(\int_Jf(x,y)dy\right)dx$ et $\int_J\left(\int_If(x,y)dx\right)dy$. + - Pour $\alpha\in\,]0,1[$ et $f\colon\R\ra\R$ de classe $\mc C^1$ telle que $f^2$ et $f^{' 2}$ sont intégrables sur $\R$, on note $\|f\|_{\alpha}^2=\int_{\R}\left(\int_{\R}\frac{|f(x)-f(y)|^ {2}}{|x-y|^{1+2\alpha}}dy\right)dx$. Montrer que $\|\ \|_{\alpha}$ définit une norme sur l'espace vectoriel $\{f\in\mc C^1(\R,\R),\ (f,f')\in L^2( \R,\R)^2\}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 364] +Soient $n\in\N^*$, $(a_{i,j})_{1\leq i,j\leq n}$ des éléments de $\R^{+*}$, $f_1,\ldots,f_n$ des fonctions dérivables de $\R^+$ dans $\R$ tendant vers $0$ en $+\i$ telles que, pour tout $i\in\db{1,n}$, $f'_i=\sum_{j=1}^na_{i,j}f_j$. Montrer que la famille $(f_1,\ldots,f_n)$ est liée. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 365] + - Soit $f\in\mc C^1([0,\pi],\R)$ telle que $f(0)=f(\pi)=0$. Montrer que $\int_0^{\pi}f^2\leq\frac{\pi^2}{8}\int_0^{\pi}(f')^2$. + - Soit $f,q\in\mc C^0([0,\pi],\R)$ telle que $\forall x\in[0,\pi],\ q(x)\lt \frac{8}{\pi^2}$. Soient $a,b\in\R$. Montrer qu'il existe une unique fonction $y\in\mc C^2([0,\pi],\R)$ telle que $y''+qy=f,\ y(0)=a,\ y(\pi)=b$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 366] +Pour $f\in\mc C^{\i}(\R,\R)$, on pose $H(f):x\mapsto x^2f(x)-f''(x)$, $A_-(f):x\mapsto-f'(x)+xf(x)$ et $A_+(f):x\mapsto f'(x)+xf(x)$. + - Déterminer $A_-\circ A_+$ et $A_+\circ A_-$. + - Montrer qu'il existe une unique $\phi_0\in\mc C^{\i}(\R,\R)$ de carre intégrable, telle que $H(\phi_0)=\phi_0$ et $\phi_0(0)=1$.On pose, pour $n\in\N^*$, $\phi_n=A_-^n(\phi_0)$. + - Montrter que, pour tout $n\in\N$, $H(\phi_n)=(2n+1)\phi_n$. + - Montrter que $\phi_n$ s'écrit sous la forme $P_n\times\phi_0$ avec $P_n$ polynomiale. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 367] + - Soit $f$ une fonction croissante de $[a,b]$ dans $[a,b]$. Montrter que $f$ possede un point fixe. + - On s'interesse à l'équation différentielle $(E)$ $x'(t)=\cos(x(t))+\cos(t)$. On admet que, pour tout $a\in[0,\pi]$, il existe une unique solution $\phi_a$ définie sur $\R$ telle que $\phi_a(0)=a$, et de plus que s'il existe $t$ tel que $\phi_a(t)=\phi_b(t)$ alors $a=b$. + +Montrter qu'il existe une unique solution $\phi_a$ de $(E)$ qui est $2\pi$-periodique. + +_Ind._ Montrter que toute solution $\phi_a$ est à valeurs dans $[0,\pi]$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 368] + - Soit $x$ de classe $\mc C^1$ au voisinage de $+\i$. On suppose qu'il existe $\tau\gt 0$ et $\lambda\gt 0$ tels qu'on ait $x'(t)+\lambda x(t-\tau)\leq 0$ et $x(t)\geq 0$ au voisinage de $+\i$. + +Démontrer que $x(t-\tau)\leq\frac{4}{(\lambda\tau)^2}x(t)$ au voisinage de $+\i$. + - Soient $x$ de classe $\mc C^1$ sur $\R$, $m$ et $n$ dans $\N^*$, $\lambda_1,\ldots,\lambda_n,\mu_1,\ldots,\mu_m$ des réels, $\tau_1,\ldots,\tau_n$, des réels strictement positifs, $\sigma_1,\ldots,\sigma_m$ des réels positifs. + +On suppose que $\forall t\in\R,\ x'(t)+\sum_{i=1}^n\lambda_ix'(t- \tau_i)+\sum_{i=1}^m\mu_ix(t-\sigma_i)=0$. + +Démontrer qu'il existe $c$ et $K$ réels tels que, pour $t$ au voisinage de $+\i$, $|x'(t)|\leq Ke^{ct}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 369] + - Soient $f,g\colon\R^+\ra\R$ des fonctions continues et $K$ un réel strictement positif. On suppose que, pour tout $t\in\R^+$, $f(t)\leq g(t)+K\int_0^tf(u)\,du$. + +Montrter que, pour tout $t\in\R^+$, $f(t)\leq g(t)+K\int_0^te^{K(t-u)}g(u)\,du$. + - Soient $A,B\colon\R^+\ra\M_n(\R)$ des fonctions continues, et $M,N\colon\R^+\ra\M_n(\R)$ de classe $\mc C^1$. On suppose que $\forall t\in\R^+$, $M'(t)=A(t)M(t)$, $N'(t)=B(t)N(t)$ et que $M(0)=N(0)=I_n$. + +On suppose de plus que $\|A(t)\|\leq K$ et $\|A(t)-B(t)\|\leq\eta$ pour tout $t\in[0,T]$, ou $K,\eta,T$ sont des réels strictement positifs, et $\|\ \|$ une norme subordonnée sur $\M_n(\R)$. + +Montrer que, pour tout $t\in[0,T]$, $\|M(t)-N(t)\|\leq e^{Kt}\left(e^{\eta t}-1\right)$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 370] +On munit $\R^2$ de la norme euclidienne canonique. Soit $P\colon\R^2\ra\R$ une fonction polynomiale à valeurs positives. + - La fonction polynomiale $P$ atteint-elle nécessairement un minimum? + - On suppose que $P(x,y)\underset{\|(x,y)\|\ra+\i}{\longrightarrow}+\i$. La fonction polynomiale $P$ atteint-elle nécessairement un minimum? + - On garde l'hypothese précédente. On note $S(0,1)$ le cercle unite. + +Montrer que $\colon\forall(x,y)\in S(0,1),\exists C(x,y)\in\R^{+*}\cup\{+\i\}, \lim_{t\ra+\i}\frac{P(tx,ty)}{t^2}=C(x,y)$. + - Montrer que $C$ est à valeurs dans $\R^{+*}$ ou qu'il n'existe qu'un nombre fini de couples $(x,y)$ tels que $C(x,y)\lt +\i$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 371] +Soient $u_0,u_1\in\mc C^{\i}(\R,\R)$. Déterminer les fonctions $u\colon\R^2\ra\R$ de classe $\mc C^{\i}$ telles que $\frac{\partial^2u}{\partial x^2}-\frac{\partial^2u}{\partial t^2}= \left(\frac{\partial u}{\partial x}\right)^2-\left(\frac{\partial u}{ \partial t}\right)^2,$ avec $u(t=0,\cdot)=u_0$ et $\frac{\partial u}{\partial t}(t=0,\cdot)=u_1$. + +Ind. On utilisera la fonction $U=f\circ u$ avec $f\in\mc C^{\i}(\R,\R)$ convenable. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 372] +Soient $n\in\N^*$ et $r\in\db{0,n}$, $\mc{P}$ l'ensemble des projecteurs orthogonaux de $\R^n$ sur un sous-espace de dimension $r$ et $p\in\mc{P}$. Déterminer l'ensemble des vecteurs tangents à $\mc{P}$ en $p$. +#+end_exercice + + +** Geometrie + +#+begin_exercice [X MP 2024 # 373] +Soit $P$ un polynôme réel de degre $6$. Une droite $D$ est tangente à la courbe $C_P$ en trois points $A,B,C$ d'abscisses $a\lt b\lt c$. + - On suppose que $AB=BC$. Montr er que les aires delimitées par $[BC]$ et $C_P$ d'une part, et par $[AB]$ et $C_P$ d'autre part, sont egales. + - On pose : $q=\frac{BC}{AB}$ et $Q=\frac{A_1}{A_2}$ avec $A_1$ et $A_2$ les aires susmentionnées. Montr erque : $\frac{2}{7}q^5\leq Q\leq\frac{7}{2}q^5$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 374] +On se place dans le plan $\R^2$. Soient $e_0=(1,0)$, $e_1=(0,1)$ $e_2=(-1,0)$, $e_3=(0,-1)$ et, pour $k\geq 4$, $e_k=e_{k\bmod 4}$. Soit $P\in\R[X]$. On écrit $P=c_0X^n-c_1X^{n-1}+\cdots+(-1)^nc_n$. On pose $M_{-1}(P)=(0,0)$, et pour $k\in\db{0,n}$, $M_k(P)=M_{k-1}(P)+c_k\,e_k$. Pour $k\in\N$, soit $D_k$ la droite passant par $M_k(P)$ dirigée par $e_k$. Soit $\lambda\in\R$. On pose $\Delta_1(\lambda)$ la droite passant par $(0,0)$ de pente $\lambda$, $\Delta_0(\lambda)$ la perpendicularaire à $\Delta_1(\lambda)$ et passant par $(0,0)$ et, pour $k\geq 2$, $\Delta_k(\lambda)=\Delta_{(k\bmod 2)}(\lambda)$. + +On pose $\mu_0=(0,0)$. Pour $k\in\N^*$, $\mu_k$ est l'intersection de $D_k$ et de la parallele à $\Delta_k(\lambda)$ passant par $\mu_{k-1}$. + - On suppose dans cette question que $P=X^3-2X^2-5X+6$. + - Déterminer les racines de $P$. + - Pour chaque racine $\lambda$ de $P$, construire $M_3$ et $\mu_3$. + - Que peut-on conjecturer? + - En notant $\delta_k$ la distance algebrique selon $e_k$ de $M_k$ à $\mu_k$, montrer que $M_n=\mu_n$ si et seulement si $P(\lambda)=0$. +#+end_exercice + + +** Probabilités + +#+begin_exercice [X MP 2024 # 375] +Soient $v_1,\ldots,v_n$ des vecteurs unitaires d'un espace euclidien. Montrer qu'il existe $(\eps_1,\ldots,\eps_n)\in\{-1,1\}^n$ tel que $\left\|\sum_{i=1}^n\eps_iv_i\right\|\leq\sqrt{n}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 376] +Soit $E$ un ensemble fini. Dénombrer les triplets $(A,B,C)$ de parties de $E$ telles que $A\subset B\subset C$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 377] +Soit $r\in\N^*$. Combien y a-t-il de facon d'apparier les entiers de $1$ à $2r$? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 378] +Soit $n\in\N^*$. + - Dénombrer les décompositions $n=n_1+\cdots+n_r$ ou $r\geq 1$ est arbitraire, et $n_1,\ldots,n_r$ sont des entiers naturels non nuls. + - On fixe $r\in\N^*$. Dénombrer les décompositions $n=n_1+\cdots+n_r$ ou $n_1,\ldots,n_r$ sont des entiers naturels non nuls. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 379] + - Dénombrer les triplets $(A,B,C)$ de parties deux à deux disjointes de $\db{1,n}$. + - Soit $N\in\N^*$. Dénombrer les fonctions $f\colon\db{0,2N]\!]\ra[\![0,2N]\!]$ telles que $f(0)=f(2N)=0$ et $\forall k\in[\![0,2N-1},\;|f(k+1)-f(k)|=1$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 380] +Soit $(X_n)_{n\geq 1}$ une suite de variables aléatoires i.i.d. de loi uniforme sur $\{-1,1\}$. + +On pose $S_n=\sum_{k=1}^nX_k$, et on note $N\colon\omega\mapsto\op{card}\{n\in\N^*,\;S_n(\omega)=0\} \in\N\cup\{+\i\}$. + - Montrer que $\mathbf{E}(N)=+\i$. + - Exprimer $\mathbf{P}(N\geq 2)$ en fonction de $\mathbf{P}(N\geq 1)$. + - Montrer que $\mathbf{P}(N=+\i)=1$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 381] +Soit $n\in\N^*$. Déterminer esperance et variance du nombre de points fixes d'une permutation de $\db{1,n}$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 382] +On munit $\mc{S}_n$ de la loi uniforme et on considére $X_n$ la variable aléatoire qui associe à une permutation le nombre d'orbites de cette permutation. + - Calculer $\mathbf{P}(X_n=1)$ et $\mathbf{P}(X_n=n)$. + - Déterminer la fonction generatrice de $X_n$. + - En déduire des équivalent de $\mathbf{E}(X_n)$ et $\mathbf{V}(X_n)$ quand $n\ra+\i$. + - Comment peut-on déterminer la loi de $X_n$? +#+end_exercice + + +#+begin_exercice [X MP 2024 # 383] + - Déterminer le nombre de listes de $k$ entiers non consécutifs dans l'intervalle d'entiers $\db{1,n}$. + - On place aléatoirement des couples $(A_i,B_i)$, ou $i\in\{1,\ldots,n\}$, autour d'une table ronde à $2n$ places, de sorte qu'aucun des $A_i$ ne soit assis à cote d'un autre $A_j$. On cherche la probabilité $p_n$ que $A_i$ et $B_i$ ne soient pas à cote. Montrer que, si la configuration des $A_i$ est fixée, la probabilité que $A_i$ et $B_i$ ne soient pas à cote est inchangée. En déduire une expression sommatoire de $p_n$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 384] +Soit $s$ un $\op{\mathsf{r}\acute{e}el}\gt 1$. On munit $\N^*$ de la probabilité $\mathbf{P}_s$ définie par $\mathbf{P}_s(\{n\})=\frac{1}{n^s\zeta(s)}$ pour tout $n\geq 1$. On note par ailleurs $\mc{P}$ l'ensemble des nombres premiers. Pour tout $p\in\mc{P}$ on note $X_p$ la variable aléatoire définie sur $\N^*$ telle que $X_p(n)=1$ si $p$ divise $n$, et 0 sinon. + - Montrer que les variables aléatoires $X_p$ sont mutuellement indépendantes. + - En déduire que $\zeta(s)=\prod_{p\in\mc{P}}\frac{1}{(1-p^{-s})}$. + - Pour $p\in\mc{P}$ et $n\in\N^*$, on note $v_p(n)$ la plus grande puissance de $p$ qui divise $n$. Déterminer la loi de $v_p$ étudier l'indépendance mutuelle des variables aléatoires $v_p$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 385] +On joue à pile ou face avec probabilité $p\in]0,1[$ d'obtenir pile. On decoupe la succession des lancers en sequences maximales de résultats identiques. Déterminer l'esperance de la longueur de la deuxieme sequence. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 386] +Une grille $\{1,2,3\}\times\{1,2,...,n\}$ modelise un tuyau vertical. On depose à l'instant $t=0$ une goutte d'eau au point $(2,n)$. à chaque instant, si elle se trouve au milieu (i.e. en un point $(2,k)$), la goutte descend d'un niveau avec probabilité $\frac{1}{2}$ ou se deplace à droite (resp. gauche) avec probabilité $\frac{1}{4}$; si elle se trouve sur un bord, elle descend avec probabilité $\frac{1}{2}$ ou va au milieu avec probabilité $\frac{1}{2}$. + - Calculer la probabilité pour que la goutte sorte du tuyau à un instant $t$. + - Calculer l'esperance du temps d'attente pour que l'eau sorte du tuyau. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 387] + - Soient $n\in\N^*$ et $X,Y$ deux variables aléatoires indépendantes suivant la loi uniforme sur les entiers pairs entre $2$ et $2n$. Déterminer $\mathbf{P}(|X-Y|\leq 1)$ et $\mathbf{P}(|X-Y|\leq 2)$. + - Soient $n\in\N^*$ et $X_1,...,X_n$ des variables aléatoires à valeurs dans $\Z$, indépendantes et identiquement distribuées. Pour $m\in\N$, on pose : + + $S_m(n)=\big{|}\{(i,j)\in\db{1,n}^2\,;\;|X_i-X_j| \leq m\}\big{|}$. + +Montrer que $\mathbf{E}(S_m(n))=n+n(n-1)\mathbf{P}(|X_1-X_2|\leq m)$. + - Soit $(x_n)\in\Z^{\N^*}$. + +Pour $n\in\N^*$ et $m\in\N$, on pose : $s_m(n)=\big{|}\{(i,j)\in\db{1,n}^2\,;\;|x_i-x_j| \leq m\}\big{|}$. + +Montrer que pour tout $n\in\N^*$, $s_2(n)\leq 3s_1(n)$. + - En déduire que, si $X,Y$ sont deux variables aléatoires à valeurs dans $\Z$, indépendantes et de même loi, alors $\mathbf{P}(|X-Y|\leq 2)\leq 3\,\mathbf{P}(|X-Y|\leq 1)$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 388] +Soit, pour $n\in\N^*$, $X_n$ une variable aléatoire qui suit la loi uniforme sur $\db{1,n}$. On pose : $A_n=\big(\sqrt{X_n}$ admet $1$ pour 1er chiffre apres la virgule $\big)$ et + + $B_n=\big(\sqrt{X_n}$ admet $1$ pour 1er chiffre $\big)$, $C_n=\big(2^{X_n}$ admet $1$ pour 1er chiffre $\big)$. + - Étudier l'existence et, le cas echeant, calculer la limite de la suite $(\mathbf{P}(A_n))$. + - Étudier l'existence et, le cas echeant, calculer la limite de la suite $(\mathbf{P}(B_n))$. + - Étudier l'existence et, le cas echeant, calculer la limite de la suite $(\mathbf{P}(C_n))$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 389] +On dit qu'une variable aléatoire $Y$ est $k$-divisible lorsqu'elle à la même loi que la somme de $k$ variables indépendantes et identiquement distribuées. + - On suppose que $Y\sim\mc{B}(n,p)$. Pour quels entiers $k\gt 0$ la variable $Y$ est-elle $k$-divisible? + - Construire une variable aléatoire $Y$ non constante infiniment divisible. + - Soit $Y$ une variable aléatoire bornée infiniment divisible. Montrer que $Y$ est constante presque surement. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 390] +Soient $\alpha\gt 0$ et $(B_i)_{i\in\N^*}$ une suite de variables aléatoires indépendantes telle que $\mathbf{P}(B_i=1)=1-\mathbf{P}(B_i=0)=\frac{1}{i^{\alpha}}$. Soit $S=\{n\in\N^*,B_n=1\}$. + - Donner une condition sur $\alpha$ pour que $S$ soit infini presque surement, puis pour que $S$ soit fini presque surement. - On suppose $\alpha\lt 1$. On pose $\beta\gt 0$ et $N=\max\{n\in\N^*,S\cap\db{n,n+n^{\beta}}=\emptyset\}$. Donner des conditions sur $\beta$ pour que $\mathbf{P}(N=+\i)=1$ et pour que $\mathbf{P}(N=+\i)=0$. + - Montrer que, presque surement, il existe un rationnel $\gamma$ tel que $\lfloor\gamma^{2^n}\rfloor\not\in S$ pour tout $n\in\N$. +#+end_exercice + + +#+begin_exercice [X MP 2024 # 391] +Soient $N\geq 1$, $\mu$ une distribution de probabilité sur $\db{1,N}$ telle que $\mu(1)\gt 0$, $(X_n)_{n\geq 1}$ une suite i.i.d. de variables aléatoires suivant la loi $\mu$. On pose $S_0=0$ et, pour $n\in\N^*$, $S_n=X_1+\cdots+X_n$. Soit $E=\{S_m,\ m\in\N\}$. + - Pour $n\in\N^*$, montrer que $\mathbf{P}(n\in E)=\sum_{k=1}^N\mu(k)\,\mathbf{P}(n-k\in E)$. + +On pose $F:z\mapsto\sum_{n=0}^{+\i}\mathbf{P}(n\in E)z^n$ et $G:z\mapsto\sum_{k=1}^N\mu(k)z^k$. + - On pose $\mathbb{D}=\{z\in\C,\ |z|\lt 1\}$. Montrer que $\colon\forall z\in\mathbb{D}$, $F(z)=\frac{1}{1-G(z)}$. + - Montrer que $1$ est un pole simple de $F$ et tous les autres poles de $F$ ont un module strictement supérieur à 1. + - Montrer que $\mathbf{P}(n\in E)\underset{n\ra+\i}{\longrightarrow}\frac{1}{\mathbf{E}( X_1)}$. +#+end_exercice + + +* X - PSI :autre: + +** Algèbre + +#+begin_exercice [X PSI 2024 # 392] +Soit $n\in\N^*$. On pose $P_0=1$ et, pour $1\leq k\leq n$, $P_k=\prod_{j=0}^{k-1}(X-j)$. Montrer qu'il existe $(s_0,\ldots,s_n)\in\Z^{n+1}$ tel que $X^n=\sum_{k=0}^ns_kP_k$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 393] +Soit $E$ un $\R$ espace vectoriel de dimension finie. + - Quels sont les endomorphismes de $E$ qui commutent avec tous les projecteurs? + - Quels sont les éléments de $\op{GL}(E)$ qui commutent avec tous les éléments de $\op{GL}(E)$? +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 394] +Soient $A_1,\ldots,A_m$ des matrices distinctes de $\M_n(\R)$, commutant entre elles et telles que $\colon\forall i\in\db{1,m}\,,A_i^2=I_n$. Montrer que $m\leq 2^n$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 395] +Soient $A,B\in\M_n(\C)$. Montrer que $A$ et $B$ ont une valeur propre commune si et seulement s'il existe $C\in\M_n(\C)$ non nulle telle que $AC=CB$. +#+end_exercice + + +** Analyse + +#+begin_exercice [X PSI 2024 # 396] +Soit $n\in\N^*$. Montrer qu'il existe $c\gt 0$ tel que, pour tout $(v_1,\ldots,v_n)\in(\R^n)^n$, $|\det(v_1,\ldots,v_n)|\leq c\prod_{j=1}^n\|v_i\|_{\i}$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 397] +Déterminer les $f\in{\cal C}^0({\R},{\R})$ telles que: $\forall(x,y)\in{\R}^2$, $f(x)f(y)=\int_{x-y}^{x+y}f(t)\,{\rm d}t$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 398] +On note ${\cal S}({\R})=\{\phi\in{\cal C}^{\i}({\R},{\R}) \;;\;\forall k\in{\N},\,\forall j\in{\N},\,x\mapsto x^k \phi^{(j)}(x)$ est bornée}. + - Montrer que $\forall k\in{\N},\,\forall j\in{\N},\,x\mapsto x^k\phi^{( j)}(x)$ est intégrable sur ${\R}$. + - Soit $\phi\in{\cal S}({\R})$. Déterminer une condition nécessaire et suffisante sur $\phi$ pour que $\phi$ possede une primitive appartenant à ${\cal S}$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 399] +On munit ${\R}^n$ de son produit scalaire canonique. Soient $U$ et $W$ deux sous-espaces vectoriels de même dimension $m$. On suppose qu'il existe un vecteur $u\in U$ tel que $u\in W^{\perp}$, $u\neq 0$. Montr er qu'il existe $v\neq 0\in W$ tel que $v\in U^{\perp}$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 400] +Soit $f\in{\cal C}^0([0,\pi/2],{\R})$. On pose, pour $n\in{\N}^*$, $I_n=(n+1)\int_0^{\pi/2}x\,f(x)\cos(x)^n\,{\rm d}x$. + +Déterminer la limite de $(I_n)_{n\geq 0}$. + +Ind. Utiliser $J_n=(n+1)\int_0^{\pi/2}\sin(x)\cos(x)^ndx$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 401] +Soient $f\in{\cal L}^1({\R})$ et $F:x\in{\R}\mapsto\int_{-\i}^{+\i}f(t)\,e^{it^2x}{\rm d}t$. Montr er que $F$ est définie et qu'elle tend vers 0 en $+\i$._Ind._ Traiter le cas ou $f$ est ${\cal C}^1$, le cas ou $f$ est nulle sur ${\R}$ prive de $[-a,a]$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 402] +Soit $(E):y''+\frac{t}{1+t^3}y=0$. Montr er que $(E)$ admet une solution non bornée. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 403] +On considére l'équation différentielle $(E):y''(t)+\phi(t)y(t)=0$, avec $\phi$ continue $2\pi$-periodique et on note $Sol$ l'ensemble des solutions de $(E)$ de classe ${\cal C}^2$ à valeurs complexes. + - Montr er qu'il existe $y_1\in Sol$ telle que $y_1(0)=1$, $y'_1(0)=0$, et $y_2\in Sol$ telle que $y_2(0)=0,y'_2(0)=1$. + +Montr er que toute solution de $(E)$ est combinaison lineaire de $y_1$ et $y_2$. + - Pour $y\in Sol$, on note $\Psi(y)$ la fonction $t\mapsto y(t+2\pi)$. Montr er que $\Psi(y)\in Sol$. + +Déterminer la nature de l'application $\Psi$. + - Montr er que, si $z\in Sol$ avec $z\neq 0$ est telle que $\forall t\in{\R},z(t+2\pi)=\lambda z(t)$ avec $\lambda\in{\C}$, alors $\lambda$ est racine du polynôme $X^2-(y_1(2\pi)+y'_2(2\pi))X-y'_1(2\pi)y_2(2\pi)+y_ {1}(2\pi)y'_2(2\pi)$. + +Étudier la réciproque. + - Montr er que $\lambda$ ne peut être nul puis que $\det(\phi)=1$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 404] +Soient $E_{n,d}=\{(i_1,...,i_d)\in{\N}^d,\;i_1+\cdots+i_d=n\}$ et + + $V_{n,d}=\{f:(x_1,...,x_d)\in{\R}^d\mapsto x_1^{i_1}\ldots x_d^{i_d},\;(i_1,\ldots,i_d)\in E_{n,d}\}$. + - Montr er que $V_{n,d}$ forme une famille libre et déterminer son cardinal. + - Soit $\Delta:f\in{\rm Vect}(V_{n,d})\mapsto\Delta(f)=\frac{\partial^2f}{\partial x_1^2}+\cdots+\frac{\partial^2f}{\partial x_d^2}$. Déterminer ${\rm Ker}\,\Delta$. +#+end_exercice + + +** Geometrie + +#+begin_exercice [X PSI 2024 # 405] +Soient $n\in\N^*$ et $x_1,\ldots,x_{2n}$ des points distincts de $\R^2$. Montrer qu'il existe toujours une droite separant ces $2n$ points en deux groupes de $n$ points. +#+end_exercice + + +** Probabilités + +#+begin_exercice [X PSI 2024 # 406] +Soit $X$ une variable aléatoire de loi $\mc{G}(1/2)$. Pour tout $k\in\N^*$, on note $A_k$ l'evenement \lt \lt $X$ est multiple de $k$\gt \gt . + - Soient $(p,q)\in(2\N^*)^2$. Les evenements $A_p$ et $A_q$ sont-ils indépendants? + - Même question pour $p$ et $q$ quelconques dans $\N^*$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 407] +Soient $X$ et $Y$ deux variables aléatoires. On suppose que $X$ suit la loi geometrique de paramêtre $p$ et que, pour tout $N\in\N^*$, la loi conditionnelle de $Y$ sachant $(X=N)$ est la loi binomiale $\mc{B}(N,p)$. Déterminer $\mathbf{E}(Y)$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 408] +Soit $M\,=\,\begin{pmatrix}X_1&1&0\\ 0&X_2&1\\ 0&0&X_3\end{pmatrix}$ ou $X_1$, $X_2$, $X_3$ sont des variables aléatoires indépendantes suivant la loi geometrique de paramêtre $p$. Calculer la probabilité que $M$ soit diagonalisable. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 409] +Soient $(a_{i,j})_{1\leq i,j\leq n}$ une famille de variables aléatoires indépendantes suivant la loi uniforme sur $\{-1,1\}$ et $A=(a_{i,j})_{1\leq i,j\leq n}$. Déterminer $\mathbf{E}(\det(A))$ et $\mathbf{V}(\det(A))$. +#+end_exercice + + +#+begin_exercice [X PSI 2024 # 410] +Soit $Y$ une variable aléatoire à support fini inclus dans $\R^+$. Déterminer à quelle condition on a $\mathbf{E}(Y^{1/2^n})=\mathbf{E}(Y)^{1/2^n}$ pour tout entier naturel $n$. +#+end_exercice + + +* X - ESPCI - PC :autre: + +** Algèbre + +#+begin_exercice [X PC 2024 # 411] +Un graphe est un couple $G=(S,A)$ ou $S$ est un ensemble fini et $A$ un ensemble de paires de $S$. Les éléments de $S$ s'appellent les sommets de $G$ et ceux de $A$ les arêtes de $G$. Soient $G=(S,A)$ et $G'=(S',A')$ deux graphes et $f$ une application de $S$ dans $S'$. On dit que $f$ est un morphisme de $G$ dans $G'$ si $\forall(u,v)\in S^2,\{u,v\}\in A\Rightarrow\{f(u),f(v)\}\in A'$. On dit que $f$ est un isomorphisme de $G$ dans $G'$ si + + $\forall(u,v)\in S^2,\{u,v\}\in A\iff\{f(u),f(v)\}\in A'$. + +Donner une majoration du nombre de graphes à $n$ sommets et $k$ arêtes deux à deux non isomorphes. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 412] +Soient $n\geq 2$ et $a_0,\ldots,a_n\in\R$. Montrer qu'il existe un entier $i\in\db{0,n}$ tel que l'on + +$$\text{ait}\left|\sum_{k=0}^ia_k-\sum_{k=i+1}^na_k\right|\leq \sup_{0\leq k\leq n}|a_k|.$$ +#+end_exercice + + +#+begin_exercice [X PC 2024 # 413] +Soit $P=X^2+c_1X+c_0$ à coefficients dans $\N$. Déterminer les suites d'entiers naturels $(a_n)$ telles que, pour tout $n\in\N$, $P\left(a_n\right)=a_{n+1}a_{n+2}$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 414] +Soit $k\in\N$. Déterminer les suites $(a_n)_{n\in\N}$ à valeurs dans $\N$ pour lesquelles il existe un polynôme $P$ à coefficients dans $\N$, unitaire et de degre $k$ tel que $\forall n\in\N$, $P(a_n)=\prod_{j=1}^ka_{n+j}$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 415] +Soient $A$ et $B$ deux éléments de $\R[X]$ dont toute combinaison lineaire réelle est scindée ou nulle, $x$ et $y$ deux racines de $A$ telles que $x\lt y$. Montrer que $B$ à une racine dans $[x,y]$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 416] +Calculer $\sum_{z\in\mathbb{U}_n}\frac{1}{2-z}$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 417] +Soit $n\in\N$ avec $n\geq 2$. Soient $u_1,\ldots,u_n$ des nombres complexes de module 1. Montrer que $\prod_{i\neq j}|u_i-u_j|^{\frac{1}{n(n-1)}}\leq n^{\frac{1}{n}}$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 418] +Pour $n\in\N^*$, calculer le module de $\sum_{k=0}^{n-1}\exp\bigg(2i\pi\frac{k^2}{n}\bigg)$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 419] +Soit $P\in\R[X]$ scindé sur $\R$. Soit $a\in\R$. Montrer que le polynôme $\op{Re}\big(P(X+ia)\big)$, polynôme dont les coefficients sont les parties réelles du polynôme $P(X+ia)$, est scindé sur $\R$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 420] +On note $\mathbb{D}=\{z\in\C\;;\;|z|\leq 1\}$ et $\|P\|=\sup_{z\in\mathbb{D}}|P(z)|$ pour $P\in\C[X]$. Pour $P\in\C[X]$, on définit la suite $(P_n)_{n\geq 0}$ en posant $P_0=P$ puis $P_{n+1}=(P_n')^2$ pour tout $n\in\N$. Montrer qu'il existe un réel $\eps\gt 0$ tel que, si $\|P\|\lt \eps$, alors $\lim_{n\ra+\i}\|P_n\|=0$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 421] +Soit $F$ un polynôme non constant à coefficients dans $\Z$. Montrer qu'il existe une infinite d'entiers $n\in\Z$ tels que $F(n)$ ne soit pas premier. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 422] +Montrer que $\R^n$ ne s'écrit pas comme reunion finie de sous-espaces vectoriels strictly. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 423] +Montrer que, pour tout $n\in\N^*$, il existe une matrice $M\in\M_n(\R)$ telle que, pour n'importe quelle permutation de ses $n^2$ coefficients, on obtienne toujours une matrice inversible. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 424] +Soient $E$ et $F$ deux $\C$-espaces vectoriels. Une application $f:E\mapsto F$ est dite antilineaire si $\forall x,y\in E,\forall\lambda\in\C,f(x+\lambda y)=f(x)+\overline{ \lambda}f(y)$ Pour quels entiers $n$ existe-t-il $f\colon\C^n\mapsto\C^n$ antilineaire telle que $f\circ f=-\op{id}$? +#+end_exercice + + +#+begin_exercice [X PC 2024 # 425] +Soient $n\geq 2$ et $A=\left(\begin{array}{cccc}0&1&\cdots&1\\ 1&0&\ddots&\vdots\\ \vdots&\ddots&\ddots&1\\ 1&\cdots&1&0\end{array}\right)$. Montrer que $A\in\op{GL}_n(\R)$. Trouver les valeurs propres de $A$ et leurs multiplicités. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 426] +Soient $(a_1,\ldots,a_n)\in\R^n$, $(b_1,\ldots,b_n)\in\R^n$ et $A=(a_i+\delta_{i,j}b - {1\leq i,j\leq n}\in\M_n( \R)$. + - Calculer $\det(A)$. + - La matrice $A$ est-elle diagonalisable? +#+end_exercice + + +#+begin_exercice [X PC 2024 # 427] +Soient $A$ et $B\in\M_n(\C)$. Montrer que les assertions suivantes sont équivalentes ; + +(i) $A$ et $B$ admettent au moins une valeur propre commune, + +(ii) il existe $P\in\M_n(\C)$ non nulle telle que $PA=BP$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 428] +Soient $A,B\in\M_n(\R)$ telles que $A^2=B^2=-I_n$. Montrer que $A$ et $B$ sont semblables. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 429] +Soit $n$ un entier naturel impair. Soient $A,B\in\M_n(\R)$ telles que $AB+BA=A$. Montrer que $A$ et $B$ ont un vecteur propre commun. Le résultat persiste-t-il pour $n$ pair? +#+end_exercice + + +#+begin_exercice [X PC 2024 # 430] +Soient $A$ et $B$ des matrices de $\M_n\left(\R\right)$ telles que $AB$ est diagonalisable. + - Est-ce que que $BA$ est diagonalisable? + - Montrer que : + + $\dim\left(\op{Ker}\left(AB\right)\right)\leq\dim\left( \op{Ker}\left(B\left(AB\right)A\right)\right)\leq\dim\left( \op{Ker}\left(A\left(BABA\right)B\right)\right)\leq\dim\left( \op{Ker}\left(AB\right)\right)$. + - Est-ce que que $\left(BA\right)^2$ est diagonalisable? +#+end_exercice + + +#+begin_exercice [X PC 2024 # 431] +Soient $A$ et $B$ dans $\M_n(\R)$. On suppose que les valeurs propres complexes de $A$ ont une partie réelle strictement negative et que celles de $B$ ont une partie réelle negative. Soit $C\in\M_n(\R)$. Montrer qu'il existe une unique matrice $M\in\M_n(\R)$ telle que $C=AM+MB$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 432] +Dans $\M_n(\C)$, soient $S$ et $S'$ diagonalisables, $N$ et $N'$ nilpotentes. On suppose $NS=SN$ et $N'S'=S'N'$ et $S+N=S'+N'$. Montrer que $S=S'$ et $N=N'$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 433] +Montrer que, pour toute matrice $A\in\mc{S}_n(\R)$, il existe un unique couple $(B,C)$ de matrices symétriques positives telles que $A=B-C$ et $BC=CB=0$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 434] + - Montrer que toute matrice réelle de taille $n$ symétrique positive admet une racine carrée symétrique positive. + - Soient $S$ et $A$ deux matrices de taille $n$ avec $S$ symétrique définie positive et $A$ antisymétrique. Montrer que $AS$ est $\C$-diagonalisable. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 435] +Soient $A\in\mc{S}_n(\R)$ et $k\in\N^*$. Pour $H\in\mc{S}_n(\R)$, on pose $\phi_k(H)=\sum_{i=0}^{k-1}A^iHA^{k-1-i}$. + - Montrer que $\phi_k$ est un endomorphisme de $\mc{S}_n(\R)$. + - à quelle condition $\phi_k$ est-elle injective? surjective? bijective? +#+end_exercice + + +#+begin_exercice [X PC 2024 # 436] +Soit $f\in\mc{L}\left(S_n(\R),\R\right)$ telle que $\forall M\in\mc{S}_n^+(\R),\ f(M)\geq 0$. Montrer que $f$ est une combinaison lineaire des formes lineaires $\phi_X:M\mapsto X^TMX$ avec $X\in\M_{n,1}(\R)$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 437] +Soit $n$ un entier naturel impair. Soient $A$ et $B$ dans $S_n(\R)$. On note $C(A)$ (resp. $C(B)$) l'ensemble des matrices de $\M_n(\R)$ qui commutent avec $A$ (resp. $B$).Montrer que $C(A)\cap C(B)=\RI_n$ si et seulement s'il n'existe pas deux sous-espaces $F$ et $G$ de $\R^n$, stables par $A$ et $B$, de dimension $\geq 1$, tels que $F\oplus G=\R^n$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 438] +Soient $A,B\in\mc{S}_n(\R)$ deux matrices dont les valeurs propres sont strictement supérieures à $1$. Montrer que les valeurs propres de $AB$ sont strictement supérieures à $1$. +#+end_exercice + + +** Analyse + +#+begin_exercice [X PC 2024 # 439] +On note $E$ l'ensemble des polynômes non nuls à coefficients dans $\{-1, 0, 1\}$ et $A$ l'ensemble des racines des polynômes appartenant à $E$. Déterminer l'adhérence de $A$. +#+end_exercice + +#+BEGIN_exercice [X PC 2024 # 440] +Chercher les fonctions $f\colon \R^2\ra\R^2$ bijectives, continues, dont la réciproque est continue, et telle que, pour tout droite $\mc D$, $f(\mc D)$ est une droite. +#+END_exercice + +#+BEGIN_exercice [X PC 2024 # 446] +On note $a = \sqrt{2}$. Pour $n\geq 1$, soit $S_n = \frac{1}{n}\sum_{a \lt \frac{k}{n}\lt a+1} \frac{1}{\sqrt{\frac{k}{n} - a}}$. Étudier la convergence de $(S_n)$. +#+END_exercice + + +#+begin_exercice [X PC 2024 # 449] +Soit $(a_n)$ une suite de réels de $]0,1[$ telle que la série $\sum\frac{a_n}{\ln(1/a_n)}$ converge. Montrer que la série $\sum\frac{a_n}{\ln(n)}$ converge. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 450] +Soit $(a_n)_{n\in\N}$ une suite complexe vérifiant, pour $n\in\N$, $a_{n+1}=a_n+\frac{1}{(n+1)^2}\sum_{k=0}^na_n$. + - Trouver $\alpha$ tel qu'il existe $C$ vérifiant $\forall n\in\N^*$, $|a_n|\leq Cn^{\alpha}$ + - On suppose $a_0\gt 0$. Montrer que $\sum a_n$ diverge. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 451] +Prouver que la série de terme general $2^{-2^n}$ converge et que sa somme $\sum_{n=0}^{+\i}2^{-2^n}$ est irrationnelle. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 452] +Soit $(a_n)$ une suite de réels strictement positifs telle que $\sum a_n$ converge. Soit $(u_n)$ une suite réelle. On pose, pour $n\in\N$, $v_n=\frac{\sum_{k=0}^na_ku_{n-k}}{\sum_{k=0}^na_k}$. + - Montrer que, si $\sum u_n$ converge absolument, alors $\sum v_n$ converge. + - Est-ce toujours le cas si $\sum u_n$ ne converge pas absolument? +#+end_exercice + + +#+begin_exercice [X PC 2024 # 453] +Soit $f\in[\,0\,;+\i\,[\,\ra\R$ de classe $\mc C^1$ telle que $\int_0^{+\i}|f'(t)|\,\dt$ converge. Montrer que + +$$\int_0^{+\i}f(t)\dt$$ converge si et seulement si + +$$\sum f(n)$$ +#+end_exercice + + +#+begin_exercice [X PC 2024 # 454] +Soient $k\in\N^*$ et $x_1,\ldots,x_k\in\R^{+*}$. Montrer l'inegalite $\prod_{i=1}^k(1+x_i^k)\geq\left(1+\prod_{i=1}^kx_i \right)^k$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 455] +Déterminer les fonctions continues $f\colon\R\ra\R$ telles que : + + $\forall(a,b)\in\R^2,a\lt b$, $f\left(\frac{a+b}{2}\right)=\frac{1}{b-a}\int_a^bf(t)\dt$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 456] +Soient $n\in\N$ et $\lambda\in\,]0,1[$ distinct de $\frac{1}{n+2}$. + - Trouver toutes les fonctions $f$ de classe $\mc C^{n+1}$ telles que, pour tous réels $a$ et $b$, on ait $f(b)=\sum_{k=0}^n\frac{(b-a)^k}{k!}f^{(k)}(a)+\frac{(b-a)^{n+1}}{(n+1)!}f^ {(n+1)}(\lambda b+(1-\lambda a))$. + - Étudier le cas $\lambda=\frac{1}{n+2}$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 457] +Soient $a_1,\ldots,a_n$ des réels et $P:x\mapsto\sum_{k=1}^na_k\sin(kx)$. Pour tout entier $r\in\N$, on suppose que $(-1)^rP^{(2r)}$ est positive sur $[\,0\,;\pi\,]$. Montrer que $P$ est la fonction $x\mapsto a_1\sin(x)$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 458] +Soit $(u_{k,n})_{(k,n)\in\N\times\N}$ une suite doublement indexée à valeurs complexes. On suppose que, pour toute suite complexe $(v_n)_{n\in\N}$ bornée, $\lim_{k\ra+\i}\sum_{n=0}^{+\i}v_nu_{k,n}=0$. + +Montrer que $\lim_{k\ra+\i}\sum_{n=0}^{+\i}|u_{k,n}|=0$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 460] :todo: +Soient $f,g\in\mc C^0([0,1], \R)$ telles que $\int_0^1 fg = 0$. + - Montrer que $$\int_0^1 f^2 \left(\int_0^1 g\right)^2 + \int_0^1 g^2 \left(\int_0^1 f\right)^2 \geq 4 \left(\int_0^1 f \int_0^1 g\right)^2.$$ + - Montrer que $$\int_0^1 f^2 \int_0^1 g^2 \geq 4\left(\int_0^1 f \int_0^1 g\right)^2.$$ +#+end_exercice + + +#+begin_exercice [X PC 2024 # 464] +Pour $f:[0,1]\ra\R$ et $n\in\N^*$, on pose $P_n:x\mapsto\sum_{k=0}^n\binom{n}{k}f\Big(\frac{k}{n}\Big)\,x^k(1-x)^{ n-k}$. On admet que, si $f$ est continue, alors $(P_n)$ tend uniformément vers $f$ sur $[0,1\,]$. + +Déterminer une condition nécessaire et suffisante sur $f:[\,0,1\,]\ra\R$ afin qu'il existe une suite de polynômes à coefficients entiers qui converge uniformément vers $f$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 465] +Soit $f:z\mapsto\sum_{n=1}^{+\i}\frac{1}{n^5\left(1+\frac{i}{n^3}-z\right)}$. + - Montrer que $f$ est développable en série entiere au voisinage de 0. + - Montrer que la restriction de $f$ à l'ensemble des nombres complexes de module 1 n'est pas continue. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 466] +Soit $\mc{S}$ l'ensemble des $f\in\mc C^1(\R,\R)$ telles que, pour tout $x\in\R$, $f\left(x\right)=xf'\left(x/2\right)$. + - Chercher les $f\in\mc{S}$ développables en série entiere. + - L'espace $\mc{S}$ est-il de dimension finie? +#+end_exercice + + +#+begin_exercice [X PC 2024 # 467] +Soit $\left(u_n\right)_{n\geq 0}\in\C^{\N}$ une suite qui tend vers $0$. Pour $t\in\,]-1,1[$, on pose $f(t)=\sum_{n=0}^{+\i}u_nt^n$. + - Vérifier que $f$ est bien définie sur $]-1\,;1\,[$. + - Montrer que $\lim\limits_{t\ra 1^-}tf(t)=0$. + - On suppose de plus qu'il existe des réels $a_1,\ldots,a_r$ et $0\lt \theta_1\lt \cdots\lt \theta_r\lt \pi$ tels que $\forall n\in\N,\,u_n=\sum_{k=1}^ra_k\cos(n\theta_k)$. Montrer que $a_k=0$ pour tout $k\in\db{1,r}$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 468] +La fonction $f:x\mapsto\sum_{k=0}^{+\i}(-1)^kx^{k!}$ admet-elle une limite lorsque $x$ tend vers $1^-\,$? +#+end_exercice + + +#+begin_exercice [X PC 2024 # 469] +Soit $(a_{k,n})_{(k,n)\in\N^2}$ une famille de nombres complexes telle que, pour tout $n\in\N$, la série entiere $f_n:z\mapsto\sum_{k=0}^{+\i}a_{k,n}z^k$ à un rayon de convergence supérieur ou egal à 1. On note $B$ l'ensemble des nombres complexes de module $\leq 1$. On suppose que la suite $(f_n)$ converge simplement sur $B$ et qu'il existe $M\in\R^+$ tel que, pour tous $n\in\N$ et $z\in B$, $|f_n(z)|\leq M$. Montrer que la suite $(f_n)$ converge uniformément sur $\{z\in\C,\;|z|\leq r\}$ pour tout $r\lt 1$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 470] +Soient $U$ un voisinage de $0$ dans $\C$, $k\in\N$ et $f$ une fonction de $U$ dans $\C$ développable en série entiere au voisinage de $0$ telle que $f(z)\underset{z\ra 0}{=}O(z^k)$. Montrer que, pour $r\gt 0$ assez petit, il existe au moins $2k$ nombres complexes $z$ de module $r$ tels que $f(z)\in\R$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 471] +Pour $x\geq 0$, on pose $I(x)=\int_0^{\pi/2}\cos(x\cos\theta)\,d\theta$. + - Écrire $I(x)$ sous la forme d'une série. + - Montrer que $I(x)=\mc{O}(x^{-1/4})$ quand $x$ tend vers $+\i$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 472] +On admet le theoreme d'approximation de Weierstrass. Soit $f\colon\R\ra\R$ une fonction continue. Soient $a,b\gt 0$. On suppose que $f(x)=0$ pour tout $x\in\R\setminus[\,-a\,;a\,]$. Pour $x\in\R$, on pose $\hat{f}(x)=\int_{-\i}^{+\i}f(t)e^{-ixt}\dt$. + - On suppose que $\hat{f}(x)=0$ pour tout $x\in[\,-b\,;b\,]$. Montrer que $f=0$. + - On suppose que $\hat{f}(x)=0$ pour tout $x\in\R\setminus[\,-b\,;b\,]$. Montrer que $f=0$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 473] +Déterminer les solutions sur $\R$ de l'équation différentielle : $xy''+y'-4xy=0$. + +Ind. Chercher les solutions développables en série entiere. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 474] +Soient $p\colon\R\ra\R$ intégrable et $y\colon\R\ra\R$ de classe $\mc C^2$ vérifiant $(E):y''-py=0$. + - Montrer que $\lim_{x\ra+\i}y'(x)=0$. + - On admet que, pour tout $(a,b)\in\R^2$, il existe $y$ vérifiant $(E)$ et $(y(0),y'(0))=(a,b)$. + +Montrer que $(E)$ admet une solution non bornée. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 475] +Soit $X\colon\R\mapsto\R^{2n}$ de classe $\mc C^1$ telle que $X'(t)=JSX(t)$, ou $J=\left(\begin{array}{cc}O_n&-I_n\\ I_n&O_n\end{array}\right)$ et $S\in S_n^{++}(\R)$. Montrer que $X$ est bornée sur $\R$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 476] +Déterminer les extrema globaux et locaux de $f:M\in\text{SO}_4(\R)\mapsto\op{tr}(A)$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 477] +Soient $d\in\N$ et $\Omega\in\mc C^2(\R^d,\R)$. On suppose que $\nabla(\Omega)(0)=0$ et on note $D_a^2(\Omega)$ la hessienne en $a$ de $\Omega$. On suppose que $\op{Im}(D_a^2(\Omega))=F$, ou $F$ est indépendant de $a$ et de rang $p$. + +Montrer qu'il existe un changement de coordonnées $f$ (c'est-a-dire une application de $\R^d$ dans $\R^d$) tel que, pour tout $(x_1,\ldots,x_d)\in\R^d$, $(\Omega\circ f)(x_1,\ldots,x_d)$ ne depende que de $(x_1,\ldots,x_p)$. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 478] +Soient $N\in\N^*$ et $f\in\mc C^0(\R^N,\R)$. Montrer qu'il existe une suite $(f_n)$ de fonctions dans $\mc C^{\i}(\R^N,\R)$ et une suite $(x_n)$ d'éléments de $\R^N$ qui tend vers $0$ telles que, pour tout $n\in\N$, la fonction $f-\phi_n$ admette un minimum local en $x_n$. +#+end_exercice + + +** Probabilités + +#+begin_exercice [X PC 2024 # 479] +On lance une piece une infinite de fois. On note $S_n$ le nombre de successions de deux pile consécutifs dans les $n$ premiers lancers. + - Trouver $\mathbf{E}(S_n)$ et $\mathbf{V}(S_n)$. + - On pose $T=\min\{n\in\N,\ S_n=1\}$. Calculer $G_T(t)$ et en déduire sa loi. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 480] +Soit $f:[\,0\,;1\,]\ra\R$ une fonction croissante. Pour $n\in\N^*$, montrer que la fonction $p_n:x\mapsto\sum_{k=0}^n\binom{n}{k}f\Big(\frac{k}{n}\Big)\,x^k(1- x)^{n-k}$ est croissante sur $[0,1]$. Interpreter d'un point de vue probabiliste. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 481] +On étudie un groupe de cellules. à l'instant initial, $n=0$, il y en à une. à chaque instant, chaque cellule peut de facon equiprobable : mourir, raster telle qu'elle est, se diviser en 2, se diviser en 3. Calculer la probabilité que le groupe disparaisse. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 482] +Soient $p\in\left]0,1\right[$, $(X_n)_{n\in\N}$ une suite de variables aléatoires définie par $X_0=0$ et, pour $n\in\N$, $X_{n+1}=X_n+1$ avec une probabilité $p$ et $X_{n+1}=0$ avec probabilité $1-p$. + +Déterminer la loi de $X_n$, son esperance et sa variance. +#+end_exercice + + +#+begin_exercice [X PC 2024 # 483] +Soit $\Omega$ un ensemble. On dit que $\M\subset\mc{P}(\Omega)$ est une classe monotone si elle vérifie : + +(i) $\Omega\in\M,$ (ii) $\M$ est stable par union croissante, + +(iii) si $A,B\in\M$ et $B\subset A$, alors $A\setminus B\in\M$. + - Montrer qu'une intersection de classes monotones est une classe monotone. + - Montrer qu'une classe monotone stable par intersection finie est une tribu. + - Soit $C\subset\mc{P}(\Omega)$ stable par intersection finie. Montrer que la classe monotone $D$ engendrée par $C$ (c'est-a-dire la plus petite classe monotone contenant $C$) est une tribu. +#+end_exercice + +* Mines - Ponts - MP + +** Algèbre + +#+begin_exercice [Mines MP 2024 # 484] +Soit $n\in\N\setminus\{0,1\}$. Calculer $S_n=\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2k}(-3)^k$ et $T_n=\sum_{k=0}^{\left\lfloor\frac{n}{3}\right\rfloor}\binom{n}{3k}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 485] +Soient $(a_1,\ldots,a_n),(b_1,\ldots,b_n)\in\R^n$. Montrer que l'application définie sur l'ensemble des permutations de $\db{1,n}$ par $f(\sigma)=\sum_{i=1}^na_ib_{\sigma(i)}$ admet un minimum et un maximum à expliciter. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 486] +On note $\phi$ la fonction indicatrice d'Euler. + - Calculer $\phi(7)$ et $\phi(37044)$. + - Montrer que : $\forall n\in\N^*,\phi(n)\geq\frac{n\ln 2}{\ln n+\ln 2}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 487] +Soient $a$ et $b$ dans $\N^*$. Montrer que $a\wedge b=1$ si et seulement si, pour tout $n\geq ab$, il existe $u,v\in\N$ tels que $au+bv=n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 488] +Pour $n\in\N$, soit $F_n=2^{2^n}+1$. + - Montrer que, si $m$ et $n$ sont deux entiers naturels distincts, $F_m\wedge F_n=1$. + - Retrouver à l'aide de la question précédente que l'ensemble des nombres premiers est infini. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 489] +Soit $n\in\N^*$. Déterminer et dénombrer les sous-groupes de $\Z/n\Z$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 490] +Soit $G$ un groupe fini non reduit à l'élément neutre et tel que : $\forall g\in G,\ g^2=e$. + - Montrer que $G$ est abelien. - Soit $H$ un sous-groupe strict de $G$ et $a\in G\setminus H$. Montrer que $H\cup aH$ est un sous groupe de $G$ et que l'union est disjointe. + - Montrer que le cardinal de $G$ est une puissance de 2. + - Calculer le produit des éléments de $G$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 491] +Soient $G$ un groupe fini et $\Omega=G^2$ que l'on munit de la probabilité uniforme. + +On pose : $C=\{(x,y)\in G^2\;;\;xy=yx\}$ et $p=\mathbf{P}(C)$. + - Montrer que $p\gt 0$. Que dire si $p=1$? + +Dans la suite, on suppose que $G$ n'est pas commutatif. + - Calculer $p$ lorsque $G=\mc{S}_3$ puis lorsque $G=\mc{S}_4$. + - On définit la relation $\sim$ sur $G^2$ par : $x\sim y\Longleftrightarrow\exists g\in G,x=gyg^{-1}$. Montrer que $\sim$ est une relation d'équivalence. + - On note $s$ le nombre de classes d'équivalence. Montrer que : $p=\frac{s}{\op{card}G}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 492] +Soit $G$ un groupe abelien. Soient $a$ et $b$ deux entiers naturels non nuls premiers entre eux, et $x\in G$ d'ordre $a$ et $y\in G$ d'ordre $b$. Montrer que $xy$ est d'ordre $ab$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 493] +Soit $G$ un ensemble muni d'une loi de composition interne $\cdot$ associative, telle qu'il existe $e\in G$ vérifiant $xe=x$ pour tout $x\in G$, et, pour tout $x\in G$, il existe $x'\in G$ tel que $xx'=e$. Montrer que $(G,\cdot)$ est un groupe. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 494] +Soit $\alpha=e^{i\theta}$ un nombre complexe de module $1$. Calculer $\prod_{k=0}^n(\alpha^{2^{-k}}+\bar{\alpha}^{2^{-k}})$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 495] +Soit $n$ un entier $\geq 2$. On pose $Q=1+2X+\cdots+nX^{n-1}$. Calculer $\prod_{\zeta\in\mathbb{U}_n}Q(\zeta)$, ou $\mathbb{U}_n$ designe le groupe des racines $n$-iemes de l'unite. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 496] +Soient $m\in\N^*$, $x_1\lt x_2\lt \cdots\lt x_m$ des nombres réels, $\alpha_1,\ldots,\alpha_m$ des éléments de $\N^*$ et $P=\prod_{k=1}^m(X-x_k)^{\alpha_k}$. Quel est le nombre de racines réelles distinctes de $P'$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 497] + - Soit $n\in\N$. Montrer qu'il existe un unique polynôme $P_n\in\Z[X]$ tel que + + $\forall x\in\R^*,\;P_n\left(x+\frac{1}{x}\right)=x^n+\frac{1}{x ^n}$. + - Soit $a\in\Q$ tel que $\cos(a\pi)\in\Q$. Montrer que $2\cos(a\pi)\in\Z$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 498] +Soient $0\lt a_0\lt \cdots\lt a_n$, $P=\sum_{k=0}^na_kX^k$ et $Q=(X-1)P$. + - Soient $p\geq 2$ et $z_1$, $\ldots$, $z_p\in\C^*$ tels que $|z_1+\cdots+z_p|=|z_1|+\cdots+|z_p|$. Montrer qu'il existe $\lambda\in\R^{+*}$ tel que, pour tout $k\in\db{1,p}$, $z_k=\lambda z_1$. + - Justifier que, pour tout $z\in\C$, $|Q(z)|\leq Q(|z|)$. + - Montrer que les racines de $P$ sont de module strictement inférieur à $1$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 499] +Soit $\mathbb{K}$ un sous-corps de $\C$. Déterminer les $P\in\C[X]$ tel que $P(\mathbb{K})\subset\mathbb{K}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 500] +Soit $P\in\C[X]$. + - à quelle condition a-t-on $P(\C)=\C$? + - à quelle condition a-t-on $P(\R)=\R$? + - à quelle condition a-t-on $P(\Q)=\Q$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 501] +Soit $P\in\R[X]$ un polynôme non constant. On note $r^+(P)$ le nombre de racines de $P$ dans $\R^{++}$ et $N(P)$ le nombre de coefficients non nuls de $P$. + - Que dire de $P$ si $N(P)=1$? si $N(P)=2$? + - Montr per que : $r^+(P)\leq r^+(P')+1$. + - On suppose que $P(0)=0$. Montr per que : $r^+(P)\leq r^+(P')$. + - Montr per que : $r^+(P)\leq N(P)-1$. + - Soit $n\in\N$. Soient $0\lt x_1\lt \cdots\lt x_n$ des réels et $0\leq p_1\lt \cdots p_n$ des entiers. Montr per que : $\det\left(x_i^{p_j}\right)_{1\leq i,j\leq n}\gt 0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 502] +Soit $P$ un polynôme à coefficients complexes. + - Donner la décomposition en éléments simples de $P'/P$. + - Montr per que l'enveloppe convexe des racines de $P'$ est incluse dans l'enveloppe convexe des racines de $P$. Que dire si $P$ est un polynôme à coefficients réels scindé dans $\R$? + - Montr per que si un demi-plan ferme $H$ contient une racine de $P'$ alors $H$ contient une racine de $P$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 503] + - Soient $a,b,c\in\Z$ premiers entre eux, montrer que $A=\begin{pmatrix}a&b&c\\ 2c&a&b\\ 2b&2c&a\end{pmatrix}$ est inversible. + - On pose $\alpha=2^{1/3}$. Soit $(a,b,c)\in\Q^3$ tel que $a+b\alpha+c\alpha^2=0$. + +Montrer que $a=b=c=0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 504] +Soient $E$ un $\mathbb{K}$-espace vectoriel et $u\in\mc{L}(E)$. + - On suppose que $E$ est de dimension finie. Montr per que les propriétés suivantes sont équivalentes : (i) $\op{Ker}u=\op{Ker}u^2$ ; (ii) $\op{Im}u=\op{Im}u^2$ ; (iii) $\op{Ker}u\oplus\op{Im}u=E$. + - En dimension infinie, donner des contre-exemples. + - En dimension finie ou infinie, montrer que : (iii) $\Longleftrightarrow$ ((i) et (ii)). +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 505] +Soit $f\in\mc{L}(\R^3)$ tel que $f^2=0$. Montr per que, si $F$ est un plan vectoriel de $\R^3$ stable par $f$, on a $\op{Im}(f)\subset F$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 506] +Soit $\phi$ une forme lineaire sur $\M_n(\mathbb{K})$. Montr per qu'il existe $A\in\M_n(\mathbb{K})$ telle que $\phi=M\mapsto\op{tr}(AM)$. En déduire que tout hyperplan de $\M_n(\mathbb{K})$ contient une matrice inversible. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 507] +Soient $E$ un espace vectoriel de dimension $n\geq 2$, $p_1,\ldots,p_n\in\mc{L}(E)\setminus\{0\}$ tels que : $\forall i,j,\ p_i\circ p_j=\delta_{i,j}p_i$. Montr per que les $p_i$ sont de rang 1 et que $E=\bigoplus_{i=1}^n\op{Im}(p_i)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 508] +Soient $E$ et $F$ deux $\mathbb{K}$-espaces vectoriels de dimension finie. + - Soient $u\in\mc{L}(E,F)$ et $v\in\mc{L}(F,E)$ tels que $uvu=u$ et $vuv=v$. Montrer que $E=\op{Ker}(u)\oplus\op{Im}(v)$. + - Soient $u\in\mc{L}(E,F)$, $E_1$ un supplementaire de $\op{Ker}u$ dans $E$, $F_1$ un supplementaire de $\op{Im}(u)$ dans $F$. Montrer qu'il existe un unique $v\in\mc{L}(F,E)$ tel que $\op{Ker}v=F_1$, $\op{Im}v=E_1$, $uvu=u$ et $vuv=v$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 509] +Soit $E$ un $\mathbb{K}$-espace vectoriel de dimension finie. Soient $u,v\in\mc{L}(E)$. + - Montrer que : $\op{rg}u+\op{rg}v-\dim E\leq\op{rg}(u \circ v)\leq\min(\op{rg}u,\op{rg}v)$. + - On suppose que $u\circ v=0$ et $u+v\in\op{GL}(E)$. Montrer que $\op{rg}u+\op{rg}v=\dim E$, $\op{Im}v=\op{Ker}u$, $E=\op{Ker}u\oplus\op{Im}u$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 510] +Soient $a,b\in\C$ distincts, $E$ un $\C$-espace vectoriel de dimension finie et $u\in\mc{L}(E)$ vérifiant $(u-a\op{id})\circ(u-b\op{id})=0$. On pose $p=\frac{1}{b-a}(u-a\op{id})$ et $q=\frac{1}{a-b}(u-b\op{id})$. + +Déterminer $p^2$, $q^2$, $p\circ q$, $q\circ p$ et $p+q$ puis montrer que $E=\op{Ker}(p)\oplus\op{Ker}(q)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 511] +Soient $E$ un $\mathbb{K}$-espace vectoriel de dimension infinie dénombrable, $(e_n)_{n\geq 0}$ une base de $E$. Soit $u\in\mc{L}(E)$ tel que : $\forall n\in\N$, $u(e_n)=e_{n+1}$. Soit $\Phi$ l'endomorphisme de $\mc{L}(E)$ tel que : $\forall v\in\mc{L}(E)$, $\Phi(v)=uv-vu$. + - Montrer que $\Phi$ n'est pas injectif et que la dimension de $\op{Ker}\Phi$ est infinie. + - Soient $x_0\in E$ et $w\in\mc{L}(E)$. Montrer qu'il existe un unique $v\in\mc{L}(E)$ tel que $\Phi(v)=w$ et $v(e_0)=x_0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 512] +Soient $E$ un $\mathbb{K}$-espace vectoriel et $\mc{A}$ une sous-algèbre de $\mc{L}(E)$ telle que les seuls sous-espaces vectoriels stables par tous les éléments de $\mc{A}$ sont $E$ et $\{0\}$. Montrer que, pour tout $x\in E$ non nul et tout $y\in E$, il existe $u\in\mc{A}$ tel que $u(x)=y$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 513] +Soient $E$ un espace vectoriel de dimension $n$ et $u\in\mc{L}(E)$ nilpotent de rang $n-1$. Montrer que $u$ admet exactement $n+1$ sous-espaces stables. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 514] +Soit $E$ un $\R$-espace vectoriel de dimension finie. Trouver les endomorphismes de $E$ qui commutent avec tous les automorphismes de $E$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 515] + - Soient $n\geq 2$ et $B=(b_{i,j})_{1\leq i,j\leq n}\in\M_n(\R)$ à coefficients entiers telle que, pour tout $i$, $b_{i,i}$ soit impair et, pour tout $(i,j)$ avec $i\neq j,b_{i,j}$ soit pair. Montrer que $B$ est inversible. + - La propriété est-elle encore vérifiée lorsqu'on intervertit \lt \lt pair \gt \gt et \lt \lt impair \gt \gt ? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 516] +Soit $A\in\M_n(\R)$ telle que $A^2=0$. Déterminer une condition nécessaire sur $n$ et $A$ pour qu'il existe $B\in\M_n(\R)$ telle que $A=B^2$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 517] +Soient $A,B\in\M_n(\C)$ telles que $A=B^3$. On suppose que $A$ est de rang $1$. Donner une relation entre $\op{tr}A$ et $\op{tr}B$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 518] +Soit $A\in\M_n(\R)$. Montrer qu'il existe une matrice $D\in\M_n(\R)$ diagonale à coefficients diagonaux éléments de $\{-1,1\}$ telle que $A+D$ soit inversible. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 519] +Soient $n\in\N$ et $x_1\lt x_2\lt ...\lt x_n$ réels. On note $V=(x_i^{j-1})_{1\leq i,j\leq n}$. + - Calculer le déterminant de la matrice $V$. + - Montrer que $V$ est inversible et calculer son inverse. + +Ind. On pourra interpreter $V$ comme matrice de passage dans $\R_{n-1}[X]$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 520] +Soient $n\in\N^*$ et $P_1,\ldots,P_n\in\mathbb{K}[X]$. Montrer que la famille $(P_1,\ldots,P_n)$ est libre si et seulement s'il existe $a_1,\ldots,a_n\in\mathbb{K}$ tels que la matrice $(P_i(a_j))_{1\leq i,j\leq n}$ soit inversible. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 521] +Soient $E$ un $\mathbb{K}$-espace vectoriel et $f,g\in\mc{L}(E)$ tels que : $fg-gf=\op{id}$. + - Montrer que : $\forall P\in\mathbb{K}[X],\,fP(g)-P(g)f=P'(g)$. + - Montrer que $(g^n)_{n\in\N}$ est une famille libre. + - Si $E=\R[X]$, donner un exemple de couple $(f,g)$ vérifiant les relations précédentes. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 522] +Soient $n\geq 2$ et $E$ un ensemble à $n$ éléments. On pose $N=2^n-1$ et $E_1,\ldots,E_N$ les parties non vides de $E$. Soit $A=(a_{i,j})_{1\leq i,j\leq N}\in\M_N(\R)$ ou $a_{i,j}=1$ si $E_i\cap E_j\neq\emptyset$, et 0 sinon. Calculer $\det A$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 523] +Soient $n\in\N^*$ et $f_1,...,f_n$ des fonctions de $\R$ dans $\R$. + +Montrer que la famille $(f_1,...,f_n)$ est libre si et seulement s'il existe $(x_1,...,x_n)\in\R^n$ tel que $\det\left((f_i(x_j))_{1\leq i,j\leq n}\right)\neq 0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 524] +Soient $E$ un $\C$-espace vectoriel et $f_1,...,f_p$ des formes lineaires sur $E$. + +Montrer que les propriétés suivantes sont équivalentes : + - $(f_1,...,f_p)$ est libre, + - l'application $\phi:x\mapsto(f_1(x),...,f_p(x))$ est surjective de $E$ sur $\C^p$, + - il existe $x_1,...,x_p\in E$ tels que $\det\left((f_i(x_j))_{1\leq i,j\leq p}\right)\neq 0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 525] +Soient $A,M\in\M_n(\C)$ avec $A$ inversible et $M$ de rang 1. + - On suppose que $\det(A+M)=0$. Que dire de $\op{tr}\left(A^{-1}M\right)$? + - On suppose que $\det(A+M)\neq 0$. Donner une expression de $(A+M)^{-1}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 526] +Soient $A\in\M_n(\C)$ et $M=\begin{pmatrix}I_n&A\\ A&I_n\end{pmatrix}$. Étudier l'inversibilité de $M$, et le cas echeant, déterminer $M^{-1}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 527] +Soient $A,B\in\M_n(\R)$ avec $B$ nilpotente et $AB=BA$. + - Montrer que $A\in\op{GL}_n(\R)$ si et seulement si $A+B\in\op{GL}_n(\R)$. + - Calculer $(A+B)^{-1}$ quand $A$ est inversible. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 528] +Soit $A\in\M_n(\mathbb{K})$. Montrer que $A^2=0$ si et seulement si $A$ est semblable à une matrice de la forme $\begin{pmatrix}0&I_r\\ 0&0\end{pmatrix}$ ou $2r\leq n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 529] +Pour $n\in\N^*,$ soit $P_n=X^n-X+1$. + - + - Montrer que, pour tout $n\in\N^*$, $P_n$ admet au plus une racine réelle. + - Donner les racines des $P_n'$.. + - Montrer que les $P_n$ sont à racines simples. + - Notons $r_1,r_2,r_3$ les racines de $P_3$. Calculer $\begin{pmatrix}r_1+1 & 1 & 1 \\ 1 & r_2+1 & 1 \\ 1 & 1 & r_3 + 1 \end{pmatrix}$ +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 530] + - Soient $E$ un $\mathbb{K}$-espace vectoriel de dimension finie $n$ et $p\in\db{1,n-1}$. Soit $u\in\mc{L}(E)$, qui stabilise tous les sous-espaces de dimension $p$. Montrer que $u$ est une homothetie. + - Soient $A,M\in\M_n(\C)$. On suppose que $A$ n'est pas scalaire et que $M$ commute avec toutes les matrices semblables à $A$. Que dire de $M$? + - Même question pour deux matrices réelles. + +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 531] +Soient $\mathbb{K}$ un sous-corps de $\C$, $A$ et $B$ dans $\M_n(\mathbb{K})$. Si $A$ et $B$ sont semblables, montrer que $\text{Com}(A)$ et $\text{Com}(B)$ le sont aussi. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 532] +Soit $A\in\M_n(\R)$. Montrer que, si $t\in\R^+$, $\det(A^2+tI_n)\geq 0$. + +#+end_exercice + +#+begin_exercice [Mines MP 2024 # 533] +Soit $N\in\M_n(\C)$ nilpotente. Montrer que $G=\{P(N),P\in\C[X]\text{ et }P(0)=1\}$ est un sous-groupe de $\text{GL}_n(\C)$. +#+end_exercice + +#+begin_exercice [Mines MP 2024 # 534] +Soient $n\geq 2$ et $A,B\in\M_n(\C)$ non inversibles telles que $(AB)^n=0$. + - Montrer que $(BA)^n=0$. + - On suppose que $(AB)^{n-1}\neq 0$ et $(BA)^{n-1}\neq 0$. + Montrer que, pour tout $k\in\db{1,n}$, $\text{Ker}((AB)^k)=\text{Ker}(B)$ et $\text{Ker}((BA)^k)=\text{Ker}(A)$. + - Conclure + +#+end_exercice + +#+begin_exercice [Mines MP 2024 # 535] +Soient $n\geq 2$ et $A\in\M_n(\C)$ non nulle et non inversible. + - Montrer qu'il existe $p\in\N^*$ tel que $\C^n=\text{Im}(A^p)\oplus\text{Ker}(A^p)$. + - Montrer qu'il existe $r\in\db{1,n-1}$, $A_0\in\text{GL}_r(\C)$ et $N\in\M_{n-r}(\C)$ nilpotente tels que $A$ est semblable à $\left(\begin{array}{c|c}A_0&0\\ \hline 0&N\end{array}\right)$ + - On suppose qu'il existe $m\geq 2$ et $B\in\M_n(\C)$ tels que $A^mB=A$. + Montrer que $A^m B= A^{m-1} BA = \dots = BA^m$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 536] +Soient $n\in\N$, $P\in\mathbb{K}[X]$ de degre $n$, $\alpha_0,\ldots,\alpha_n$ des éléments distincts de $\mathbb{K}$. + - Calculer le déterminant de la matrice $(P^{(i)}(\alpha_j))_{0\leq i,j\leq n}$. + - Montrer que $(P(X+\alpha_j))_{0\leq j\leq n}$ est une base de $\mathbb{K}_n[X]$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 537] +Soient $n\gt 2$, $m=2^n-2$, $E=\db{1,n}$ et $\mc{F}=\mc{P}(E)\setminus\{\emptyset,E\}$. + - Montrer qu'il existe une unique bijection $g\colon\mc{F}\ra\mc{F}$ telle que $\forall\alpha\in\mc{F}$, $g(\alpha)\cap\alpha=\emptyset$. + - On se donne une enumeration $\alpha_1,...,\alpha_m$ de $\mc{F}$. Soit $A=(a_{i,j})\in\M_m(\R)$ la matrice définie par $a_{i,j}=-1$ si $\alpha_i\cap\alpha_j=\emptyset$ et $0$ sinon. Calculer $\det(A)$. +#+end_exercice + +#+begin_exercice [Mines MP 2024 # 538] +Soient $n\in\N^*$, $E$ un $\mathbb{K}$-espace vectoriel de dimension $3n$ et $f\in\mc{L}(E)$. On suppose que $f^3=0$ et $\op{rg}(f)=2n$. Montrer qu'il existe une base dans laquelle la matrice de $f$ est egale à $\left(\begin{array}{c|c}0&I_n&0\\ \hline 0&0&I_n\\ \hline 0&0&0\end{array}\right)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 539] +Soit $G$ un sous-groupe de $\op{GL}_n(\R)$ vérifiant $\forall M\in G,\ M^2=I_n$. Montrer que $G$ est fini. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 540] +Soit $I$ l'ensemble des matrices inversibles de $\M_n(\Z)$ et $A\in\M_n(\Z)$. + - Preciser la structure algebrique de $I$. + - Montrer que $A\in I$ si et seulement si $\det A\in\{-1,1\}$. + - Pour toute colonne $X$ à coefficients entiers, on note $\alpha(X)$ le pgcd de ses coefficients. Montrer que $A\in I$ si et seulement si, pour toute colonne $X$ à coefficients entiers, $\alpha(AX)=\alpha(X)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 541] +Déterminer les parties $G\subset\M_n(\C)$ telles que $(G,\times)$ est un groupe multiplicatif mais pas un sous-groupe de $\op{GL}_n(\C)$. Montrer que toutes les matrices de $G$ ont même rang. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 542] +Soit $f\in\op{GL}\left(\M_n(\R)\right)$ vérifiant : $\forall A,B\in\M_n(\R),f(AB)=f(A)f(B)$. + - Calculer $f(I_n)$. + - On pose $\Delta=\text{Diag}(1,\ldots,n)$. Montrer qu'il existe une matrice $P\in\op{GL}_n(\R)$ telle que $f(\Delta)=P\Delta P^{-1}$. Montrer que, pour toute matrice diagonale $D$, on a : $f(D)=PDP^{-1}$. + - Expliciter $f$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 543] +Soient $A,B\in\M_n(\C)$. On suppose qu'il existe $c\in\C$ tel que $AB-BA=cA$. + - Montrer que $\forall k\in\N$, $(A-cI_n)^kB=BA^k$. + - Montrer que $\forall t\in\R$, $e^{-ct}e^{tA}B=Be^{tA}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 544] +Pour $M\in\M_n(\R)$, on dit que $M$ est_stochastique_ si : $\forall(i,j)\in\db{1,n]\!]^2,m_{i,j}\geq 0$ et $\forall i\in[\![1,n},\sum_{j=1}^nm_{i,j}=1$. Soit $A\in\M_n(\R)$. Trouver une condition nécessaire et suffisante pour que $\exp(tA)$ soit stochastique pour tout $t\in\R^+$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 545] + - Soient $M\in\M_{n,p}(\mathbb{K})$ et $N\in\M_{p,n}(\mathbb{K})$. Trouver une relation entre $\chi_{MN}$ et $\chi_{NM}$. + - Soit $A\in\op{GL}_n(\mathbb{K})$. On pose $B=(1+a_{i,j})_{1\leq i,j\leq n}$, on écrit $A^{-1}=(s_{i,j})_{1\leq i,j\leq n}$ et on pose enfin $S=\sum_{1\leq i,j\leq n}s_{ij}$. Trouver une relation entre $\det A$, $\det B$ et $S$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 546] +Soient $J=$ $\begin{pmatrix}0&1&0&\cdots&0\\ 0&\ddots&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&0\\ 0&\cdots&0&\ddots&1\\ 1&0&\cdots&0&0\end{pmatrix}$, $A=$ $\dfrac{1}{2}$ $\begin{pmatrix}0&1&0&\cdots&0&1\\ 1&\ddots&\ddots&\ddots&\vdots&0\\ 0&\ddots&\ddots&\ddots&0&\vdots\\ \vdots&\ddots&\ddots&\ddots&\ddots&0\\ 0&\cdots&0&\ddots&\ddots&1\\ 1&0&\cdots&\cdots&1&0\end{pmatrix}\in\M_n(\R)$. + - Montrere que $J$ est diagonalisable dans $\M_n(\C)$, et preciser ses éléments propres. + - Déterminer les éléments propres de la matrice $A$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 547] +Soient $(a_1,\ldots,a_n)\in\C^n$ et $M=$ $\begin{pmatrix}0&\ldots&0&a_n\\ a_1&\ddots&\vdots&0\\ \vdots&\ddots&0&\vdots\\ 0&\ldots&a_{n-1}&0\end{pmatrix}$. à quelle condition + + $M$ est-elle diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 548] +Soient $a,b\in\R$ avec $a^2\neq b^2$. Diagonaliser si possible la matrice $A\in\M_{2n}(\R)$ telle que $a_{i,j}=a$ si $i+j$ est pair et $a_{i,j}=b$ sinon. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 549] +Soit $A=$ $\begin{pmatrix}0&1&0&0\\ 1&k&1&1\\ 0&1&0&0\\ 0&1&0&0\end{pmatrix}\in\M_4(\C)$. + - Justifier que $A$ est diagonalisable lorsque $k\in\R$. + - Montrere que $\chi_A=X^2(X-u_1)(X-u_2)$ avec $u_1+u_2=k$ et $u_1^2+u_2^2=k^2+6$. + - à quelle condition $A$ est-elle diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 550] +Soit $n\in\N^*$. Soit $A=(a_{i,j})\in\M_n(\R)$ définie par $a_{i,j}=j$ si $i\neq j$ et $0$ sinon. + - Calculer $\det(A+kI_n)$ pour $k\in\{1,2,...,n\}$. + - - Montrere que $A$ à $n$ valeurs propres distinctes. + - Pour $\lambda$ valeur propre de $A$, montrer que $\sum_{k=1}^n\dfrac{k}{\lambda+k}=1$. + - Déterminer la somme et le produit des valeurs propres de $A$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 551] +Soit $A\in\M_n(\R)$ une matrice diagonalisable dans $\M_n(\C)$. Montrere que les matrices $A$ et $A^T$ sont semblables dans $\M_n(\R)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 552] +Soit $\omega$ un nombre complexe non réel + - Montrere qu'il existe un unique couple $(\alpha,\beta)\in\R^2$ tel que $\omega^2=\alpha\omega+\beta$. + - Montrere que, si $z\in\C$, il existe un unique $(\lambda,\mu)\in\R^2$ tel que $z=\lambda+\mu\omega$. + - Soient $E$ un $\R$-espace vectoriel de dimension finie $2n$ et $u\in\mc{L}(E)$. On suppose que $u^2=\alpha u+\beta\op{id}_E$. On pose $(\lambda+\mu\omega)*x=\lambda x+\mu u(x)$ pour tous $(\lambda,\mu)\in\R^2$ et $x\in E$. Montrere que $(E,+,*)$ est un $\C$-espace vectoriel de dimension finie. + - Soit $(e_1,\ldots,e_p)$ une base de ce $\C$-espace vectoriel.Montrer que $e=(e_1,u(e_1),\ldots,e_p,u(e_p))$ est une base du $\R$-espace vectoriel $E$. + - Quelle est la matrice de $u$ dans $e\,?$ Son polynôme caractéristique? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 553] +Soient $E$ un espace vectoriel de dimension finie, $H$ un hyperplan de $E$, $u\in\op{GL}(E)\setminus\{\op{id}\}$ tel que $\forall x\in H$, $u(x)=x$. Montrer l'équivalence des conditions suivantes : + - pour tout supplementaire $S$ de $H$ dans $E$, il existe $x\in S$ tel que $u(x)\neq x$ ; + - $u$ est diagonalisable ; + +_(iii)_ $u$ admet une valeur propre autre que $1$ ; + - $\det(u)\neq 1$ ; + - l'image de $u-\op{id}$ n'est pas contenue dans $H$ ; + - il existe $\lambda\neq 1$ et une base de $E$ dans laquelle la matrice de $u$ est $\op{Diag}(1,\ldots,1,\lambda)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 554] +Soient $E$ un espace vectoriel de dimension finie et $s\in\mc{L}(E)$ une symétrie. + +Soit $\Phi:u\in\mc{L}(E)\mapsto\dfrac{su+us}{2}$. Déterminer les éléments propres de $\Phi$ puis étudier sa diagonalisabilité. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 555] +Soient $A,B\in\M_n(\R)$ des matrices non nulles. Soit $f$ l'endomorphisme de $\M_n(\R)$ défini par $f(M)=M+\op{tr}(AM)B$ pour tout $M\in\M_n(\R)$. + - Déterminer un polynôme annulateur de degre $2$ de $f$. + - Donner une condition nécessaire et suffisante pour que $f$ soit diagonalisable. + - Déterminer les éléments propres de $f$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 556] +Soit $B\in\M_3(\R)$. Donner une condition nécessaire et suffisante sur $B$ pour que l'équation $A^3=B$ admette au moins une solution. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 557] +Pour $P\in\R[X]$, on pose $L(P)\in\R[X]$ le polynôme associe à la fonction polynomiale $x\mapsto\int_0^{+\i}P(x+t)\,e^{-t}dt$. + - Montrer que $L$ définit un endomorphisme de $\R[X]$. + - Montrer que $L=\sum_{k=0}^{+\i}D^k$ ou $D$ est l'endomorphisme de derivation de $\R[X]$. + - Déterminer les éléments propres de $L$. + - Déterminer le commutant de $L$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 558] +Soient $E=\mc C^0(\R,\R)$ et $\phi$ tel que, pour tout $f\in E$ et tout $x\in\R\colon\phi(f)(x)=\dfrac{1}{2x}\int_{-x}^xf(u)\,du$ si $x\neq 0$, $\phi(f)(0)=f(0)$. + - Montrer que $\phi$ est un endomorphisme de $E$. + - Trouver les éléments propres de $\phi$. + - Montrer que $\phi$ stabilise $\R_n[X]$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 559] +Soient $E=\mc C^0([-1,1],\C)$ et $g\in\mc C^0([-1,1],[-1,1])$ surjective et croissante. Soit $\Phi\in\mc{L}(E)$ définie par : $\forall f\in E$, $\Phi(f)=f\circ g$. On considére $F\neq\{0\}$ un sous-espace de dimension finie de $E$ stable par $\Phi$. + - Montrer que $\Phi_F$ est un automorphisme. - Montrer que 1 est l'unique valeur propre de $\Phi_F$. + - Montrer que $u=\Phi_F-\mathrm{id}_F$ est nilpotent. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 560] +Soient $E$ un $\C$-espace vectoriel de dimension finie, $v\in\mc{L}(E)$ diagonalisable et $P\in\C[X]$ non constant. Montrer qu'il existe $u\in\mc{L}(E)$ tel que $v=P(u)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 561] +Quelles sont les $M\in\M_n(\C)$ telles que l'ensemble $\{M^k\;;\;k\in\N\}$ soit fini? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 562] +Trouver les $A\in\M_n(\C)$ telles que $PA$ est diagonalisable pour tout $P\in\mathrm{GL}_n(\C)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 563] +Soit $A=\begin{pmatrix}aM&bM\\ bM&cM\end{pmatrix}$ avec $M\in\M_n(\R)$ et $a$, $b$, $c\in\R$. Étudier la diagonalisabilité de $A$ en fonction de $a$, $b$, $c$ et $M$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 564] +Soient $A$, $B$, $C\in\M_n(\C)$ telles que $AB=BC$. Déterminer une condition nécessaire et suffisante pour que $\begin{pmatrix}A&B\\ 0&C\end{pmatrix}$ soit diagonalisable. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 565] +Soit $A\in\M_n(\Z)$ tel que $A^p=I_n$ ( $p\in\N^*$). Soit $m\geq 3$. On suppose que les coefficients de $A-I_n$ sont divisibles par $m$. Montrer que $A=I_n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 566] +Soit $M=(m_{i,j})_{1\leq i,j\leq n}\in\M_n(\C)$. On note $\overline{M}=\big(\overline{M_{i,j}}\big)_{1\leq i,j\leq,n}$. + - Montrer qu'il existe $\alpha\in\mathbb{U}$ tel que $\alpha M+\overline{\alpha}I_n\in\mathrm{GL}_n(\C)$. + - Montrer l'équivalence entre : + +(i) $M\overline{M}=\lambda I_n$ avec $\lambda\geq 0$, (ii) $\exists P\in\mathrm{GL}_n(\C)$, $\exists\mu\in\C$, $M=\mu P\overline{P}^{-1}$. + - Montrer l'équivalence entre : (i) $M\overline{M}$ est diagonalisable et $\text{Sp}\left(M\overline{M}\right)\subset\R^+$, + +(ii) $M=PD\overline{P}^{-1}$ avec $P\in\mathrm{GL}_n(\C)$ et $D\in\M_n(\C)$ diagonale. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 567] + - Montrer l'existence et l'unicité d'une suite $(P_n)_{n\geq 0}$ de polynômes telle que $P_0=2$, $P_1=X$ et $\forall n\in\N$, $P_{n+2}=XP_{n+1}-P_n$, $\deg(P_n)=n$. + - Soit $n,N\in\N^*$. Soit $A\in\M_N(\C)$ telle que $P_n(A)=0$. Montrer que $A$ est diagonalisable. + - Soit $n\geq 2$. Résoudre le systeme $\forall i\in\db{1,n}$, $x_i=x_{i-1}+x_{i+1}$ en convenant que $x_0=x_{n+1}=0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 568] +Soit $A\in\M_n(\R)$ diagonalisable sur $\C$. Montrer que $A$ est semblable sur $\R$ à une matrice diagonale par blocs dont les blocs diagonaux sont soit de taille $1$, soit de la forme $\left(\begin{array}{cc}a&-b\\ b&a\end{array}\right)$ avec $(a,b)\in\R\times\R^*$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 569] +Soient $A$ et $B$ deux matrices non cotrigonalisables de $\M_2(\C)$. Montrer qu'il existe $P\in\text{GL}_2(\C)$ telle que $P^{-1}AP$ soit triangulaire supérieure et $P^{-1}BP$ triangulaire inférieure. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 570] +Soient $E$ un $\mathbb{K}$-espace vectoriel de dimension finie et $f\in\mc{L}(E)$. + - Soit $F$ un plan stable par $f$. Montrer qu'il existe $P\in\mathbb{K}[X]$ non nul de degre au plus $2$ tel que : $F\subset\mathrm{Ker}\,P(f)$. - Soit $P\in\mathbb{K}[X]$ non nul de degre $2$ divisant le polynôme minimal de $f$. Montrer qu'il existe un plan $F$ stable par $f$ tel que $F\subset\op{Ker}P(f)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 571] +Soient $\mathbb{K}$ un corps et $E$ un $\mathbb{K}$-espace vectoriel de dimension $n$. Soit $u\in\mc{L}(E)$. Donner une condition nécessaire et suffisante sur $\chi_u$ pour que les seuls sous-espaces stables par $u$ soient $\{0\}$ et $E$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 572] +Soient $A$ et $B$ dans $\M_n(\C)$. Montrer l'équivalence entre : (i) $BA=0$ et $B$ nilpotente, (ii) $\forall M\in E$, $\chi_{AM+B}=\chi_{AM}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 573] +Soient $A,B$ dans $\M_n(\C)$ telles que $\op{sp}A\cap\op{sp}B=\emptyset$. + - Montrer que $\chi_A(B)$ est inversible. + - Soit $M\in\M_n(\C)$. Montrer qu'il existe une unique matrice $X$ telle que $AX-XB=M$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 574] +Quelles sont les $A$ de $\M_n(\C)$ qui commutent avec chaque matrice de leur classe de similitude? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 575] +Soient $A,B\in\M_n(\C)$. + - On suppose que $AB-BA=\alpha A$ avec $\alpha\in\C$. Montrer que $A$ et $B$ sont cotrigonalisables. + - On suppose que $AB-BA=\alpha A+\beta B$. Montrer que $A$ et $B$ sont cotrigonalisables. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 576] +Soient $A,B\in\M_n(\mathbb{K})$. + - On suppose que $A$ et $B$ admettent une valeur propre commune $\lambda$. Montrer qu'il existe $C\in\M_n(\mathbb{K})$ non nulle telle que $AC=CB=\lambda C$. + - On suppose qu'il existe $C\in\M_n(\mathbb{K})$ non nulle telle que $AC=CB$, et on note $r$ le rang de $C$. Montrer que $\chi_A$ et $\chi_B$ admettent un diviseur commun de degre $r$. + - Étudier la réciproque. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 577] +Pour $A\in\M_n(\C)$, soit $C(A)$ la sous-algèbre des matrices de $\M_n(\C)$ qui commutent avec $A$. + - On suppose que $A$ est diagonalisable. Calculer la dimension de $C(A)$. à quelle condition a-t-on $C(A)=\C[A]$? + - Montrer que, sans hypothese sur $A$, la dimension de $C(A)$ est supérieure ou egale à $n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 578] +Pour $A\in\M_n(\C)$, soit $C(A)$ la sous-algèbre des matrices de $\M_n(\C)$ qui commutent avec $A$. à quelle condition sur $A$ est-il vrai que $C(A)$ ne contient aucune matrice nilpotente non nulle? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 579] +Soient $n\in\N^*$, $A$, $B\in\M_n(\R)$, $P\in\R[X]$ et $M=\begin{pmatrix}A&B\\ 0&A\end{pmatrix}$. + - Supposons $\deg P\geq 2$. Montrer que, si $P$ est scindé à racines simples, $P'$ l'est egalement. + - Calculer $P(M)$ en fonction de $P(A)$, $P'(A)$ et $B$. + - Montrer que $M$ est diagonalisable dans $\R$ si et seulement si $A$ est diagonalisable dans $\R$ et $B=0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 580] +Soient $E$ un espace prehilbertien réel et $(e_1,\ldots,e_n)$ une famille libre de vecteurs de $E$ telle que $\|x\|^2=\sum_{i=1}^n\left\langle x,e_i\right\rangle^2$ pour tout $x\in E$. + - Montrer que $(e_1,\ldots,e_n)$ est une base orthonormale de $E$. + - On remplace l'hypothese $\lnot(e_1,\ldots,e_n)$ est libre $\triangleright$ par $\lnot\lnot$ les vecteurs $e_1,\ldots,e_n$ sont non-nuls $\triangleright$. Le résultat subsiste-t-il? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 581] +On munit $\R^n$ de son produit scalaire canonique. Soient $\delta\gt 0$ et $A$ une partie de $\R^n$ vérifiant : $\forall(x,y)\in A^2,x\neq y\implies\|x-y\|=\delta$. + - Soient $p\in\N$ et $u_0,\ldots,u_p\in A$ distincts. On considére la matrice $M\in\M_p(\R)$ définie par : $m_{i,j}=\left\langle u_i-u_0,u_j-u_0\right\rangle$. Montrer que $M$ est inversible. + - Montrer que $A$ est finie. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 582] + - Montrer que $(P,Q)\mapsto\int_{-1}^1\frac{P(t)\,Q(t)}{\sqrt{1-t^2}}dt$ est un produit scalaire sur $\R[X]$. + - - Montrer que, pour tout $n\in\N$, il existe un unique polynôme $T_n$ tel que $\forall x\in\R,\ T_n(\cos(x))=\cos(nx)$. + - Donner, pour $n\in\N^*$, degre et coefficient dominant de $T_n$. + - Soit $n\in\N^*$. On note $U_n$ l'ensemble des polynômes réels unitaires de degre $n$. + +Calculer $\min_{P\in U_n}\int_{-1}^1\frac{P^2(t)}{\sqrt{1-t^2}}dt$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 583] +Soit $M\in\M_n(\R)$. Montrer que : $|\det M|\leq\prod_{j=1}^n\sqrt{\sum_{i=1}^nm_{i,j}^2}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 584] +Soient $E$ un espace euclidien et $f\in\mc{L}(E)$ tel que $\|f(x)\|\leq\|x\|$ pour tout $x\in E$. Étudier la convergence de la suite $(u_n)_{n\geq 0}$ définie par $u_n=\frac{1}{n+1}\sum_{k=0}^nf^k$ pour tout $n\geq 0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 585] +Soient $E$ un espace euclidien et $f\in\mc{L}(E)$ un endomorphisme $1$-lipschitzien. Montrer que : $E=\mathrm{Ker}(f-\mathrm{id})\overset{\perp}{\oplus}\mathrm{Im}(f-\mathrm{id})$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 586] +Soit $M\in\M_n(\R)$ une matrice nilpotente non nulle. + +Déterminer l'image de l'application $\phi:x\in\R^n\mapsto x^TMx$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 587] +Soit $E$ un espace vectoriel euclidien. Montrer que l'application $f:x\in E\mapsto\frac{x}{\max(\|x\|,1)}$ est $1$-lipschitzienne. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 588] +Soit $(a,b,x_0)$ une famille libre d'un espace euclidien $E$. Trouver une condition nécessaire et suffisante pour qu'il existe un endomorphisme $u$ de $E$ tel que $u(x_0)=a$ et $u^*(x_0)=b$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 589] +Soient $E$ un espace euclidien, $p$ et $q$ dans ${\cal L}(E)$ des projecteurs orthogonaux. Montrer que $q\circ p$ est un projecteur si et seulement si c'est un projecteur orthogonal. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 590] +Soient $E$ un espace euclidien, $u$ et $v$ dans ${\cal O}(E)$ telles que $\det(u)\det(v)\lt 0$. Calculer $\|v-u\|_{_{\rm op}}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 591] +Pour ${\mathbb{K}}={\R}$ ou ${\mathbb{K}}={\C}$, on appelle $d_n({\mathbb{K}})$ la dimension du plus grand sous-espace vectoriel de ${\cal M}_n({\mathbb{K}})$ dont tous les éléments sont diagonalisables. + - Que peut-on dire du spectre réel d'une matrice antisymétrique? + - Déterminer $d_n({\R})$. + - Déterminer $d_2({\C})$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 592] +Soit $n\geq 3$. Soient $A,B\in{\R}^n$ non colineaires. On pose : $M=AB^T+BA^T$. + - Montrer que $M$ est diagonalisable. + - Déterminer ${\rm rg}\,M$. + - Déterminer les valeurs propres et les sous-espaces propres de $M$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 593] +Soit $J=\begin{pmatrix}0_n&I_n\\ I_n&0_n\end{pmatrix}\in{\cal M}_{2n}({\R})$. Soit $G=\{M\in{\cal M}_{2n}({\R}),M^TJM=J\}$. + - Montrer que $G$ est un sous-groupe de ${\rm GL}_{2n}({\R})$. + - Caractériser les éléments de ${\cal O}_{2n}({\R})\cap G$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 594] +Décrire $\left\{e^A\ ;\ A\in{\cal A}_n({\R})\right\}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 595] +Soient $f:{\cal M}_n({\R})\ra{\R}^{+*}$ continue et $A\in{\cal A}_n({\R})$. Montrer que $\inf_{x\in{\R}}f(e^{xA})\gt 0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 596] + - Trouver toutes les applications $f$ de ${\R}^n$ dans ${\rm GL}_n({\R})$ telles que + + $\forall x\in{\R}^n,\forall P\in{\rm GL}_n({\R}),f(Px)=Pf( x)P^{-1}$. + - Même question en remplacant ${\rm GL}_n({\R})$ par ${\cal\tilde{O}}_n({\R})$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 597] + - Soit $A\in{\cal A}_n({\R})$. Montrer que ${\rm Sp}_{{\C}}(A)\subset i{\R}$. + - On note ${\cal L}$ l'ensemble des matrices $M\in{\rm SO}_n({\R})$ telles que $-1\notin{\rm Sp}(M)$. Montrer que l'application $\phi:{\cal A}_n({\R})\ra{\cal L},M\mapsto(I_n+M)(I_n-M)^{-1}$ est une bijection. + - Soit $Q\in{\rm SO}_2({\R})$. + +Résoudre l'équation : $(I_n+X)(I_n-X)^{-1}=Q$ d'inconnue $X\in{\cal A}_2({\R})$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 598] +Soit $A=(a_{i,j})_{1\leq i,j\leq n}\in{\cal O}_n({\R})$. Montrer que + + $\Big{|}\sum_{1\leq i,j\leq n}a_{i,j}\Big{|}\leq n \leq\sum_{1\leq i,j\leq n}|a_{i,j}|\leq n\sqrt{n}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 599] +On munit ${\R}^3$ de sa structure euclidienne canonique. + +Soient $e_1,e_2\in{\R}^3$ et $f:x\mapsto\langle x,e_1\rangle\,e_2+\langle x,e_1\rangle\,e_1$. + - Si $e_1$ et $e_2$ sont lineairement indépendants, montrer qu'il existe une base orthonormée de ${\R}^3$ dans laquelle la matrice de $f$ est ${\rm Diag}(\lambda_1,\lambda_2,0)$ avec $\lambda_1,\lambda_2\in{\R}^*$. + - Étudier la réciproque. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 600] +Soit $E$ un espace réel de dimension $n\geq 2$. Lorsque $\Phi$ est un produit scalaire sur $E$, on note $\mc{O}_{\Phi}(E)$ le groupe des isométries pour $\Phi$, et $\mc{S}_{\Phi}^{++}(E)$ l'ensemble des endomorphismes autoadjoints définis positifs pour $\Phi$. + - On fixe un produit scalaire $\Phi$. Montrer que les propositions suivantes sont équivalentes : + +(i) $\Psi$ est un produit scalaire, (ii) $\exists a\in\mc{S}_{\Phi}^{++}(E),\Psi(x,y)=\Phi(a(x),y)$. + - Soit $u\in\mc{O}_{\Phi}(E)$. Déterminer une condition nécessaire et suffisante pour que $u\in\mc{O}_{\Psi}(E)$ (on utilisera l'endomorphisme $a$ de la question précédente). + - Soit $P$ l'ensemble des produits scalaires sur $E$. Déterminer $\bigcap_{\Psi\in P}\mc{O}_{\Psi}(E)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 601] +Soit $M\in\M_n(\R)$. Montrer qu'il existe une base orthonormée $(e_1,\ldots,e_n)$ de $\R^n$ telle que $(Me_1,\ldots,Me_n)$ soit orthogonale. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 602] +Soit $k$ un réel fixe. On pose $A=$ $\begin{pmatrix}k&1&0&\cdots&0\\ 1&\ddots&\ddots&\ddots&\vdots\\ 0&\ddots&\ddots&\ddots&0\\ \vdots&\ddots&\ddots&\ddots&1\\ 0&\cdots&0&1&k\end{pmatrix}\in\M_n(\R)$. + +Montrer que $\max_{\lambda\in\op{Sp}A}\lambda\geq k+1$ et $\min_{\lambda\in\op{Sp}A}\lambda\geq k-1$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 603] +Soit $\in\mc{S}_n(\R)$. + - Montrer l'équivalence des enonces suivants : (i) $x^TSx\geq 0$ pour tout $x\in\R^n$, + +(ii) $\op{Sp}S\subset\R^+$, (iii) il existe $T\in\mc{S}_n(\R)$ telle que $S=T^2$. + +Desormais, on suppose ces conditions realisées. + - Montrer que, pour tous $1\leq i\neq j\leq n$ et $x,y\in\R$, $s_{i,i}x^2+2s_{i,j}xy+s_{j,j}y^2\geq 0$. En déduire que $s_{i,j}^2\leq s_{i,i}s_{j,j}$. + - On suppose de plus les coefficients de $S$ non nuls, et on pose $T=\left(\frac{1}{s_{i,j}}\right)_{1\leq i,j\leq n}$. Montrer que $T\in\mc{S}_n^+(\R)$ si et seulement si $\op{rg}S=1$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 604] +Soit $A\in\M_n(\R)$. + - Donner une condition nécessaire et suffisante pour qu'il existe $S\in\mc{S}_n(\R)$ telle que $A=S^2+S+I_n$. + - à quelle condition la matrice $S$ est-elle unique? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 605] +Soient $A,C\in\mc{S}_2(\R)$ et $B\in\mc{A}_2(\R)$. + - Montrer que $M=\begin{pmatrix}A&-B\\ B&C\end{pmatrix}$ est diagonalisable. + - On suppose ici que $B=0$. Donner une base de diagonalisation de $M$ construite à partir de vecteurs propres de $A$ et $C$. + - Montrer que, pour tous $E\in\op{GL}_2(\R)$ et $G\in\M_2(\R)$, $\op{rg}(EG)=\op{rg}(GE)=\op{rg}(G)$. + - On suppose ici que $A$ est inversible. On pose $P=\begin{pmatrix}I_2&A^{-1}B\\ 0&I_2\end{pmatrix}$. Calculer $MP$. En déduire le rang de $M$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 606] +Soit $A=\left(\frac{1}{i+j}\right)_{1\leq i,j\leq n}$. Montrer que $A$ est diagonalisable et que son spectre est inclus dans $\R^{+*}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 607] +Soit $A_n=\left(\frac{1}{i+j+1}\right)_{0\leq i,j\leq n}$. Montrer que les valeurs propres de $A_n$ sont dans $]0,\pi[$ et que la plus petite valeur propre de $A_n$ est inférieure à $\frac{1}{2n+1}$. On pourra montrer que, pour $P\in\R[X]$, on a $\int_{-1}^1P(t)\dt+\int_0^{\pi}P(e^{i\theta})ie^{i\theta}\, d\theta=0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 608] +Soient $E$ un espace euclidien, $u\in\mc{S}(E)$, $a$ et $b$ deux réels tels que $a\lt b$, $P\in\R[X]$ tel que $\forall x\in[a,b],P(x)\gt 0$. On suppose que $\forall x\in E,\ a\|x\|^2\leq\langle u(x),x\rangle\leq b\|x\|^2$. Montrer que $P(u)\in\mc{S}^{++}(E)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 609] +Soit $M\in\M_n(\R)$. Montrer que $M$ est combinaison lineaire de quatre matrices orthogonales. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 610] +Soit $A\in\M_n(\R)$ telle que $A^TA=A^TA$. Montrer que si $F$ est un sous-espace de $\R^n$ stable par $A$ alors $F^{\perp}$ est stable par $A^T$. On suppose $n=3$. Montrer que $A$ est soit diagonalisable, soit semblable à une matrice de la forme $\begin{pmatrix}\lambda&0&0\\ 0&\alpha&\beta\\ 0&-\beta&\alpha\end{pmatrix}$ avec $\beta\neq 0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 611] +Soit $M\in\mathrm{GL}_n(\R)$. + - Montrer qu'il existe un unique couple $(O,S)\in\mc{O}_n(\R)\times S_n^{++}(\R)$ tel que $M=OS$. + - Déterminer $\sup_{A\in\mc{O}_n(\R)}\mathrm{tr}(AM)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 612] +Soit $(E,\langle\,\ \rangle)$ un espace euclidien. + - Soit $u\in\mc{S}(E)$. Montrer que $E=\mathrm{Ker}(u)\oplus\mathrm{Im}\,u$. + - Soit $u\in\mc{S}^+(E)$. Montrer qu'il existe $h\in\mc{S}^+(E)$ tel que $u=h^2$. + - Soient $f,g\in\mc{S}^+(E)$ tels que $\mathrm{Ker}(f+g)=\mathrm{Ker}\,f\cap\mathrm{Ker}\,g$. + +Montrer que $\mathrm{Im}(f+g)=\mathrm{Im}\,f+\mathrm{Im}\,g$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 613] +Soient $S\in\mc{S}_n^{++}(\R)$ et $A\in\M_n(\R)$ qui commute avec $S^2$. Montrer que $A$ commute avec $S$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 614] +Soient $A,B\in\mc{S}_n^+(\R)$ telles que $A^2B^2=B^2A^2$. Montrer que $AB=BA$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 615] +Soient $n,k\in\N^*$. Étudier l'injectivite et la surjectivite de l'application $f\colon\mc{S}_n(\R)\ra\mc{S}_n(\R)$ définie par $f(A)=A^k$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 616] +Soit $M\in\mathrm{GL}_n(\R)$. + - Montrer qu'il existe $P\in\mc{O}_n(\R)$ et $D=\mathrm{Diag}(\lambda_1,\ldots,\lambda_n)$ avec $\lambda_i\gt 0$ pour tout $i$ telles que $P^TM^TMP=D^2$. + - On note $V_1,\ldots,V_n$ les colonnes de $MP$.Soit $Q$ la matrice dont les colonnes sont $\frac{1}{\lambda_1}V_1,\ldots,\frac{1}{\lambda_n}V_n$. Montrer que $Q\in\mc{O}_n(\R)$. + - Montrer qu'il existe $O,O'$ dans $\mc{O}_n(\R)$ telles que $M=ODO'$. + - Montrer le même résultat si $M$ est non inversible. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 617] +Soit $n\geq 2$ + - Déterminer le sous-espace vectoriel engendre par $\mc{S}_n^{++}(\R)$. + - Déterminer le plus petit sous-anneau de $\M_n(\R)$ contenant $\mc{S}_n^{++}(\R)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 618] +Soit $n\geq 2$. + - Soit $S\in\mc{S}_n^{++}(\R)$. Montrer qu'il existe $P\in\mathrm{GL}_n(\R)$ tel que $S=P^TP$. + - Déterminer le sous-espace vectoriel engendre par $\mc{S}_n^{++}(\R)$. + - Soient $A_1$,..., $A_k\in\mc{S}_n^{++}(\R)$, $\alpha_1$,..., $\alpha_k\in\R$. + +Montrer que $|\mathrm{det}(\alpha_1A_1+\cdots+\alpha_kA_k)|\leq\mathrm{det}(| \alpha_1|A_1+\cdots+|\alpha_k|A_k)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 619] +Soient $E$ un espace euclidien et $u\in\mc{L}(E)$. + - Montrer que $|\!|\!|u|\!|\!|=\sup\limits_{\|x\|=1}\|u(x)\|=\sup\limits_{\|x\|\leq 1} \|u(x)\|$. + - Montrer que $|\!|\!|u|\!|\!|=\sup\limits_{\|x\|=1,\|y\|=1}|\langle u(x),y\rangle|=\sup \limits_{\|x\|\leq 1,\|y\|\leq 1}|\langle u(x),y\rangle|$. + - On suppose $u$ symétrique. Montrer que $|\!|\!|u|\!|\!|=\sup\limits_{\|x\|=1}|\langle u(x),x\rangle|=\sup\limits_{\|x \|\leq 1}|\langle u(x),x\rangle|$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 620] +On munit $\M_n(\R)$ de la norme subordonnée à la norme euclidienne canonique. + +Soit $A\in\mathrm{GL}_n(\R)$. On note $r$ la plus petite valeur propre de $A^{\rap}A$ et $R$ la plus grande. Montrer que $\|A\|^2=R$ et $\|A^{-1}\|^{-2}=r$. +#+end_exercice + + +** Analyse + +#+begin_exercice [Mines MP 2024 # 621] +Soient $E=\mc C^1([0,1],\R)$ et $N:f\mapsto\sqrt{f(0)^2+\int_0^1f'(t)^2\dt}$. + - Montrer que $N$ est une norme sur $E$. + - Compare $N$ à la norme $\|\!\|\i$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 622] +Pour $a\in\R$ et $P\in\R[X]$, on pose $N_a(P)=|P(a)|+\int_0^1|P'|$. + - Montrer que, pour tout $a\in\R$, $N_a$ est une norme. + - Soient $a,b\in\R$. Les normes $N_a$ et $N_b$ sont-elles équivalentes? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 623] +On munit $\R^n$ de la norme euclidienne canonique. Soit $f\in\mc C^0([a,b],\R^n)$. Montrer que $\left\|\int_a^bf\right\|=\int_a^b\|f\|$ si et seulement s'il existe $\Phi\in\mc C^0([a,b],\R^+)$ et $u\in\R^n$ tels que $\forall t\in[a,b],\,f(t)=\Phi(t)u$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 624] +On pose $E=\{f\in\mc C^2([0,1],\R),\;f(0)=f'(0)=0\}$. + - Montrer que $\|f\|=\|f+2f'+f''\|_{\i}$ définit une norme sur $E$. + - Les normes $\|\!|\!|$ et $\|\!|\!|\!|_{\i}$ sur $E$ sont-elles équivalentes? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 625] +Soit $Q\in\R[X]$. Construire une norme $N$ sur $\R[X]$ telle que : $N(X^n-Q)\underset{n\ra+\i}{\longrightarrow}0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 626] +Pour tout $P\in\R[X]$, on pose $N(P)=\sup\limits_{t\in[0,1]}|P(t)|$. Pour $n\in\N$, on note $E_n$ l'ensemble des polynômes unitaires de $\R_n[X]$ et $a_n=\inf\limits_{P\in E_n}N(P)$. + - Montrer que $a_n\gt 0$; calculer $a_0$ et $a_1$. + - Montrer que $(a_n)_{n\in\N}$ est decroissante et de limite nulle. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 627] +Déterminer les sous-groupes compacts de $(\C^*,\times)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 628] +Soit $Q\in\R[X]$ non nul. Pour $P\in\R[X]$, on pose $\|P\|_Q=\sup\limits_{x\in[-1,1]}|PQ(x)|$. + - Montrer que $\|\ \|_Q$ est une norme sur $\R[X]$. + - à quelle condition sur $Q$ la norme $\|\ \|_Q$ est-elle équivalente à $\|\ \|_1$ (norme associée au polynôme egal à 1)? + - Soit $c\in[-1,1]$ une racine de $Q$. Trouver $P\in\R[X]$ tel que $P(c)=1$, $P'(c)=0$ et $\forall x\in[-1,1]\setminus\{c\}$, $0\leq P(x)\lt 1$. + - Montrer que $\|P^n\|_Q\longrightarrow 0$ quand $n\ra+\i$. + - Qu'en déduire? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 629] +Soient $(E,\|\ \|)$ un espace vectoriel norme, $A$ une partie de $E$, $f:[0,1]\ra E$ continue. On suppose que $f(0)\in A$ et $f(1)\in E\setminus A$. Montrer que $f([0,1])\cap\text{Fr}(A)\neq\emptyset$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 630] +On munit $E=\mc C^0([a,b],\R)$ de la norme de la convergence uniforme. Soit $(x - {1\leq i\leq n}$ des points distincts de $[a,b]$ et $(y - {1\leq i\leq n}$ des réels. Montrer que l'adherence de l'ensemble $\{P\in\R[X];\forall i\in\db{1,n]\!],P(x_i)=y_i\}$ est $\{f\in E;\forall i\in[\![1,n},f(x_i)=y_i\}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 631] +On munit $\C[X]$ de la norme $\|P\|=\max|p_k|$ ou $P=\sum\limits_{k=0}^{+\i}p_kX^k$. + +Déterminer les valeurs $b\in\C$ pour lesquelles $f:P\mapsto P(b)$ est continue. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 632] +Soient $C$ une partie convexe d'un espace norme $E$, $X$ une partie de $E$ telle que $C\subset X\subset\overline{C}$. Montrer que $X$ est connexe par arcs. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 633] + - Soit $P\in\R[X]$ unitaire de degre $n$. Montrer que $P$ est scindé sur $\R$ si et seulement si, pour tout $z\in\C$, $|\op{Im}(z)|^n\leq|P(z)|$. + - On note $\mc{T}$ l'ensemble des matrices trigonalisables sur $\R$ et $\mc{D}$ l'ensemble des matrices diagonalisables. Montrer que $\mc{T}$ est un ferme de $\M_n(\R)$ et que l'adherence de $\mc{D}$ est $\mc{T}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 634] + - Montrer que l'image par une fonction continue d'une partie connexe par arcs est connexe par arcs. + - Montrer qu'une fonction continue injective de $\R$ dans $\R$ est strictement monotone. + - Soient $f\colon\R\ra\R$ continue et $F\colon\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\M_2(\R)\mapsto\begin{pmatrix}f(a)&f(b)\\ f(c)&f(d)\end{pmatrix}$. On suppose que $F$ envoie toute matrice inversible sur une matrice inversible. + - Montrer que $f$ est injective et ne s'annule pas sur $\R^*$. - Montrer que $f(0)=0$. +#+end_exercice + + + +#+begin_exercice [Mines MP 2024 # 643] +Déterminer la limite de $u_n=\frac{1}{16^n}\sum_{k=n}^{3n}\left(\begin{matrix}4n\\ k\end{matrix}\right)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 644] +Soit $(u_n)$ la suite définie par $u_0=x\gt 0$ et $\forall n\in\N$, $u_{n+1}=\frac{e^{u_n}}{n+1}$. + - Soit $k\in\N$. Montrer que, si $u_{k+1}\leq u_k$, alors la suite $(u_n)_{n\geq k+1}$ est strictement decroissante. + - Montrer que, si la suite $(u_n)$ est croissante, alors sa croissance est stricte. + +Que dire de sa limite? + - On admet que $e^{e-2}\lt 9/4$. Montrer que, pour $x$ suffisamment petit, la suite $(u_n)$ converge. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 645] +Soit $\alpha\gt 1$. On considére l'équation : $(E_n)\colon\prod_{k=1}^n(kx+n^2)=\alpha n^{2n}$. + - Montrer que pour tout $n\in\N^*$, $(E_n)$ possede une unique solution strictement positive. On la note $x_n$. + - Montrer que : $\forall n\in\N^*,x_n\lt 2\alpha$. + - Montrer la convergence et calculer la limite de la suite $(x_n)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 646] +Soit $f\colon\R^+\ra\R$ une fonction de classe $\mc C^1$ telle que $f(x)\ra+\i$ et $f'(x)\ra 0$ quand $x\ra+\i$. Montrer que $\left\{e^{if(n)},n\in\N\right\}$ est dense dans le cercle unite. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 647] +Soient $f\in\mc C^1(\R,\R)$ et $(u_n)$ une suite vérifiant $u_{n+1}=f(u_n)$ pour tout $n$. + - Montrer que si $(u_n)$ converge alors sa limite $\ell$ est un point fixe de $f$. Dans la suite on considére $a$ un point fixe de $f$. + - On suppose que $|f'(a)|\gt 1$. Montrer qu'il existe $\eta\gt 0$ et $k\gt 1$ tel que $|f'(x)|\geq k$ pour $x\in]a-\eta,a+\eta[$. Si $|f'(a)|\gt 1$ décrire les suites $(u_n)$ qui convergent vers $a$. + - On suppose que $|f'(a)|\lt 1$. Montrer qu'il existe $\eta\gt 0$ et $k\in[0,1[$ tel que $|f'(x)|\leq k$ pour $x\in]a-\eta,a+\eta[$. Montrer que la suite $(u_n)$ converge vers $a$ si et seulement s'il existe un rang $p$ tel que $u_p\in]a-\eta,a+\eta[$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 648] +Soit $(u_n)_{n\in\N}$ définie par $u_0\in\R^{+*}$ et, pour $n\in\N$, $u_{n+1}=\sqrt{u_n}+\frac{1}{n+2}$. + +Montrer qu'il existe un entier naturel $N$ tel que $u_N\gt 1$. + - Montrer qu'il existe $n_0\gt N$ tel que $(u_n)_{n\geq n_0}$ est decroissante. + - La suite $(u_n)$ est-elle convergente? Si oui, trouver sa limite. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 649] +Soit $f\colon\R^+\ra\R^+$ une fonction bornée et telle que $|f(x)-f(y)|\lt |x-y|$ pour tous $x,y\in\R^+$ tels que $x\neq y$. On considére une suite $(u_n)_{n\geq 1}$ telle que $u_1\in\R^+$ et $u_{n+1}=f(u_n)+\frac{1}{n}$ pour tout $n\geq 1$. On pose enfin $a_n=|u_{n+1}-u_n|$ pour tout $n\geq 1$. + - Soient $p$ et $q$ des entiers tels que $1\leq p\lt q$. Montrer que $a_q-a_p\leq\frac{1}{p}$. + - Montrer que la suite $(a_n)_{n\geq 1}$ est convergente. + - Montrer que la suite $(u_n)_{n\geq 1}$ est convergente. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 650] +Soit $(u_n)$ la suite définie par $u_0\gt -1$ et $\forall n\in\N^*,\;u_{n+1}=u_n+u_n^2$. + - Montrer que la suite $(u_n)$ converge. + - On suppose $u_0\gt 0$ et on pose $v_n=\frac{\ln(u_n)}{2^n}$ pour $n\in\N$. + - Montrer la convergence de la suite $(v_n)$ vers un réel $\alpha$ puis que $0\leq\alpha-v_n\leq\frac{1}{2^nu_n}$. + - Donner un équivalent de $u_n$. + - Donner un équivalent de $u_n$ dans le cas $u_0\in]-1,0[$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 651] +Soit $(u_n)_{n\geq 1}$ à valeurs dans $[0,1]$. On dit que $(u_n)$ est equirepartie si et seulement si, pour tous $\alpha\lt \beta$ dans $[0,1]$, on a $\frac{1}{n}\op{card}\big{\{}k\in\db{1,n},\alpha\lt u_n\lt \beta\big{\}}\mathop{\longrightarrow}\limits_{n\ra+\i}\beta-\alpha$. + - On suppose $(u_n)$ equiperaptie. Montrer que $(u_n)$ diverge. Montrer que $\{u_n,n\in\N^*\}$ est dense dans $[0,1]$. + - Montrer l'équivalence entre : + +( ii) $\forall f\in\mc C^0([0,1],\C),\lim\frac{1}{n}\sum_{k=1}^nf(u_k)=\int_0^1f(t)\dt$, + +(iii) $\forall m\in\N^*,\lim\frac{1}{n}\sum_{k=1}^ne^{2\pi imu_k}=0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 652] +Soit $(u_n)_{n\geq 0}$ une suite réelle decroissante de limite nulle. Quelle est la nature de la série $\sum(-1)^{\lfloor n/2\rfloor}u_n$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 653] +Existe-t-il une bijection $f\colon\N^*\ra\N^*$ telle que la série $\sum\frac{f(n)}{n^2}$ converge? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 654] +Soient $(u_n)$ une suite de réels non nuls et $\lambda\in\R$. + +On suppose que : $\frac{u_{n+1}}{u_n}=1-\frac{\lambda}{n}+O\left(\frac{1}{n^2}\right)\cdot$ Étudier la nature de $\sum u_n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 655] +Soit $f\colon\R\ra\R$ telle que $f(x)\mathop{=}\limits_{x\ra+\i}a_0+\frac{a_1}{x}+\cdots+\frac{a_p}{ x^p}+o\left(\frac{1}{x^p}\right)$. + - à quelle condition la série de terme general $u_n=f(n)$ converge-t-elle? + - à quelle condition la suite de terme general $v_n=\prod_{k=1}^nf(k)$ converge-t-elle? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 656] + - Soit $f:[1,+\i[\ra\R$ de classe $\mc C^1$. + +Montrer que, pour tout $n\geq 1$, $\left|f(n)-\int_n^{n+1}f(t)\dt\right|\leq\frac{1}{2}\max_{t \in[n,n+1]}|f'(t)|$. + - Quelle est la nature de la série $\sum\frac{\sin(\ln n)}{n}$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 657] +Pour tout $n\geq 0$, on pose $u_n=\int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}}\sin(x^2)dx$. - Étudier le signe de $u_n$. + - Montr'er que la série $\sum u_n$ est semi-convergente. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 658] +Existe-t-il une suite réelle $(u_n)$ telle que $\sum u_n$ converge et $\sum u_n^3$ diverge? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 659] +Soit $(u_n)_{n\in\N}$ à valeurs dans $\R^+$. + - On suppose $\sum u_n$ convergente et on pose $R_n=\sum_{k=n+1}^{+\i}u_k$. construire à partir de $R_n$ une suite $v_n\gt 0$ croissante tendant vers $+\i$ telle que $\sum u_nv_n$ converge. + - On suppose $\sum u_n$ divergente. construire $v_n$ decroissante qui tend vers $0$ telle que $\sum u_nv_n$ diverge. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 660] +Étudier la convergence de la série $\sum\sin(\pi en!)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 661] +Soit $(u_n)_{n\geq 0}$ une suite réelle telle que la série $\sum n(\ln n)^2u_n^2$ converge. Montr'er que la série $\sum u_n$ converge. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 662] +On considére la suite réelle définie par $x_0=0$ et $x_{n+1}=\sqrt{\frac{1+x_n}{2}}$ pour tout $n\geq 0$. + +Étudier la convergence de la série $\sum(1-x_n)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 663] +Soient $\alpha\in\R^+$ et $(u_n)_{n\geq 1}$ vérifiant $u_1\in\R^{+*}$ et, pour $n\in\N^*$, $u_{n+1}=u_n+\frac{1}{n^{\alpha}u_n}$. + - Déterminer les valeurs de $\alpha$ pour lesquelles $(u_n)$ converge. + - Trouver alors un équivalent de $\ell-u_n$, ou $\ell$ designe la limite de la suite. + - Donner un équivalent de $u_n$ lorsque $(u_n)$ diverge. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 664] +Soit $\sum u_n$ une série à termes positifs divergente. On pose $v_n=\frac{u_n}{\prod_{k=0}^n(1+u_k)}$ pour tout $n\geq 0$. Montr'er que la série $\sum v_n$ est convergente et calculer sa somme. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 665] +Soient $(u_n)_{n\geq 0}$ définie par $u_0\gt 0$ et, pour tout $n\in\N$, $u_{n+1}=\ln((\exp(u_n)-1)/u_n)$. + - Déterminer la limite eventuelle de $(u_n)$. + - En déduire la nature de $\sum u_n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 666] +Soit $T$ l'endomorphisme de $\R^{\N}$ qui à la suite $u$ associe $Tu$ telle que : + + $\forall n\in\N$, $(Tu)_n=\frac{1}{n+1}\sum_{k=0}^nu_k$. + - Si $u$ converge vers $\ell$, montr'er que $Tu$ converge vers $\ell$. + - On suppose que $u$ est à valeurs positives. + +On note $\sqrt{u}$ la suite telle que : $\forall n$, $(\sqrt{u})_n=\sqrt{u_n}$. Si $Tu$ tend vers 0, montr'er que $T\sqrt{u}$ tend egalement vers 0.On suppose $u$ positive et decroissante. + - On pose $w_n=\sqrt{n}\,u_n$. Montrer que $Tw$ tend vers 0 si et seulement si $w$ tend vers 0. + +On pose, pour $n\in\N$, $s_n=\sum_{k=0}^nu_k$ et $v_n=nu_n$. + - Montrer que $s-Ts=Tv$. + - On suppose que $Ts$ converge. Montrer que $Tv$ tend vers 0 si et seulement si la série $\sum u_n$ converge. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 667] +Soit $(u_n)$ une suite de réels positifs convergeant vers 0. On pose, pour tout $n\in\N$, $S_n=\sum_{k=0}^nu_k$ et on suppose $u_0\gt 0$ et $(|S_n-nu_n|)$ majorée. On suppose enfin $\sum u_n$ divergente. + - Montrer que $\ln S_n\sim\ln n$. + - Montrer que $\forall n$, $S_n\geq\sqrt{n}$. + - Montrer que $\lim u_n\gt 0$. Conclusion? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 668] +Soit $f$ une fonction de classe $\mc C^1$ de $\R^+$ dans $\R^{+*}$ telle que $\dfrac{f'(x)}{f(x)}\underset{x\ra+\i}{\ra}-\i$. Montrer que $\sum f(k)$ converge et donner un équivalent de $\sum_{k=n}^{+\i}f(k)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 669] +Soit $(u_n)_{n\geq 1}$ une suite réelle decroissante de limite nulle. Montrer que la série $\sum\dfrac{u_n}{n}$ converge si et seulement si la série $\sum(u_n-u_{n+1})\ln n$ converge. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 670] +Soient $\alpha\gt 0$ et $(a_n)$ définie par $a_1\gt 0$, $a_1+a_2\gt 0$ et $\forall n\geq 2$, $a_{n+1}=\dfrac{(-1)^n}{n^{\alpha}}\sum_{i=1}^na_i$. + +Déterminer les valeurs de $\alpha$ pour lesquelles la série $\sum a_n$ converge. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 671] +Nature et somme de la série de terme general $u_n=\sum_{k=n}^{+\i}\dfrac{(-1)^k}{k^2}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 672] +Soit $x\in\R\setminus(-\N)$. Montrer que $\sum_{n=0}^{+\i}\dfrac{1}{x(x+1)\cdots(x+n)}=e\sum_{n=0}^{+ \i}\dfrac{(-1)^n}{n!(x+n)}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 673] + - Pour $n\in\N^*$, soit $d(n)$ le nombre de diviseurs de $n$. + +Pour $\alpha\gt 1$, montrer que $\sum_{n=1}^{+\i}\dfrac{d(n)}{n^{\alpha}}=\zeta(\alpha)^2$. + - Pour $\alpha\gt 2$, montrer que $\sum_{n=1}^{+\i}\dfrac{\phi(n)}{n^{\alpha}}=\dfrac{\zeta(\alpha-1)}{ \zeta(\alpha)}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 674] + - Étudier la convergence de la suite $(u_n)$ définie par $u_0\gt 0$ et $\forall n\in\N$, $u_{n+1}=u_n^{u_n}$. On choisit desormais $u_0\in\N\setminus\{0,1\}$. + - Montrer que $\forall N\in\N,\ \forall n\in\db{0,N},\ u_n\mid u_N$. + - Montrer que, pour $N,k\in\N$, $u_{N+k}\geq u_N^{k+1}$. + - Montrer la convergence de la série $\sum\dfrac{1}{u_n}$. + - Montrer que $u_N\sum_{n=N+1}^{+\i}\dfrac{1}{u_n} \longrightarrow 0$ quand $N\ra+\i$. + - Montrer que $\sum_{n=0}^{+\i}\dfrac{1}{u_n}\notin\Q$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 675] +Soit $f\colon\R\ra\R$ croissante. Montrer que l'ensemble des points de discontinuité de $f$ est au plus dénombrable. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 676] +Soit $f\colon\R\ra\R$ de classe $\mc C^2$. Montrer que $f$ est convexe si et seulement si, pour tous $x\in\R$ et $r\in\R^+$, $2rf(x)\leq\int_{x-r}^{x+r}f(t)\dt$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 677] +Soit $I$ un intervalle non trivial de $\R$. Montrer que toute fonction de classe $\mc C^2$ sur $I$ est la différence deux fonctions convexes. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 678] +Soit $f(t)=\dfrac{1}{\sqrt{1+t^2}}$. Montrer que la derivée $n$-ieme de $f$ s'écrit sous la forme $\dfrac{P_n(t)}{(1+t^2)^{n+\frac{3}{2}}}$ ou $P_n\in\R[X]$. Trouver une relation lineaire entre $P_{n+2},P_{n+1}$ et $P_n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 679] +Soient $E$ un $\R$-espace vectoriel de dimension finie, $f\colon\R\ra E$ continue en 0. Montrer que $f$ est dérivable en 0 si et seulement si $x\mapsto\dfrac{f(2x)-f(x)}{x}$ possede une limite quand $x\ra 0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 680] +Soient $I=\left]-3,9\right[$ et $f$ une fonction de classe $\mc C^2$ de $I$ dans $\R$. Pour $x\in I\setminus\{3\}$, on pose $g(x)=\tan\left(\dfrac{\pi x}{6}\right)f(x)$. à quelle condition la fonction $g$ se prolonge-t-elle continument à $I$? Le prolongement est-il de classe $\mc C^1$ sur $I$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 681] +Une fonction de classe $\mc C^{\i}$ de $[0,1]$ dans $\R$ est-elle nécessairement monotone par morceaux? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 682] +Soit $f\colon\R^+\ra\R$ de classe $\mc C^{\i}$ telle que $f(0)\gt 0$, $f'(0)\gt 0$ et $\lim\limits_{x\ra+\i}f(x)=0$. + - Montrer qu'il existe $x_1$ tel que $f'(x_1)=0$. + - Montrer qu'il existe une suite $(x_n)$ strictement croissante telle que, pour tout $n\in\N$, $f^{(n)}(x_n)=0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 683] +Montrer que la fonction $x\mapsto e^{x^2}$ n'admet pas de primitive de la forme $x\mapsto f(x)e^{x^2}$, ou $f\colon\R\ra\R$ est une fonction rationnelle. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 684] +Soit $f\colon\R^+\ra\R$ de classe $\mc C^1$ telle que $f'(t)+f(t)\ra 0$ quand $t\ra+\i$. Montrer que $f(t)\ra 0$ quand $t\ra+\i$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 685] +Posons $f:x\neq 0\mapsto e^{-\frac{1}{x^2}}$ prolongée par continuité en $0$. + - Montrer que $f$ est de classe $\mc C^{\i}$ sur $\R$. + - Montrer que $f$ n'est solution d'aucune équation différentielle lineaire homogène. + - Pour $n\in\N$, soit $P_n\in\R[X]$ tel que $\forall x\neq 0$, $f^{(n)}(x)=P_n\left(\frac{1}{x}\right)f(x)$. Déterminer degre et coefficient dominant de $P_n$. + - Montrer que les polynômes $P_n$ sont scindés dans $\R$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 686] +Soient $I$ un intervalle non trivial de $\R$, $M\in\R^{+*}$ et $f$ une fonction de classe $\mc C^1$ de $I$ dans $\C$ non identiquement nulle et telle que $|f'|\leq M|f|$. Montrer que $f$ ne s'annule pas. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 687] +Soit $E=\mc C^0(\R^+,\R)$. + - Soit $f\in E$. Montrer $v:x\in\R^{+*}\mapsto\frac{1}{x^{p+1}}\int_0^xt^pf(t)\dt$ se prolonge par continuité en $0$. + +On note $u(f)$ ce prolongement. + - Montrer que $u$ ainsi défini est un endomorphisme injectif de $E$. + - Déterminer son spectre. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 688] +Soit $f\in\mc C^0([0,1],\R)$. Montrer $\int_{-1/2}^{3/2}f(3x^2-2x^3)\dx=2\int_0^1f(3x^2-2x^3)\, dx$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 689] +Donner un équivalent de $f(x)=\int_1^xt^tdt$ lorsque $x$ tend vers $+\i$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 690] +Soit $f\colon\R^+\ra\R^+$ une fonction continue, strictement croissante telle que $f(0)=0$. + - On suppose que $f$ est de classe $\mc C^1$. + +Montrer que $\forall x\gt 0,\ \int_0^xf(t)dt+\int_0^{f(x)}f^{-1}(t)dt=xf(x)$. + - - Soit $x\gt 0$. Pour $n\in\N^*$ et $i\in\db{0,n}$, on note $x_{i,n}=\frac{ix}{n}$. + +Montrer que $\sum_{i=0}^{n-1}x_{i,n}(f(x_{i+1,n})-f(x_{i,n}))\longrightarrow \int_0^{f(x)}f^{-1}(t)dt$ quand $n\ra+\i$. + - Montrer l'egalite vue en -. + - Soient $a\in\R^+$ et $b\in f\colon\R^+\ra\R^+$ continue et bijective. + +Montrer que $\int_0^af(t)dt+\int_0^bf^{-1}(t)dt\geq ab$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 691] +Soit $f$ continue et strictement positive sur $[a,b]$. + - Soit $n\in\N^*$. Montrer qu'il existe une unique subdivision $(x_{0,n},\ldots,x_{n,n})$ de $[a,b]$ telle que $\forall k\in\db{1,n},\int_{x_{k-1,n}}^{x_k,n}f=\frac{1}{n}\int_a^bf$ - Déterminer la limite de la suite de terme general $\frac{1}{n}\sum_{k=1}^nf(x_{k,n})$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 692] +Soit $f\in\mc C^n(\R,\R)$. On suppose que $f$ et $f^{(n)}$ sont bornées sur $\R$. + - Pour tout $p\in\db{1,n}$, on pose : $\phi_p:x\mapsto f(x+p)-\int_x^{x+p}\frac{f^{(n)}(t)}{(n-1)!}(x+p-t)^{n-1}\, dt$. + +Montrer que $\phi_p$ est bornée sur $\R$. + - En déduire que $f',\ldots,f^{(n-1)}$ sont bornées sur $\R$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 693] + - Soit $f\in\mc C^0([0,1],\R)$. On suppose que, pour toute fonction $\phi\in\mc C^1([0,1],\R)$ vérifiant $\phi(0)=\phi(1)=0$, l'on ait $\int_0^1f(t)\phi(t)dt=0$. Montrer que $f=0$. + - Soient maintenant $f,g\in\mc C^0([0,1],\R)$ telles que, pour tout $\phi\in\mc C^1([0,1],\R)$ vérifiant $\phi(0)=\phi(1)=0$, l'on ait $\int_0^1f(t)\phi(t)\dt=\int_0^1g(t)\phi'(t)\, dt$. Montrer que $g$ est de classe $\mc C^1$ et déterminer sa derivée. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 694] +Soient $h\gt 0$, $f\in\mc C^2([a,b],\R)$ avec $f''\geq m^2\gt 0$, et $E=\{x\in[a,b],|f'(x)|\gt h\}$. + - On suppose que $[c,d]$, avec $c\lt d$, est inclus dans $E$. Montrer que $\left|\int_c^de^{if(x)}\dx\right|\leq\frac{3}{h}$. + - Montrer que $\left|\int_Ée^{if(x)}\dx\right|\leq\frac{6}{h}$. + - Montrer que $\left|\int_a^be^{if(x)}\dx\right|\leq\frac{6}{h}+ \frac{2h}{m^2}$. + - Montrer que $\left|\int_a^be^{if(x)}\dx\right|\leq\frac{8}{m}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 695] +Déterminer la nature de $\int_2^{+\i}\frac{\cos x}{\ln x}dx$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 696] +Soit $a\gt 0$. Calculer $\int_0^{+\i}\frac{\ln t}{a^2+t^2}\dt$. Que dire de $\int_0^{+\i}\frac{\ln t}{a^p+t^p}\dt$ pour $p\geq 2$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 697] +Soit $E$ l'ensemble des $f\in\mc C^1([0,1],\R)$ telles que $f(0)=f(1)=0$. + - Pour $f\in E$, montrer la convergence de $I_1=\int_0^1f(t)f'(t)\,\mathrm{cotan}(\pi t)dt$ et de $I_2=\int_0^1f^2(t)\;(1+\mathrm{cotan}^2(\pi t))dt$. Comparer $I_1$ et $I_2$. + - Montrer que, si $f\in E$, $\int_0^1(f')^2\geq\pi^2\int_0^1f^2$. Pour quelles $f$ y-a-t-il egalite? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 698] +Convergence et calcul de $\int_0^1\sqrt{\frac{x}{1-x}}\ln(x)dx$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 699] +Soit $\alpha\in\R^{+*}$. Nature de l'intégrale $\int_0^{+\i}\exp(-t^{\alpha}\sin^2(t))dt$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 700] +Montrer que $f:x\mapsto\int_x^{+\i}\frac{dt}{t(e^{\sqrt{t}}-1)}$ est définie, continue et intégrable sur $]0,+\i[$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 701] + - Calculer $\int_0^{\pi/2}\frac{\sin(x)}{1+\sqrt{\sin(2x)}}dx$. + - Soit $f$ une fonction continue de $[0,1]$ dans $\R$. + +Montrer que $\int_0^{\pi/2}f(\sin(2x))\sin(x)dx=\int_0^{\pi/2}f(\cos^2(y)) \cos(y)dy$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 702] + - Soit $(a,\eps)\in(\R^{+*})^2$. Apres avoir simplifie $\ln\left(\frac{1-e^{2ax}}{1-e^{ax}}\right)$, montrer que + +$$\int_{\eps}^{+\i}\frac{\ln(1+e^{ax})}{x}dx=-\int_1^2 \frac{a\eps}{e^{a\eps y}-1}\ln(y)dy.$$ + - Montrer que $\int_1^2\frac{\ln(1-e^{-a\eps y})}{y}dy=\ln(2)\ln(1-e^{-2 a\eps})-\int_1^2\ln(y)\frac{a\eps}{e^{a\eps y}-1} dy$. + - En déduire la valeur de $\int_0^{+\i}\frac{\ln(x)}{e^{ax}-1}dx$. + - Retrouver le résultat précédent par un calcul direct de $\int_0^{+\i}\frac{\ln(x)}{e^x-1}dx$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 703] +Soit $F$ définie sur $\R^{+*}$ par $\colon\forall x\gt 0,F(x)=\int_x^{+\i}\frac{\sin t}{t^2}dt$. + - Montrer que $F$ est bien définie. + - Montrer que $F$ est intégrable sur $\R^{+*}$. + - Calculer $\int_0^{+\i}F(x)\dx$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 704] +Soit $f\colon\R^+\ra\R^+$ une fonction continue. On note $F$ la primitive de $f$ qui s'annule enE $0$. Montrer que les intégrales $\int_0^{+\i}\frac{F(t)}{(t+1)^2}dt$ et $\int_0^{+\i}\frac{f(t)}{t+1}dt$ sont de même nature. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 705] + - Soit $f:[1,+\i[\ra\R$ continue. On suppose que l'intégrale $\int_1^{+\i}f$ est convergente. + +Montrer que $\int_1^{+\i}\frac{f(t)}{t}\dt$ est une intégrale convergente. + - Soit $\sum u_n$ une série convergente. Montrer que $\sum\frac{u_n}{n}$ est une série convergente. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 706] +Trouver un équivalent simple de $\int_0^x\frac{|\sin t|}{t}\dt$ quand $x$ tend vers $+\i$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 707] +Soit $f\colon\R\ra\C$ une fonction continue et $T$-periodique. - à quelle condition $f$ admet-elle une primitive $T$-periodique? + - On suppose à present que $\int_0^Tf(x)\dx\neq 0$, et on fixe un réel $a\in]0,1]$. Donner un équivalent de $\int_1^x\frac{f(t)}{t^a}\dt$ quand $x\ra+\i$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 708] +Quelles sont les fonctions de $[0,1]$ dans $\R$ qui sont limite uniforme sur $[0,1]$ d'une suite de polynômes convexes? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 709] +Soit $f$ continue sur $[0,\pi]$ telle que $\forall n,\int_0^{\pi}\cos(nt)f(t)dt=0$. Que dire de $f$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 710] +Soit $f:x\mapsto\sum_{n=0}^{+\i}\frac{\sin{(2^nx)}}{2^n}$. + - Montr re que $f$ est définie sur $\R$. + - Montr re que $f$ n'est pas dérivable en $0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 711] +Soit $\alpha\in\R^{+*}$. + - Montr re qu'en posant $\forall x\in\R^{+*},\ f(x)=\sum_{n=1}^{+\i}e^{-n^{\alpha}x}$, on définit une fonction de classe $\mc C^{\i}$ de $\R^{+*}$ dans $\R$. + - Donner la limite puis un équivalent simple de $f$ en $+\i$. + - Donner la limite puis un équivalent simple de $f$ en $0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 712] +Déterminer le domaine de définition et un équivalent simple en $1^-$ de $f:x\mapsto\sum_{n=0}^{+\i}x^{n^2}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 713] +Pour $n\geq 0$, soit $u_n:x\mapsto\prod_{i=0}^n\frac{1}{x+i}$. + - Montr re que $S=\sum_{n=0}^{+\i}u_n$ est définie et continue sur $\R^{+*}$. + - Ex primer $S(x+1)$ en fonction de $S(x)$ et de $x$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 714] +On pose $f:x\mapsto\sum_{n=1}^{+\i}\frac{e^{-nx}}{n+x}$. + - Déterminer le domaine de définition $D$ de $f$. + - Étudier la continuité de $f$ sur $D$. + - Déterminer des équivalents de $f$ aux extremites de $D$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 715] + - Soit $x\in[0,1[$ Justifier la convergence de $f(x)=\prod_{n=0}^{+\i}\left(\frac{1+x^n}{1+x^{n+1}}\right)^{x^n}$. - Montrer que, pour tout $x\in]0,1[$, $\ln f(x)=\frac{x-1}{x}\sum_{n=1}^{+\i}x^n\ln(1+x^n)+\ln 2$. + - En déduire que, pour tout $x\in[0,1[$, $\ln f(x)=\ln 2+\sum_{m=1}^{+\i}\frac{(-1)^m}{m}\frac{x^m}{1+x+ \cdots+x^m}$. + - Montrer que $f$ possede une limite finie en $1$ et la déterminer. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 716] +On pose $f:x\mapsto\sum_{p=0}^{+\i}\frac{(-1)^p}{p!(x+p)}$. + - Déterminer le domaine de définition de $f$. + - Exprimer $f(x)$ en fonction de : $g(x)=\frac{1}{x}+\sum_{k=1}^{+\i}\frac{1}{x(x+1)\cdots(x+k)}$. + - Déterminer un équivalent simple de $f$ en $+\i$. + - Déterminer un équivalent simple de $f$ en $0^+$. + - Étudier la convergence uniforme de la série de fonctions $\sum\frac{(-1)^p}{p!(x+p)}$ sur les parties du domaine de définition de $f$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 717] +Soit $f:x\mapsto\sum_{n=0}^{+\i}\left(\mbox{Arctan}(n+x)-\mbox{ Arctan}(n)\right)$. + - Donner le domaine de définition de $f$. Étudier sa régularite. + - Exprimer $f(x+1)$ en fonction de $f(x)$ et de $x$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 718] +Domaine de définition et équivalent en $+\i$ de $f:x\mapsto\sum_{n=2}^{+\i}\frac{(\ln n)^x}{n^2}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 719] +Soit $u_0$ l'identite de $[1,+\i[$ et, pour $n\in\N$, $u_{n+1}:x\in[1,+\i[\mapsto u_n(x)+\frac{1}{u_n(x)}$. + - Montrer que la suite de fonction $(u_n)$ est bien définie. + - Étudier la convergence simple de $(u_n)$ sur $[1,+\i[$. + +Pour $n\in\N$, soit $f_n:x\in[1,+\i[\mapsto\frac{(-1)^n}{u_n(x)}$. + - Montrer que la suite $(f_n)$ converge simplement sur $[1,+\i[$. + - Montrer que la somme de la série de terme general $f_n$ est continue sur $[1,+\i[$. + - A-t-on convergence normale sur $[1,+\i[$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 720] +Notons, pour $\alpha\gt 0$, $n\in\N^*$ et $x\geq 0$, $u_n(x)=\frac{x}{n^{\alpha}(1+nx^2)}$. + - Déterminer les modes de convergence de $\sum u_n$ sur $\R^+$ et $\R^{+*}$. + - Montrer que la somme $S_{\alpha}$ de cette série est continue sur $\R^{+*}$ et que si $\alpha\gt 1/2$, $S_{\alpha}$ est continue sur $\R^+$. + - Pour $\alpha\leq 1/2$, $S_{\alpha}$ est-elle continue en $0$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 721] +Pour $x$ réel convenable, on note $\zeta(x)=\sum_{n=1}^{+\i}\frac{1}{n^x}$. + - Déterminer le domaine $\mc{D}$ de définition de $\zeta$. + - Montrer que, pour $x\in\mc{D}$, $\zeta(x)=1+\frac{1}{x-1}-x\int_1^{+\i}\frac{\{t\}}{t^{x+1}} dt$, ou $\{t\}=t-\lfloor t\rfloor$. + +En déduire que $\zeta$ peut être prolongée sur un ensemble $\mc{D}'$. + - Donner un équivalent de $\zeta$ en $1$. + - Montrer que le prolongement de $\zeta$ sur $\mc{D}'$ se prolonge par continuité en $\inf(\mc{D}')$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 722] +Soit, pour $x\in\R^+$, $f(x)=\sum_{n=1}^{+\i}\frac{x^n}{1+nx}$. + - Montrer que $f$ est de classe $\mc C^1$ sur $[0,1[$. + - Donner un équivalent de $f(x)$ en $1$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 723] +Rayon de convergence et somme de $\sum_{n\geq 1}\cos\left(\frac{2\pi n}{3}\right)\frac{x^n}{n}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 724] +Montrer que la fonction $f:x\mapsto\ln(1+e^{-x})$ est développable en série entiere au voisinage de $0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 725] +Déterminer le rayon et la somme de $\sum_{n\geq 0}\frac{(2n+1)!}{(n!)^2}x^n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 726] +Soit, pour $n\in\N$, $a_n=\int_0^{\pi/2}\cos(t)^n\sin(nt)\dt$. Soit $f:x\mapsto\sum_{n=0}^{+\i}a_nx^n$. + - Calculer $a_0,a_1,a_2$. + - Calculer $f(x)$ pour $|x|\lt 1$. Preciser le rayon de convergence de $f$. + - En déduire $a_n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 727] + - Soit $z\in\C$ tel que $|z|\neq 1$. Montrer que la fonction $t\mapsto\frac{1}{e^t-z}$ est développable en série entiere au voisinage de $0$. + - Soient $F\in\C(X)$ sans pole de module $1$ et $\alpha\in\R$. Montrer que la fonction $t\mapsto F(e^{\alpha t})$ est développable en série entiere au voisinage de $0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 728] +Pour $n\in\N^*$, on note $a_n=\nu_2(n)$ (valuation 2-adique). + - Déterminer les valeurs d'adherence de $(a_n)$. + - On pose, pour $n\in\N^*$, $b_n=\frac{1}{n}\left(\sum_{k=1}^na_k\right)$. La suite $(b_n)$ possede-t-elle une limite? + - Déterminer le rayon de convergence de $\sum b_nx^n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 729] +Pour $n\in\N$, soit $a_n=\int_0^{\pi/4}\tan^n(x)\dx$. - Montrer que $(a_n)_{n\geq 0}$ tend vers $0$. + - Si $n\in\N$, exprimer $a_{n+2}$ en fonction de $a_n$. + - Déterminer le rayon de convergence $R$ de $x\mapsto\sum_{n=0}^{+\i}a_nx^n$. Calculer la somme. Étudier le comportement en $\pm R$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 730] +On pose $u_0=1$ et $u_{n+1}=\dfrac{1}{2}\sum_{k=0}^n\binom{n}{k}u_ku_{n-k}$ pour tout $n$. Trouver $u_n$ en considérant la série entiere $\sum_{n\geq 0}\dfrac{u_n}{n!}x^n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 731] +La suite $(a_n)_{n\geq 0}$ est définie par $a_0\gt 0$ et $\forall n\in\N,a_{n+1}=\ln(1+a_n)$. + - Montrer que $(a_n)_{n\in\N}$ tend vers $0$. + - Donner un équivalent de $a_n$. + - Donner le rayon de convergence $R$ de $\sum a_nx^n$. Y-a-t-il convergence pour $x=R$? pour $x=-R$? + - Donner un équivalent de $\sum_{n=0}^{+\i}a_nx^n$ quand $x$ tend vers $R^-$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 732] +Soit $t\in\R$ tel que : $\forall n\in\N^*$, $nt\not\in 2\pi\Z$. Soit $f:x\mapsto\sum_{n=0}^{+\i}\dfrac{\sin(nt)}{n}x^n$. + - Déterminer le rayon de convergence $R$ de $f$. + - Étudier la convergence en $\pm R$. Ind. Poser $S_n=\sum_{k=1}^n\sin(kt)$. + - Exprimer $f(x)$ pour $x\in]-R,R[$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 733] +Soit $(a_n)_{n\geq 0}$ une suite complexe telle que la série $\sum na_n$ converge absolument. On note $\mathbb{D}$ le disque unite ouvert de $\C$. Soit $f:z\mapsto\sum_{n=0}^{+\i}a_nz^n$. + - Montrer que le rayon de convergence de $f$ est $\geq 1$. + - On suppose que $a_1\neq 0$ et que $\sum_{n=2}^{+\i}n|a_n|\leq|a_1|$. Montrer que $f$ est injective sur $\mathbb{D}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 734] +Soit $(a_n)_{n\geq 0}$ définie par $a_0=a_1=0$, $a_2=\dfrac{1}{2}$ et $a_{n+1}=\dfrac{1}{n(n+1)}\sum_{i+j=n}a_ia_j$ pour $n\geq 2$. + - Montrer que le rayon de convergence de $\sum a_nx^n$ est supérieur ou egal à 1. + - Montrer que $x\mapsto\sum_{n=0}^{+\i}a_nx^n$ est solution de l'équation $xy''-x=y^2$ sur $]0,1[$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 735] +Soit $(a_n)$ définie par $a_0=a_1=1$ et $\forall n\geq 1,\,a_{n+1}=a_n+\dfrac{2}{n+1}a_{n-1}$. + - Montrer que $\forall n\in\N^*,1\leq a_n\leq n^2$. En déduire le rayon $R$ de $f(x)=\sum a_nx^n$. + - Montrer que $f$ est solution de $(1-x)y'-(2x+1)y=0$. Exprimer $f$ à l'aide de fonctions usuelles. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 736] +Soient $f(z)=\sum_{n=0}^{+\i}a_nz^n$ la somme d'une série entiere de rayon de convergence $R\gt 0$, et $D$ son disque ouvert de convergence. + - Montrer que, s'il existe $(z_k)\in(D\setminus\{0\})^{\N}$ de limite nulle telle que $\forall k\in\N$, $F(z_k)=0$, alors $F$ est nulle. + - On suppose que $F(0)\in\R^{+*}$ et que $|F|$ admet un maximum local en $0$. + +Montrer que $F$ est constante. + +Ind. Raisonner par contraposée et montrer l'existence de $p\in\N$ tel que pour tout $z\in D$, + +$$|F(z)|\geq|F(0)+a_pz^p|-\sum_{n=p+1}^{+\i}|a_n||z|^n.$$ +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 737] +Soit $(b_n)$ la suite définie par $b_0=1$ et $\forall n\in\N^*$, $b_n=\sum_{k=1}^n\dfrac{b_{n-k}}{k!}$. + - Montrer que $\forall n\in\N$, $b_n\leq\dfrac{1}{\ln^n(2)}$. + - Montrer que la série entiere $\sum b_nx^n$ à un rayon de convergence $R$ non nul et que + + $\forall x\in]-R,R[,\,\,\sum_{n=0}^{+\i}b_nx^n=\dfrac{1}{2-e^{ x}}$. + - En déduire une expression sommatoire explicite de $b_n$ pour $n\in\N$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 738] +Soit $f:z\in\C\setminus\{1\}\mapsto\exp\left(\dfrac{z}{1-z}\right)$. + - Montrer que $f$ est développable en série entiere au voisinage de 0 et donner son rayon de convergence. On écrit $f(z)=\sum_{n=0}^{+\i}a_nz^n$. + - Donner une expression sommatoire des $a_n$. + - Trouver une relation de récurrence vérifiée par la suite $(a_n)$. + - Donner un développement asymptotique de $\ln(a_n)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 739] +Soit $a\in\C^*$. On pose $A_0=1$ et, pour $k\in\N^*$, $A_k=\dfrac{1}{k!}X(X-ak)^k$. + - Montrer que, pour tout $P\in\C_n[X]$, $P(X)=\sum_{k=0}^nP^{(k)}(ak)A_k(X)$. + +En déduire que $\forall y\in\C^*$, $ny^{n-1}=\sum_{k=1}^n\binom{n}{k}(-ak)^k(y+ak)^{n-k}$. - Déterminer le rayon de convergence $R$ de la série entiere $\sum_{n\geq 1}\frac{(-n)^{n-1}}{n!}x^n$. + - On note $S$ sa somme. Montrer que $\forall x\in]-R,R[$, $x(1+S(x))S'(x)=S(x)$. + +Donner une expression simple de $h:x\mapsto S(x)\,e^{S(x)}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 740] +Soit $(a_n)_{n\geq 2}$ une suite réelle telle que la série entiere associée est de rayon de convergence supérieur ou egal à $1$. + +On suppose que $f:z\mapsto z+\sum_{n=2}^{+\i}a_nz^n$ est injective sur $\mathbb{D}=\{z\in\C,\;|z|\lt 1\}$. + - Montrer que, pour tout $z\in\mathbb{D}$, $f(z)\in\R$ si et seulement si $z\in\R$. + - Montrer que, pour tout $z\in\mathbb{D}$, $\op{Im}(f(z))\gt 0$ si et seulement si $\op{Im}(z)\gt 0$. + - Calculer, pour $n\in\N^*$ et $r\lt 1$, $\int_0^{2\pi}\op{Im}(f(re^{it}))\sin(nt)dt$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 741] +Soit $(a_n)$ une suite de réels positifs avec $a_0\gt 0$ et $a_1=1$. Soient $S:x\mapsto\sum_{k=0}^{+\i}a_kx^k$ et, pour $n\in\N$, $S_n:x\mapsto\sum_{k=0}^na_kx^k$. On suppose que le rayon de convergence de $S$ est $R\gt 0$. + - Soient $n\geq 1$ et $y\gt a_0$. Montrer qu'il existe un unique $x_n(y)\in\R^+$ tel que $S_n(x_n(y))=y$. Montrer que la suite $(x_n(y))$ converge vers un réel note $T(y)$. Montrer que $|T(y)|\leq R$. + - On suppose que $|T(y)|\leq R$. Calculer $(S\circ T)(y)$. Que peut-on en déduire? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 742] +Soit $\sum a_nz^n$ une série entiere de rayon de convergence $R\gt 0$ et de somme $f$. + - Montrer que, pour tout $r\in[0,R[$, $I(r)=\frac{1}{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^2d\theta=\sum_{ n=0}^{\i}|a_n|^2r^{2n}$, puis que la fonction $I$ est croissante sur $[0,R[$. + - Si $f$ n'est pas nulle, montrer que $I(r)\gt 0$ pour tout $r\in]0,R[$. + - Montrer que la fonction $t\mapsto\ln\big(I(e^t)\big)$ est convexe sur $]-\i,\ln R[$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 743] +Soient $P\in\R[X]$ de degre $2$ et $f:x\mapsto e^{P(x)}$. Montrer que $f$ est développable en série entiere en $0$ et que deux coefficients consécutifs de ce développement ne sont jamais simultanement nuls. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 744] +Soit $(p_n)$ une suite strictement croissante d'entiers naturels telle que $n=o(p_n)$. + +Soit $f:x\mapsto\sum_{n=0}^{+\i}x^{p_n}$. + - Quel est le rayon de convergence de $f$? + - Déterminer la limite en $1^-$ de $f$ puis de $x\mapsto(1-x)f(x)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 745] +Déterminer un équivalent de $p(n)=\big{|}\big{\{}(x,y,z)\in\N^3,\;x+2y+3z=n\big{\}}\big{|}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 746] +On dit que la suite $(a_n)_{n\geq 0}\in\R^{\N}$ vérifie $\mc{P}$ si le rayon de convergence de $\sum a_nx^n$ est supérieur ou egal à 1 et si $f:x\mapsto\sum_{n=0}^{+\i}a_nx^n$ possede une limite finie en $1^-$. + - Déterminer les $f\colon\R\ra\R$ continues en 0 telles que : $\forall x,y\in\R$, $f(x+y)=f(x)+f(y)$, + - Montrer que si $\sum a_n$ est absolument convergente alors $(a_n)$ vérifie $\mc{P}$. Étudier la réciproque. + - Déterminer les $f\colon\R\ra\R$ telles que, pour toute suite $(a_n)\in\R^{\N}$ vérifiant $\mc{P}$, la suite $(f(a_n))$ vérifie $\mc{P}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 747] +Soit $E$ l'ensemble des fonctions $f\in\mc C^{\i}(\R,\R)$ dont la série de Taylor en 0 à un rayon de convergence $+\i$. + - Montrer que $E$ est une $\R$ algèbre. + +Pour $f\in E$, on pose $T(f):x\mapsto f(x)-\sum_{n=0}^{+\i}\frac{f^{(n)}(0)}{n!}\,x^n$. + - Montrer que $T$ est un endomorphisme de $E$ et que $\op{Im}(T)$ est un idéal de $E$. + - Montrer que $E=\op{Im}(T)\oplus\op{Ker}(T)$. + - Déterminer le spectre de $T$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 748] +Limite de $u_n=\int_0^1\frac{dx}{\sqrt{2+\sqrt{2+\sqrt{\dots+ \sqrt{2+2x}}}}}$ (ou il y a $n$ racines car $\mathrm{e}$es)? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 749] +Soit $I_n=\int_0^{+\i}\sin(t^n)\dt$. Déterminer les $n\in\N$ pour lesquels $I_n$ est définie. Donner un équivalent de $I_n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 750] +Déterminer un développement asymptotique de $u_n=\int_0^1\frac{du}{1+u^n}$ en $o(1/n^2)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 751] +Pour tout $n\in\N$, on pose : $u_n=\int_0^1(-t^2+t-1)^n\dt$. + - Montrer que $(u_n)$ converge vers $0$. + - Montrer que $\sum_{n\geq 0}u_n$ converge et calculer sa somme. + - Trouver un équivalent simple de $u_n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 752] +Pour tout $n\in\N$, on pose : $I_n=\int_0^1\frac{t^{n+1}\ln t}{1-t^2}dt$. + - Montrer la convergence de $I_n$. + - Étudier la convergence et la limite eventuelle de $(I_n)$. + - Trouver un équivalent simple de $I_n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 753] +Exprimer sous forme de somme $\int_0^{+\i}e^{-t^2}\cos(t)dt$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 754] + +#+end_exercice + + - Justifier que $\int_0^{1/2}\frac{\ln(1-t)}{t}\dt=\int_{1/2}^1 \frac{\ln t}{1-t}\dt$. - En déduire la valeur de $\sum_{n\geq 1}\frac{1}{2^nn^2}$. +#+begin_exercice [Mines MP 2024 # 755] + +#+end_exercice + +Soient $\alpha$ et $\beta$ des réels $\gt 0$. - Montrer que la série $\sum_{n\geq 0}\frac{(-1)^n}{\alpha n+\beta}$ est convergente. - Exprimer sa somme sous forme intégrale. +#+begin_exercice [Mines MP 2024 # 756] + +#+end_exercice + +Calculer $\int_0^1\ln(t)\ln(1-t)dt$. +#+begin_exercice [Mines MP 2024 # 757] + +#+end_exercice + +Pour tout $x\in\R$, on pose $f(x)=\int_0^1\frac{e^{-x(1+t^2)}}{1+t^2}dt$. - Montrer que $f$ est de classe $\mc C^1$ sur $\R$, et exprimer sa derivée. - On pose $g(x)=f(x^2)$ pour $x\in\R$. Montrer que la fonction $x\mapsto g(x)+\left(\int_0^xe^{-t^2}\dt\right)^2$ est constante, et preciser sa valeur. - En déduire la valeur de $\int_0^{+\i}e^{-t^2}dt$. +#+begin_exercice [Mines MP 2024 # 758] + +#+end_exercice + +Pour tout réel $a\gt 0$, on pose $F(a)=\int_0^{+\i}\frac{\arctan\left(\frac{x}{a}\right)+ \arctan(ax)}{1+x^2}\dx$. Justifier l'existence de $F(a)$, puis calculer cette intégrale. +#+begin_exercice [Mines MP 2024 # 759] + +#+end_exercice + +Pour $n\in\N^*$ et $x\lt 0$, on pose $: h_n(x)=\int_0^{+\i}\frac{dt}{(t^2+x^4)^n}$. - Soit $n\in\N^*$. Montrer que $h_n$ est de classe $\mc C^1$ sur $\R^{+*}$ et $\colon\forall x\gt 0,h_n'(x)=-4nx^3h_{n+1}(x)$. - Montrer qu'il existe une suite réelle $(a_n)_{n\in\N^*}$ telle que : $\forall n\in\N^*$, $\forall x\gt 0,h_n(x)=a_nx^{2-4n}$. - Expliciter la suite $(a_n)$. +#+begin_exercice [Mines MP 2024 # 760] + +#+end_exercice + +On pose $F:x\mapsto\int_0^{+\i}\frac{\mathrm{e}^{-2t}}{x+t} dt$. - Domaine de définition de $F$? de continuité? - Donner un équivalent de $F$ en $+\i$. +#+begin_exercice [Mines MP 2024 # 761] + +#+end_exercice + +Soit $f:x\mapsto\int_0^{2\pi}\ln\left(x^2-2x\cos(t)+1\right) dt$. - Donner le domaine de définition et étudier la continuité de $f$. - Donner une expression de $f(x)$. +#+begin_exercice [Mines MP 2024 # 762] + - Déterminer le domaine de définition $D$ de $: f:x\mapsto\int_0^{+\i}\frac{\sin t}{t}e^{-xt}\dt$. - Montrer que $f$ est continue sur $\R^+$. + - Montrer que $f$ est de classe $\mc C^1$ sur $\R^{+*}$. +#+end_exercice + + - En déduire la valeur de $\int_0^{+\i}\frac{\sin t}{t}dt$. +#+begin_exercice [Mines MP 2024 # 763] + +#+end_exercice + +Soient $\alpha\gt 0$ et $f:x\in\R^{+*}\mapsto\int_0^1\frac{dt}{x^{\alpha}+t^3}$. L'application $f$ est-elle intégrable sur $\R^{+*}$? +#+begin_exercice [Mines MP 2024 # 764] +Pour $x\in\R$, calculer $f(x)=\int_{-\i}^{+\i}e^{-t^2/2}e^{-ixt}dt$ par deux methodes : + + - en déterminant le développement en série entiere de $f(x)$; + + - en montrant que $f$ est de classe $\mc C^1$ et vérifie une équation différentielle lineaire d'ordre $1$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 765] +Soit $f$ définie par : $f(x)=\int_0^{\pi/2}\sin^x(t)\dt$. + - Déterminer le domaine de définition $D_f$ de $f$. + - Montrer que $f$ est continue et decroissante. + - Pour tout $x\in D_f$, on pose $g(x)=(x+1)f(x+1)f(x)$. + +Montrer que : $\forall x\in D_f,g(x+1)=g(x)$. +#+end_exercice + + - Déterminer des équivalents simples de $f$ aux extremites de $D_f$. +#+begin_exercice [Mines MP 2024 # 766] + - Montrer la convergence de $\int_0^{+\i}\cos(t^2)dt$. + - On pose $f:x\mapsto\int_0^{+\i}\frac{e^{-(t^2+i)x^2}}{t^2+i} dt$. Montrer que $f$ est de classe $\mc C^1$ sur $\R^+$. +#+end_exercice + + - On admet que $\int_0^{+\i}e^{-t^2}dt=\frac{\sqrt{\pi}}{2}$. Calculer $\int_0^{+\i}e^{-it^2}dt$ à l'aide de $f$. +#+begin_exercice [Mines MP 2024 # 767] + +#+end_exercice + +Trouver toutes les fonctions $f\colon\R^{+*}\ra\R$ d $\mathrm{\acute{e}ivables\ \mathrm{v}\mathrm{e}rifiant}\colon\forall x\gt 0,f'(x)=f(1/x)$. +#+begin_exercice [Mines MP 2024 # 768] + +#+end_exercice + +Soit $f\colon\R^+\ra\R$ de classe $\mc C^1$, monotone et admettant une limite finie en $+\i$. Montrer que les solutions de l'équation différentielle $y''+y=f(x)$ sont bornées. +#+begin_exercice [Mines MP 2024 # 769] +On considére l'équation différentielle $(E):2xy''+y'-y=0$. + - Montrer que $(E)$ possede une unique solution $f$ sur $\R$ telle que $f(0)=1$ et qui soit la somme d'une série entiere. + - Donner une expression de $f$ à l'aide de fonctions usuelles. +#+end_exercice + + - à l'aide du changement de fonction inconnue $y=zf$, r $\mathrm{\acute{e}soudre}\ (E)$. +#+begin_exercice [Mines MP 2024 # 770] + +#+end_exercice + +Soit $\lambda\in\R$. Montrer que les solutions de : $(E):y''+(\lambda-1)x^2y=0$ sont de la forme $x\mapsto H(x)e^{-x^2/2}$ avec $H$ développable en série entiere. +#+begin_exercice [Mines MP 2024 # 771] +R $\mathrm{\acute{e}soudre}$ l'équation différentielle $(1+x^2)y''+xy'-y=0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 772] +Déterminer une solution de $(E):y''+xy'+y=1$ développable en série entiere au voisinage de 0. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 773] +Soit $(E)$ l'équation différentielle $ax^2y''+bxy'+cy=0$ sur $\R^{+*}$. + - Résoudre $(E)$ en utilisant le changement de variable $t=\ln x$. + - Résoudre $x^2y''+xy'+y=\sin(a\ln x)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 774] +Considérons l'équation différentielle $(E):x^2y'+y+x^2=0$. + - Résoudre $(E)$ sur $\R^{+*}$. + - Montrer que $(E)$ admet une unique solution qui admet une limite finie en $0$. + - Existe-t-il des solutions de $(E)$ admettant une limite finie en $+\i$? + - Déterminer les solutions de $(E)$ développables en série entiere. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 775] +Soient $n\in\N$ et $(*)$ l'équation différentielle : + + $(1+x^n)(1-x^2)y'+2x(1+x^n)y=2(1-x^2)$. + - Trouver les solutions de $(*)$ sur $]-1,1[$. + - Existe-il une solution définie sur $\R$? + - Existe-il une solution définie sur $]1,+\i[$ et bornée? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 776] +Soit $f$ une fonction continue et bornée de $\R$ dans $\R$. Déterminer les fonctions $y$ de $\R$ dans $\R$, de classe $\mc C^2$ et bornées, telles que $y''-y=f$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 777] +Soit $y$ une solution sur $\R^{+*}$ de $xy''+y'+xy=0$, + - On pose : $\forall x\gt 0$, $u(x)=\sqrt{x}\,y(x)$. Déterminer une équation différentielle dont $u$ est solution. + - Montrer que $\int_a^b\frac{u(x)v(x)}{4x^2}\dx$ = $(uv'-u'v)(b)-(uv'-u'v)(a)$ avec $v$ vérifiant $v''+v=0$. + - Montrer que, pour tout $a\gt 0$, il existe $x_a\in[a,a+\pi[$ tel que $y(x_a)=0$. + - Montrer que $f:x\mapsto\sum_{n=0}^{+\i}\frac{(-1)^nx^{2n}}{4^n(n!)^2}$ s'annule une infinite de fois. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 778] +On note $S$ l'ensemble solution de l'équation différentielle $(E):xy''+xy'-y=0$ sur $\R^{+*}$. + - Trouver $\alpha\in\R$ tel que $x\mapsto x^{\alpha}$ soit solution de $(E)$. + - Pour tout $x\gt 0$, on pose : $G(x)=\int_1^x\frac{e^{-t}}{t^2}dt$. Dresser le tableau de variation de $G$. + - Soient $f\in\mc C^2(\R^{+*},\R)$ et $s:x\mapsto xf(x)$. Montrer que $s\in S$ si et seulement si $f'$ est solution d'une certaine équation différentielle du premier ordre. Résoudre cette équation différentielle. + - Expliciter $S$ à l'aide de $G$. Étudier les limites des solutions en $0^+$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 779] +Soit $f\colon\R^+\ra\R$ une fonction monotone de classe $\mc C^1$ admettant une limite réelle en $+\i$. Montrer que les solutions de l'équation $y''+y=f$ sont bornées sur $\R^+$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 780] +Soit $f$ une fonction de classe $\mc C^2$ de $\R$ dans $\R$ telle que $f''+f\geq 0$. Montr er que, pour $t\in\R$, $f(t)+f(t+\pi)\geq 0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 781] +Résoudre les systemes différentiels + +$$\left\{\begin{array}{rcl}x'&=&2x+3y+3z+te^t\\ y'&=&3x+2y+3z+e^t\\ z'&=&3x+3y+2z+t^2e^t\end{array}.,\quad\left\{\begin{array}[] {rcl}x'&=&2y-z&+te^t\\ y'&=&3x-2y&+e^t\\ z'&=&-2x-2y+z&+t^2e^t\end{array}.$$ +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 782] +Soient $m,n\in\N^*$ et $A\in\M_n(\R)$. On considére le systeme différentiel $(S):Y^{(m)}=AY$ d'inconnue $Y\in\mc C^m(\R,\R^n)$. Montr er que $A$ est nilpotente si et seulement si toutes les solutions de $(S)$ sont polynomiales. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 783] +On munit $\R^n$ de la norme euclidienne canonique. Soit $A\in\M_n(\R)$. Montr er que $A$ est antisymétrique si et seulement si les solutions de $Y'=AY$ sont de norme constante. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 784] +Soient $T\in\R^{+*}$, $A$ une application continue et $T$-periodique de $\R$ dans $\M_n(\C)$. Montr er qu'il existe $\lambda$ dans $\C^*$ et une application $X$ de classe $\mc C^1$ non identiquement nulle de $\R$ dans $\C^n$ telle que, pour tout $t\in\R$, $X'(t)=A(t)X(t)$ et $X(t+T)=\lambda X(t)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 785] +Soit $S\in\mc{S}_n(\R)$. Montr er qu'il existe une unique fonction $M$ de $\R$ dans $\mc{S}_n(\R)$ telle que $M(0)=I_n$ et $M'(t)=SM(t)S$ pour tout $t\in\R$. à quelle condition sur $S$ la fonction $M$ est-elle bornée? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 786] +Soient $(E,\langle\,\ \rangle)$ un espace euclidien, $u\colon\R\ra$ SO $(E)$ dérivable. Montr er l'équivalence entre : (i) $\forall s,t\in\R$, $u(s+t)=u(s)\,u(t)$, (ii) $\exists a\in\mc{A}(E)$, $\forall t\in\R$, $u(t)=e^{at}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 787] +Déterminer le domaine de définition de $f:(x,y)\mapsto\sum_{n=0}^{+\i}\frac{(x+y)^n}{n^2}$. Est-elle continue? de classe $\mc C^1$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 788] +On pose $f(0,0)=0$ et, pour $(x,y)\in\R^2\setminus\{(0,0)\}$, $f(x,y)=\frac{x^3y}{x^4+y^2}$. Ethier la continuité et la différentiabilité de $f$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 789] +Soient $f\in\mc C^2(\R^{+*},\R)$ et $g$ définie sur $(\R^{+*})^2$ par $:g(x,y)=f\left(\frac{x^2+y^2}{2}\right)\cdot$ Déterminer les fonctions $f$ qui vérifient $\colon\frac{\partial^2g}{\partial x^2}+\frac{\partial^2g}{\partial y ^2}=0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 790] +On munit $\R^2$ de sa norme euclidienne canonique. + +On définit $f$ sur $\R^2$ par $\colon\forall(x,y)\in\R^2$, $f(x,y)=\Big(\frac{1}{2}\sin(x+y),\frac{1}{2}\cos(x-y)\Big)$. + - Calculer la différentielle de $f$ en tout point. + - Montr er que $\colon\forall(x,y)\in\R^2,\|df(x,y)\|_{\mathrm{op}}\leq \frac{1}{\sqrt{2}}$. + - En déduire que $f$ possede au plus un point fixe. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 791] + - Soit $f\colon\R\ra\R$ dérivable et minorée. On pose $m=\inf_{\R}f$. On suppose que $m$ n'est pas atteint. Montrer qu'il existe une suite $(x_n)_{n\geq 0}$ telle que, pour tout $n\in\N$, $f(x_n)\leq m+\frac{1}{2^n}$ et $|x_n|\geq n$. En déduire qu'il existe une suite $(u_n)_{n\geq 0}$ telle que $|u_n|\ra+\i$ et $f'(u_n)\ra 0$. + - Soient $p\geq 2$ et $f\in\mc C^1(\R^p,\R)$ minorée. Pour $\eps\gt 0$, soit $g_{\eps}:x\mapsto f(x)+\eps\|x\|$. Montrer que $g_{\eps}$ atteint son minimum (la norme est la norme euclidienne standard). En déduire qu'il existe une suite $(u_n)$ telle que $\nabla f(u_n)\ra 0$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 792] + - Soient $n\geq 2$, $U$ un ouvert de $\R^n$ et $f:U\ra\R$ différentiable. Soient $a,b\in U$ tels que $[a,b]\subset U$. Montrer qu'il existe $c\in]a,b[$ tel que $f(b)-f(a)=df_c(b-a)$. + - Application : si on souhaite connaitre la valeur de $\frac{\sqrt{2}}{e+\pi^3}$ à la precision $10^{-20}$, avec quelle precision doit-on alors connaitre $\sqrt{2},e$ et $\pi$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 793] +Soit $f\in\mc C^2(\R^n,\R)$ telle qu'en tout point $x$ le spectre de la hessienne soit inclus dans $[1,+\i[$. + - Montrer que, pour tout $x\in\R^n$ on a $f(x)\geq f(0)+\langle\nabla f(0),x\rangle+\frac{1}{2}x^Tx$. + +_Ind._ Considérer $\psi:t\mapsto f(tx)-\langle\nabla f(0),tx\rangle-\frac{1}{2}t^2x^Tx$. + - En déduire que $f$ admet un minimum. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 794] +On munit $E=\R^n$ de sa structure euclidienne canonique. Pour $x\in E\setminus\{0\}$, on note $f(x)$ l'unique vecteur $y$ positivement colineaire à $x$ vérifiant : $\|x\|\times\|y\|=1$. + - Montrer que $f$ est différentiable et calculer sa différentielle en tout point. + - Soit $x\in E\setminus\{0\}$. Interpreter $df(x)$ en faisant intervenir la reflexion d'axe $\{x\}^{\perp}$. + - En déduire que $df(x)$ conserve les angles. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 795] +Soient $\lambda\in\R$, $n\in\N^*$ et $f$ une application de classe $\mc C^1$ de $\R^n\setminus\{0\}$ dans $\R$. Montrer l'équivalence des conditions + +_(i) $\forall(t,x)\in\R^{+*}\times(\R^n\setminus\{0\},f(tx)=t^{ \lambda}f(x)$_ : + +_(ii) $\forall x\in\R^n\setminus\{0\},\sum_{i=1}^nx_i\partial_if(x)= \lambda f(x)$._ +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 796] + - Calculer la différentielle du déterminant au point $I_n$. + +La fonction det atteint-elle un extremum local en $I_n$? + - Déterminer points critiques et extrema locaux de la fonction det sur $\M_n(\R)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 797] +On pose $D=\left]0,1\right[^2$ et l'on définit $f$ sur $D$ par : + + $\forall(x,y)\in D,f(x,y)=\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{x+y}$. + - Montrer que $f$ est de classe $\mc C^1$. + - Déterminer les extrema locaux de $f$. - En etudiant la restriction de $f$ à $K=\left\{(x,y)\in D\;;\;(x,y)\in\left[0,\frac{7}{9}\right]\mbox{ et }x+y\geq\frac{2}{9}\right\}$ d'eterminer les extrema globaux de $f$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 798] +Soient $f\colon\R^2\ra\R$ de classe $\mc C^1$, $a\in\R^2$ et $\gamma$ un arc paramêtre plan de classe $\mc C^1$ tel que $\gamma(0)=a$ et, pour tout $t$, $\|\gamma'(t)\|=1$. Pour tout $\lambda\in\R$, on note $C_{\lambda}=f^{-1}(\{\lambda\})$. + - Montrer que $\nabla f(a)$ indique la direction de plus grande pente sur la surface representative de $f$ en $a$. + - Supposons $\gamma'(0)\in\R^+\nabla f(a)$. Montr per que, pour $\lambda$ suffisamment proche de $\alpha=f(a)$, il existe un unique $t_{\lambda}$ voisin de $0$ tel que $\gamma(t_{\lambda})$ appartient à $C_{\lambda}$. Donner un équivalent de $\|\gamma(t_{\lambda})-a\|$ quand $\lambda\ra\alpha$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 799] +Soit $f=(f_1,\ldots,f_n)\colon\R^n\ra\R^n$ de classe $\mc C^2$ sur $\R^n$. On considére les assertions : (i) $\forall(x,h)\in\R^n,\;\|df_x(h)\|=\|h\|$, (ii) $\forall(x,h)\in\R^n,\;\|f(x+h)-f(x)\|=\|h\|$. + - On suppose (i) et on pose, pour tous $i,j,k\in\db{1,n}$, $a_{i,j,k}=\sum_{m=1}^n\frac{\partial f_m}{\partial x_i}\cdot\frac{ \partial f_m}{\partial x_j\partial x_k}$. + +Montr per que $a_{i,j,k}=a_{i,k,j}=-a_{k,i,j}$ puis que $a_{i,j,k}=0$. + - Montr per l'équivalence des assertions (i) et (ii). +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 800] +Soit $f\colon\R^n\ra\R^n,x\mapsto(f_1(x),\ldots,f_n(x))$. + - On suppose $f$ de classe $\mc C^2$. Montr per que $J_f(x)$ est antisymétrique pour tout $x\in\R^n$ si et seulement s'il existe $A\in\mc{A}_n(\R)$ et $b\in\R^n$ tels que $f(x)=Ax+b$ pour tout $x\in\R^n$. + - On suppose $f$ de classe $\mc C^1$. Montr per que $J_f(x)$ est symétrique pour tout $x\in\R^n$ si et seulement s'il existe $\phi\colon\R^n\ra\R$ de classe $\mc C^2$ telle que $f=\nabla g$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 801] +Extrema de $f:(x,y)\in\R^2\mapsto xe^y+ye^x$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 802] +Soient $E$ un espace euclidien, $\phi\in E^*$ une forme lineaire et $f:x\mapsto\phi(x)e^{-\|x\|^2}$. Étudier les extrema de $f$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 803] +Soient $n\geq 2$ un entier et $f\colon\R^n\ra\R$ de classe $\mc C^2$ telle que la hessienne de $f$ est toujours à valeurs propres dans $[1,+\i[$. + - Soit $x\in\R^n$. Montr per que la fonction $t\mapsto f(tx)-\langle\nabla f(0),tx\rangle-\frac{t^2}{2}\|x\|^2$ est convexe. + - Montr per que $f$ admet un minimum global. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 804] +Soient $E$ un espace euclidien, $v\in E$ non nul et $f\in\mc{S}^{++}(E)$. + - Montr per qu'il existe une base $(e_1,\ldots,e_n)$ de $E$ formée de vecteurs propres de $f$. + - Montr per que, pour tout $x\in E$ non nul, $\langle f(x),x\rangle\gt 0$. + - Montr per que $g:x\mapsto\frac{1}{2}\langle f(x),x\rangle-\langle v,x\rangle$ est de classe $\mc C^1$. + - Calculer les derivées partielles de $g$ relativement à la base $(e_1,\ldots,e_n)$ et le gradient de $g$. + - Montr per que $g$ admet un unique point critique $c$. + - Montr per que $g$ admet un minimum global en $c$. Existe-t-il d'autres extrema locaux? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 805] +On munit $\R^n$ de la norme euclidienne usuelle. On note $\mc{B}$ la boule unite ouverte et $\mc{S}$ la sphere unite. Soit $f\colon\overline{\mc{B}}\ra\R$ de classe $\mc C^2$. + - On suppose que $f_{|\mc{S}}\leq 0$ et qu'il existe $\zeta\in\mc{B}$ tel que $f(\zeta)\gt 0$. + +Montrer que $\phi:x\in\overline{\mc{B}}\mapsto f(x)+\eps(\|x\|^2-1)$ admet un maximum en $\zeta_0\in\mc{B}$ pour $\eps\gt 0$ assez petit puis prouver que $\Delta f(\zeta_0)\lt 0$. + - On suppose que $\Delta f=0$. Montrer que $\min\limits_{\overline{\mc{B}}}f=\min\limits_{\mc{S}}f$ et $\max\limits_{\overline{\mc{B}}}f=\max\limits_{\mc{S}}f$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 806] +Déterminer les espaces tangents en $I_n$ aux parties $\text{SL}_n(\R)$ et $\mc{O}_n(\R)$ de $\M_n(\R)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 807] + - Soient $A$ la $\R$-algèbre des fonctions de classe $\mc C^{\i}$ de $\R^n$ dans $\R$ et $I$ l'ensemble des $f\in A$ telles que $f(0)=0$. Montrer que $I$ est un idéal de $A$ et que tout élément de $I$ s'écrit $\sum\limits_{i=1}^nf_i\theta_i$ ou les $f_i$ sont dans $A$ et les $\theta_i$ sont les formes lineaires coordonnées canoniques sur $\R^n$. + - Déterminer les $\phi$ de $A^*$ vérifiant, pour tout $(f,g)\in A^2$, $\phi(fg)=f(0)\phi(g)+g(0)\phi(f)$. + - Montrer que l'ensemble des formes lineaires de la question précédente est un sous-espace vectoriel de dimension finie de $A^*$. Quelle est sa dimension? +#+end_exercice + + +** Probabilités + +#+begin_exercice [Mines MP 2024 # 808] +On considére $n$ ampoules eteintes numerotées de 1 à $n$. L'ampoule $i$ à une probabilité $p_i$ de s'allumer à un instant donne. On note $Y$ la variable aléatoire comptant le nombre d'ampoules s'allumant. + - Exprimer $\mathbf{E}(Y)$ et $\mathbf{V}(Y)$. + - On fixe à present $m$ et on considére des $p_i$ tels que $\mathbf{E}(Y)=m$. Donner une condition nécessaire et suffisante sur les $p_i$ pour que $\mathbf{V}(Y)$ soit maximal. Donner la loi de $Y$ dans ce cas. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 809] +Un magasin dispose d'un stock de $N$ produits. Le nombre de clients qui passent dans une journée suit la loi de Poisson de paramêtre $\lambda$ et chaque client à une probabilité $p$ d'acheter le produit. Quelle est la probabilité que le magasin soit en rupture de stock avant la fin de la journée? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 810] +On lance $N$ des. à chaque tour, on relance ceux qui n'ont pas donne $6$ lors des tours précédents. Soit $S_n$ la variable aléatoire donnant le nombre total de des ayant donne $6$ au cours des $n$ premiers tours. + - Montrer que $S_n$ suit une loi binomiale dont on donnera les paramêtres. + - Montrer que $\mathbf{P}\left(\bigcup_{n=1}^{+\i}(S_n=N)\right)=1$. + - On pose $T_N=\inf\{n\in\N^*,\ S_n=N\}\in\N^*\cup\{+\i\}$. + - Donner la loi de $T_N$. + - Montrer que $T_N$ admet une esperance et la calculer. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 811] +Un peage comporte $3$ voies et $n$ voitures se presentent en choisissant aléatoirement et indépendamment une voie. On note $X_i$ le nombre de voitures qui passent par la voie $i$ pour $i\in\{1,2,3\}$. + - Déterminer la loi des $X_i$. + - Calculer $\mathbf{V}(X_1),\mathbf{V}(X_2)$ et $\mathbf{V}(X_1+X_2)$. En déduire $\op{Cov}(X_1,X_2)$. + - Les variables $X_1,X_2,X_3$ sont-elles indépendantes deux à deux? mutuellement indépendantes? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 812] +Une urne contient des boules numerotées de 0 à $n$. On en prend une poignée au hasard et on note les numeros obtenus. On effectue deux tirages indépendants. Soit $X$ la variables aléatoire correspondant au nombre de numeros communs entre les deux poignées. Déterminer la loi de $X$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 813] +Soient $m,n,p$ des entiers $\geq 1$ tels que $p\leq\min(m,n)$. Une urne contient $m$ boules mauves et $n$ boules noires. On tire simultanement $p$ boules dans l'ume et on note $X$ la variable aléatoire donnant le nombre de boules mauves tires. Quelle est la valeur la plus probable de $X$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 814] +Une urne contient $a\geq 1$ boules blanches et $b\geq 1$ boules rouges. à chaque tirage, on remet la boule tirée et on ajoute $c\geq 1$ boules de la même couleur. Soit $Y$ la variable aléatoire donnant le rang de la première boule blanche tirée. Donner sa loi. Admet-elle une esperance? Un moment d'ordre $p\geq 2$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 815] +On dispose de deux urnes $A$ et $B$, et de $2N$ boules numerotées de 1 à $2N$ reparties aléatoirement dans ces urnes. à chaque iteration, on pioche une boule au hasard et on la change d'urne. On note $X_n$ la variable aléatoire donnant le nombre de boules dans l'urne $B$ à la $n^{\text{e}}$ iteration. On pose, pour $n\in\N$, $U_n=\left(\mathbf{P}(X_n=0)\ \mathbf{P}(X_n=1)\ \cdots\ \mathbf{P}(X_n=2N) \right)^T$. + - Déterminer $M\in\M_{2N+1}(\R)$ telle que, pour tout $n$, $U_{n+1}=MU_n$. + - Soient $v_0,\ldots,v_{2N}$ des réels et $P=\sum_{k=0}^{2N}v_kX^k$. En notant $V$ le vecteur colonne défini par les coefficients $v_k$, montrer que $V\in\op{Ker}(M-\lambda I_{2N+1})\Leftrightarrow\lambda P=XP-\frac{1- X^2}{2N}P'$. + - Montrer les $X_n$ suivent la même loi si et seulement si $X_0$ suit une certaine loi à déterminer. + - La matrice $M$ est-elle diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 816] +On lance simultanement deux pieces equilibrées $n$ fois. Soit $E_n$ l'evenement \lt \lt les deux pieces donnent le me nombre de pile \gt \gt . + - - Pour $a,b,n\in\N$ tels que $n\leq a+b$, montrer que $\sum_{k=0}^n\binom{a}{k}\binom{b}{n-k}=\binom{a+b}{n}$. + - En déduire $\mathbf{P}(E_n)$. + - Déduire combien de fois en moyenne les pieces sont tombées sur Pile lorsque l'evenement $E_n$ est realise. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 817] +Soient $A$ et $B$ deux evenements. Montrer que $A$ et $B$ sont indépendants si et seulement si $\mathbf{P}\big(A\cap B\big)\mathbf{P}\big(\overline{A}\cap\overline{B} \big)=\mathbf{P}\big(A\cap\overline{B}\big)\mathbf{P}\big(\overline{A} \cap B\big)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 818] +Soit $X$ une variable aléatoire à valeurs dans $\N$ et d'esperance finie. + +Montr per que $\sum_{n\in\N^*}\mathbf{P}(X\geq n)$ converge et donner sa somme. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 819] +Soit $X$ une variable aléatoire suivant la loi de Poisson de paramêtre $\lambda$. + +Calculer $\mathbf{E}\left(\frac{1}{X+1}\right)$ et $\mathbf{E}\left(\frac{1}{(X+1)(X+2)}\right)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 820] +Soient $X,Y$ deux variables aléatoires indépendantes suivant chacune la loi geometrique de paramêtre $1/2$. On pose : $U=\max(X,Y)$ et $V=\min(X,Y)$. Déterminer la loi de $(X,Y)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 821] +A quelle condition sur $\alpha$ existe-t-il une variable aléatoire $X$ à valeurs dans $\N^*$ telle que $\mathbf{P}(X=n)=\int_1^{+\i}\frac{dt}{(1+t^{\alpha})^n}$ pour tout $n\in\N^*$. Lorsque cela est realise montrer que $X$ admet une variance et la calculer. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 822] +Soient $\alpha\in\R$ et $X$ une variable aléatoire à valeurs dans $\N^*$. Comparer $\mathbf{E}(X^{\alpha})$ et $\mathbf{E}(X)^{\alpha}$ au sens de $\overline{\R}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 823] +Soient $A=\begin{pmatrix}0&-1&0\\ 2&1&-2\\ 1&-1&-1\end{pmatrix}$ et $U=\begin{pmatrix}X\\ Y\\ Z\end{pmatrix}$ avec $X$, $Y$, $Z$ trois variables aléatoires indépendantes, $X$ et $Z$ suivant $\mc{G}(p)$ avec $p\in]0,1[$ et $Y$ suivant $\mc{P}(\lambda)$ avec $\lambda\in\R^{+*}$. Déterminer la probabilité que $U$ soit vecteur propre de $A$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 824] +Soient $X,Y$ deux variables aléatoires à valeurs dans $\R^{+*}$. Minorer aussi precisement que possible $\mathbf{E}(X/Y)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 825] +Soient $n\in\N^*$ et $X_1,...,X_n$ variables aléatoires i.i.d. à valeurs dans $\R^{+*}$. + +Calculer $\mathbf{E}\left(\frac{X_1+\cdots+X_k}{X_1+\cdots+X_n}\right)$ pour $k\in\db{1,n}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 826] +Soient $X$ une variable aléatoire suivant la loi de Poisson de paramêtre $\lambda$ et $Y=X^2+1$. + - Calculer $\mathbf{E}(Y)$. + - Calculer $\mathbf{P}(2X\lt Y)$. + - Comparer $\mathbf{P}(X\in 2\N)$ et $\mathbf{P}(X\in 2\N+1)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 827] +On suppose que la probabilité de tirer un entier $n\in\N^*$ est $\frac{1}{2^n}$. + - Calculer $\mathbf{P}(A_p)$ ou $A_p$ est l'evenement $\lnot n$ est multiple de $p$. + - Calculer $\mathbf{P}(A_2\cup A_3)$. + - On note $B$ l'evenement $\lnot n$ est premier $\lnot n$. Montr per que $\frac{13}{32}\lt \mathbf{P}(B)\lt \frac{209}{504}$. En déduire $\mathbf{P}(B)$ à $10^{-2}$ pres. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 828] +Soit $A,B$ deux variables aléatoires indépendantes qui suivent la loi geometrique de paramêtre $p\in]0,1[$. Calculer ${\bf P}(A^B\leq B^A)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 829] +Soit $X$ une variable de loi de Poisson ${\cal P}(\lambda)$, avec $\lambda\gt 0$. Soient $p\in\N^*$ et $Y=\overline{X}$ à valeurs dans $\Z/p\Z$. Quelle est la loi de $Y$? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 830] +Soit $(X_n)_{n\geq 1}$ une suite de variables i.i.d. de loi de Bernoulli ${\cal B}(p)$. Pour $n\in\N^*$, on pose $S_n=X_1+\cdots+X_n$. Montrer que $p=1/2$ si et seulement si, pour tout $n\in\N^*$ et tout $k\in\Z$, ${\bf P}(S_{2n}=k)\leq{\bf P}(S_{2n}=0)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 831] +Soit $(E_n)_{n\geq 0}$ une suite d'evenements de $(\Omega,{\cal A},{\bf P})$ et $Z=\sum_{n=0}^{+\i}{\bf 1}_{E_n}$. Montrer que si $\sum{\bf P}(E_n)$ converge alors $Z$ est d'esperance finie. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 832] + - Soit $n\in\N^*$. Donner le développement en série entiere de $f:t\mapsto\frac{1}{(1-t)^n}$. + - En déduire que $|\{(k_1,\ldots,k_n)\in\N^n,\ k_1+\cdots+k_n=s\}|=\binom{s+n- 1}{n}$. + - Soit $(X_i)_{i\geq 1}$ i.i.d. suivant la loi geometrique de paramêtre $p\in\,]0,1[$. + +Déterminer ${\bf P}\left(\bigcup_{n\geq 1}\left(X_1+\cdots+X_n=s\right)\right)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 833] +Soient $X_1,\ldots,X_n$ des variables aléatoires réelles discretes indépendantes. + - Montrer que $X_1+X_2,X_3,\ldots,X_n$ sont indépendantes. + - En déduire que, pour tout $r\in\,\db{2,n-1}$, $X_1+\cdots+X_r,X_{r+1},\ldots,X_n$ sont indépendantes. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 834] +Soient $k\in\N$ avec $k\geq 2$, $a_0,\ldots,a_{k-1}\in\,]0,1[^k$ tels que $a_0+\cdots+a_{k-1}=1$. Soit $(X_n)$ une suite de variables aléatoires à valeurs dans $\Z/k\Z$. On suppose que : ${\bf P}(X_0=0)=1$ et $\forall n\in\N,\,\forall j\in\Z/k\Z, {\bf P}(X_{n+1}=j)=\sum_{i=0}^{k-1}a_i{\bf P}(X_n=j-i)$. + - Déterminer la loi de $X_n$. + - Soit $j\in\Z/k\Z$ fixe. Étudier le comportement asymptotique de $({\bf P}(X_n=j))_{n\geq 0}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 835] +Soit $X$ une variable aléatoire suivant la loi geometrique de paramêtre $p\in]0,1[$. On pose $Y=\int_0^{2\pi}\sin(t)^Xdt$. Montrer que $Y$ possede une esperance et la calculer. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 836] +Soient $X_1,X_2$ deux variables aléatoires indépendantes qui suivent la loi geometrique de paramêtre $p\in]0,1[$. On pose $Y=|X_1-X_2|$. + - Calculer ${\bf P}(Y=0)$ puis ${\bf P}(Y=n)$ pour $n\in\N^*$. Montrer que $Y$ admet une esperance et la calculer. + - Montrer que ${\bf E}(X_1-X_2)^2=2\,{\bf V}(X_1)$. En déduire que $Y$ admet une variance et la calculer. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 837] +Soient un espace probabilise $(\Omega,{\cal A},{\bf P})$ et une variable aléatoire $X$ suivant la loi de Poisson de paramêtre $\lambda\gt 0$. + - Montrer que ${\bf P}(X\geq 2\lambda)\leq\frac{1}{\lambda}$. + - Soit $Z$ une variable aléatoire réelle centrée admettant un moment d'ordre 2. On pose ${\bf V}(Z)=\sigma^2$. + - Montrer que pour tous $a\gt 0$ et $x\gt 0$, ${\bf P}(Z\geq a)\leq\frac{\sigma^2+x^2}{(x+a)^2}$. + - En déduire que ${\bf P}(Z\geq a)\leq\frac{\sigma^2}{\sigma^2+a^2}$ et ${\bf P}(X\geq 2\lambda)\leq\frac{1}{\lambda+1}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 838] + - Rappeler le développement en série entiere au voisinage de $0$ de $\frac{1}{\sqrt{1-x}}$, ainsi que sa validite. + - Donner une condition nécessaire et suffisante sur le réel $r$ pour qu'il existe une variable aléatoire $X$ à valeurs dans $\N$ telle que ${\bf P}(X=n)=\frac{(2n)!}{2^{3n}(n!)^2}r$ pour tout $n\in\N$. + - Calculer alors l'esperance et la variance de $X$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 839] +Soit $X$ une variable aléatoire à valeurs dans $\N^*$ telle que $\forall k\in\N^*,\ {\bf P}(X=k)=\frac{1}{2^k}$. + - Justifier la bonne définition d'une telle loi et calculer l'esperance de $X$. + - Pour $n\in\N^*$, on note $A_n$ l'evenement $(n|X)$. Les evenements $A_p$ et $A_q$ sont-ils indépendants si $p$ et $q$ sont deux entiers pairs? + - Étudier l'indépendance de $A_p$ et $A_q$ pour $p$ et $q$ entiers quelconques. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 840] +Soit $p\in\,]0,1[$ et $(X_n)_{n\geq 1}$ une suite de variables aléatoires i.i.d. de loi ${\cal G}(p)$. On pose $Y_n=\min(X_1,\ldots,X_n)$ et $Z_n=\max(X_1,\ldots,X_n)$ ; $\alpha_n={\bf E}(Y_n)$ et $\beta_n={\bf E}(Z_n)$. + - Déterminer la monotonie des suites $(\alpha_n)$ et $(\beta_n)$. + - Calculer $\alpha_n$. + - Déterminer la limite de $(\beta_n)$. Donner un équivalent de $\beta_n$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 841] +Soient $\lambda\gt 0$ et $n\in\N$. Soit $X$ une variable aléatoire qui suit la loi de Poisson de paramêtre $\lambda$. On pose $Y=(X+n)!$ + - Trouver une condition sur $\lambda$ pour que $Y$ admette une esperance finie. + - On suppose que $Y\in L^1$. Montrer que : $\forall m\in\N^*,{\bf P}(Y\geq m)\leq\frac{n!}{m(1-\lambda)^{ n+1}}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 842] + - Montrer qu'il existe une variable aléatoire telle que : $\forall t\in[0,1],G_X(t)=\frac{e^{t-1}}{\sqrt{2-t}}$. + - Calculer ${\bf E}(X)$ et ${\bf V}(X)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 843] +Soit $X$ une variable aléatoire à valeurs dans $\R^{+*}$ telle que : ${\bf E}\left(1/X\right)\lt +\i$. On définit $F_X$ sur $\R^+$ par : $\forall t\in\R^+,F_X(t)={\bf E}(e^{-tX})$. + - Montrer que $F_X$ est bien définie, continue, intégrable sur $\R^+$ et calculer $\int_0^{+\i}F_X$. - Soient $Y,Z$ deux variables aléatoires indépendantes qui suivent chacune la loi geometrique de paramêtre $p\in\,]0,1[$. Calculer ${\bf E}\Big(\frac{1}{X+Y}\Big)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 844] +Soient $X_1$,..., $X_n$ des variables aléatoires indépendantes. Notons, pour tout $k$ $F_k$ la fonction de repartition associée à $X_k$. + +On note $X=\max(X_1,\ldots,X_n)$ et $Y=\min(X_1,\ldots,X_n)$. + - Montr er que $F_X=\prod_{k=1}^nF_k$ et $F_Y=1-\prod_{k=1}^n(1-F_k)$. + - Soient $x,y\in\R$ avec $y\lt x$. Montr er que ${\bf P}(y\lt Y\leq X\leq x)=\prod_{k=1}^n(F_k(x)-F_k(y))$. + - Supposons que les $X_k$ suivent des lois geometrique de paramêtre $p_k\in]0,1[$. Déterminer la loi de $Y$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 845] +Soit $X$ une variable aléatoire à valeurs dans $\N$ et admettant une variance. + - Montr er que la fonction generatrice de $X$ est convexe sur $[0,1]$. + - Prouver que ${\bf E}\left(\frac{1}{X+1}\right)\leq 1-\frac{2}{3}{\bf E}(X)+\frac{1}{6}{ \bf E}(X^2)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 846] +Soient $m,n\in\N$ tels que $n\gt m+2$. On définit une suite $(u_k)_{k\in\N}$ en fixant $u_0\in\R$ et en posant, pour tout $k\in\N$, $u_{k+1}=\frac{k+m}{k+n}u_k$. + - étudier la série $\sum\ln\left(\left(\frac{k+1}{k}\right)^{n-m}\frac{u_{k+1}}{u_k}\right)$. En déduire l'existence d'une constante $C\gt 0$ telle que $u_k\underset{k\ra+\i}{\sim}Ck^{m-n}$. + - Montr er l'existence d'une variable aléatoire réelle $X$ telle que : + + $\forall k\in\N,\,(k+n){\bf P}(X=k+1)=(k+m){\bf P}(X=k)$ + - Montr er que $X\in L^1$ et calculer ${\bf E}(X)$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 847] +Soit $(X_i)_{i\geq 1}$ une suite de variables aléatoires i.i.d. suivant la loi geometrique ${\cal G}(p)$, avec $p\in]0,1[$. + - Soit $n\in\N^*$. Donner la loi de $S_n=X_1+\cdots+X_n$. + - Déterminer un équivalent de $\max\{{\bf P}(S_n=k),k\in\N\}$ lorsque $n\ra+\i$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 848] +Soient $(X_i)_{i\geq 1}$ une suite de variables aléatoires indépendantes qui suivent la loi ${\cal G}(p)$ et $N$ une variable indépendante des $X_i$ qui suit la loi ${\cal G}(q)$. Soit $S=\sum_{k=1}^NX_k$. Montr er que $S$ est une variable aléatoire et déterminer son esperance et sa variance. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 849] +Soit $N\in\N^*$. On repartit $2N$ boules entre deux urnes $A$ et $B$. On tire successivement une boule au hasard dans l'une des urnes, et on la place dans l'autre urne. + +Pour $n\in\N$, on note $X_n$ le nombre de boules dans l'urne $B$ au $n^{\text{\`{e}me}}$ tour et on pose + + $U_n=({\bf P}(X_n=0)\cdots{\bf P}(X_n=2N))^T\in{\cal M}_{2N+1,1}(\mathbb{ R})$. + - Trouver une matrice $M\in{\cal M}_{2N+1}(\R)$ telle que $\forall n\in\N,\,U_{n+1}=MU_n$. - Soit $V=(v_0,...,v_{2N})^T\in\M_{2N+1,1}(\R)$. On note $P(X)=v_0+v_1X+...+v_{2N}X^{2N}$. + +Pour $\lambda\in\R$, montrer que $V\in\mathrm{Ker}(M-\lambda I_{2N+1})$ si et seulement si $\lambda P=XP+(1-X^2)P'$. + - Comment choisir $X_0$ pour que toutes les variables aléatoires $X_n$ soient equidistribuées? + - La matrice $M$ est-elle diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 850] +Soit $(X_i)_{i\geq 1}$ une suite de variables aléatoires i.i.d. suivant la loi geometrique $\mc{G}(p)$, avec $p\in]0,1[$. Pour $n\in\N^*$, soient $Y_n=\min(X_1,\ldots,X_n)$, $Z_n=\max(X_1,\ldots,X_n)$, $a_n=\mathbf{E}(Y_n)$ et $b_n=\mathbf{E}(Z_n)$. + - Étudier la monotonie de $(a_n)_{n\geq 1}$ et $(b_n)_{n\geq 1}$. + - Pour $n\in\N^*$, déterminer $a_n$. + - Déterminer la limite et un équivalent simple de $(b_n)_{n\geq 1}$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 851] +Soit $(X_k)_{k\geq 1}$ une suite i.i.d. de variables aléatoires de Rademacher. Pour $n\in\N^*$, soit $S_n=\sum_{k=1}^nX_k$. + - Pour $t\in{\R^+}^*$ et $n\in\N^*$, montrer que $\mathbf{E}(e^{tS_n})\leq\exp\left(\frac{nt^2}{2}\right)$. + - Pour $a\in{\R^+}^*$ et $n\in\N^*$, montrer que $\mathbf{P}(|S_n|\geq a)\leq 2\exp\left(\frac{a^2}{2n}\right)$. + - Montrer que le résultat de la question précédente subsite si $(X_k)_{k\geq 1}$ est une suite i.i.d. de variables aléatoires bornées par $1$ et centrées. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 852] +Soit $X$ une variable aléatoire réelle discrete. + - Pour $t\in\R$, justifier l'existence de $\phi_X(t)=\mathbf{E}(e^{itX})$. + - Montrer que $\phi_X$ est uniformément continue sur $\R$. + - Montrer que $\phi_X$ détermine la loi de $X$. +#+end_exercice + + +#+begin_exercice [Mines MP 2024 # 853] +Soient $k\leq n\in\N$. Un parking dispose de $n$ places consécutives numerotées de $1$ à $n$. On y dispose des vehicules nécessit chacun $k$ places consécutives pour être gares. Chaque vehicule est successivement place aléatoirement sur les emplacements disponibles jusqu'a ce qu'on ne puisse plus en garer aucun. + +Pour $j\in\db{1,n-k+1}$, $B_j$ designe l'evenement $\llcorner$ la première voiture est garée entre les emplacements $j$ et $j+k-1$, $\neq$ et $X_n$ est le nombre d'emplacements residuels libres à la fin du processus. + - Montrer que, pour $i,j$ convenables, $\mathbf{P}_{B_j}(X_n=i)=\mathbf{P}(X_{j-1}+X_{n-(j+k)+1}=i-k)$. + +En déduire que $\mathbf{E}(X_n)=k+\frac{2}{n-k+1}\sum_{\ell=0}^{n-k}\mathbf{E}(X_{\ell})$. + - Pour $n\in\N$, on pose $S_n=\mathbf{E}(X_0)+\cdots+\mathbf{E}(X_n)$. + +Montrer que la somme $f$ de la série entiere $\sum S_nt^n$ est au moins définie sur $]0,1[$ et vérifie une équation différentielle lineaire d'ordre $1$. + - Expliciter $f$ et en déduire une expression de $\mathbf{E}(X_n)$ pour $n\in\N$. +#+end_exercice + + +* Mines - Ponts - PSI :autre: + +** Algèbre + +#+begin_exercice [Mines PSI 2024 # 854] +Soit $A=\frac{1}{5}\begin{pmatrix}7&-4&0\\ 6&-7&0\\ 0&0&-5\end{pmatrix}$. + - Interpreter geometriquement $A$. + - Donner l'image du plan $P$ d'équation $x-y-z=0$ par $A$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 855] +Soit $P\in\M_n(\R)$ representant un projecteur $p$ de rang $r$ dans la base canonique de $\R^n$. Déterminer la trace de l'endomorphisme de $\M_n(\R)$ défini par : $\Psi(X)=PX-XP$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 856] +Soient $n\in\N^*,x\in\R$ et $P\in\R_{n-2}[X]$. Montrer que la matrice $A\in\M_n(\R)$ définie par : $A_{i,j}=P(x+i+j-2)$ n'est pas inversible. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 857] +Montrer que la matrice $A=\begin{pmatrix}0&1&0\\ 0&0&1\\ 0&0&0\end{pmatrix}$ n'admet pas de racine carrée. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 858] +On note $D_n$ le nombre de permutations sans point fixe de $\db{1,n}$. On note $D_0=1$. + - Soit $M=\left(\begin{pmatrix}j\\ i\end{pmatrix}\right)_{0\leq i,j\leq n}\in\M_{n+1}(\R)$. Déterminer $M^{-1}$. + - Exprimer $n!$ en fonction des $D_k$ pour $0\leq k\leq n$. + - En déduire une expression de $D_n$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 859] +Soit $r\geq 2$. + - Montrer que l'équation $X^r=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$ n'a pas de solution $X\in\M_2(\C)$. + - Déterminer les solutions $X\in\M_2(\C)$ de l'équation $X^r=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 860] +Soient $a_1,\dots,a_n$ des nombres complexes distincts. Soit $A\in\M_n(\C)$ la matrice de terme general $a_{i,j}=\left\{\begin{array}{c}0\text{ si }i=j\\ a_j\text{ si }i\neq j\end{array}.$. Soit $P\::x\mapsto\det(A+xI_n)$. + - Montrer que $P$ est un polynôme unitaire de degre $n$. + - Calculer $P(a_i)$. + - Trouver l'expression de $P$. + - Décomposer $\frac{P(X)}{(X-a_1)\cdots(X-a_n)}$ en éléments simples. + - Calculer $\det(A+I_n)$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 861] +Soient $A_1,\dots,A_n\in\M_n(\C)$ telles que $\forall i\in\db{1,n]\!]$, $\exists p\in[\![1,n]\!]$, $A_i^p=0\text{ et }\forall i,j\in[\![1,n}$, $A_iA_j=A_jA_i$. Montrer que $\prod_{i=1}^nA_i=0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 862] +On dit qu'une matrice $A\in{\cal M}_n(\R)$ admet un pseudo-inverse s'il existe $B\in{\cal M}_n(\R)$ telle que $AB=BA$, $B=BAB$ et $A=ABA$. + - Montrer que, si $A$ admet un pseudo-inverse, alors $A$ et $A^2$ sont de même rang. + - Justifier l'unicité sous reserve d'existence d'un pseudo-inverse. + - Montrer que, si $A$ et $A^2$ sont de même rang, alors $\ker(A)$ et $\op{Im}(A)$ sont supplementaires. Étudier la réciproque de la première question. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 863] +Soient $A,B\in{\cal M}_n(\C)$. On suppose qu'il existe des complexes deux à deux distincts $\lambda_0,\ldots,\lambda_n$ tels que $A+\lambda_iB$ est nilpotente pour tout $i$. + - Montrer que l'indice de nilpotence d'une matrice nilpotente de taille $n$ est inférieur ou egal à $n$. + - Montrer que : $\forall\lambda\in\C,\,(A+\lambda B)^n=0$. + - Montrer que $A$ et $B$ sont nilpotentes. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 864] +Soient $n\geq 2$ et $A\in{\cal M}_n(\R)$ telle que $\forall M\in{\cal M}_n(\R)$, $\det(A+M)=\det(A)+\det(M)$. + - Montrer que $A$ n'est pas inversible. + - Montrer que $A=0$._Ind._ Écrire $A=PJ_rQ^{-1}$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 865] + - Soit $n\in\N$. Montrer qu'il existe un polynôme $P_n$ de degre $\leq n$ tel que $X+1-P_n^2(X)$ soit divisible par $X^{n+1}$._Ind._ Penser aux développements limites. + - Soit $N\in{\cal M}_n(\C)$ une matrice nilpotente. Montrer qu'il existe une matrice $B\in\op{GL}_n(\C)$ tel que $B^2=I_n+N$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 866] +Soit $B=(b_{ij})_{1\leq i,j\leq n}\in{\cal M}_n(\R)$ telle que + + $\forall i,\,b_{i,i}\gt 0,\,\forall i\neq j,\,b_{i,j}\leq 0$ et $\forall i,\,\sum_{j=1}^nb_{i,j}\gt 0$. + - Montrer que $B$ est inversible. On prendra $X\in\op{Ker}(B)$ et on étudiera $|x_{i_0}|=\max_i|x_i|$. + - Soit $X\in\R^n$ à coefficients $\geq 0$. Montrer que $Y=B^{-1}X$ est à coefficients $\geq 0$. + - En déduire que $B^{-1}$ est à coefficients $\geq 0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 867] +Soit $E$ un espace vectoriel de dimension $n$. Soit $d\in\db{1,n-1}$. Trouver l'ensemble des endomorphismes de $E$ qui stabilisent tous les sous-espaces vectoriels de dimension $d$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 868] +Soient $A$ et $B$ dans ${\cal M}_n(\R)$. On suppose que $AB=BA$ et qu'il existe $p\in\N^*$ tel que $B^p=0$. Montrer que $\det(A+B)=\det(A)$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 869] +Soient $a,b,c\in\R$ et $A=\begin{pmatrix}0&-a&b\\ a&0&-c\\ -b&c&0\end{pmatrix}$. + - Montrer qu'il existe $d$ tel que $A^3+dA=0$. + - Déterminer $d$. Soit $n\in\N^*$, déterminer $A^{2n}$ en fonction de $d$, $n$ et $A^2$. + - Déterminer $\alpha$ et $\beta$ tels que $\sum_{k=0}^{+\i}\frac{A^k}{k!}=I_3+\alpha A+\beta A^2$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 870] +Soit $A=\begin{pmatrix}1&j&j^2\\ j&j^2&1\\ j^2&1&j\end{pmatrix}$ ou $j=e^{2i\pi/3}$. + - La matrice $A$ est-elle diagonalisable? Déterminer une matrice semblable à $A$, diagonale ou triangulaire. + - Expliciter $C_A=\{M\in\M_3(\C),\ AM=MA\}$. + - Soit $f_A$ l'endomorphisme de $\C^3$ canoniquement associe à $A$. Quels sont les sous-espaces vectoriels $f_A$-stables de $\C^3$? + - Peut-on retrouver $C_A$ par des arguments de stabilité? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 871] +Soit $k\in\C$. Soit $A=\begin{pmatrix}0&1&0&0\\ 1&k&1&1\\ 0&1&0&0\\ 0&1&0&0\end{pmatrix}$. Étudier la diagonalisabilité de $A$ en fonction de $k$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 872] +Soit $f\in\mc{L}(\C^n)$. On suppose $f^2$ diagonalisable. Montrer que $f$ est diagonalisable si et seulement si $\op{Ker}(f)\cap\op{Im}(f)=\{0\}$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 873] + - Soit $A\in\M_n(\R)$ telle que $A^2$ soit diagonalisable et $\op{Sp}(A^2)\subset]0,+\i[$. Montrer que $A$ est diagonalisable. + - Diagonaliser $A=$ $\begin{pmatrix}a&b&\ldots&b\\ b&a&\ddots&\\ &\ddots&\ddots&b\\ b&\ldots&b&a\end{pmatrix}$ et $B=$ $\begin{pmatrix}b&\ldots&b&a\\ \vdots&\ddots&\ddots&b\\ b&\ldots&\ddots&\vdots\\ a&b&\ldots&b\end{pmatrix}$ avec $a\in\R$ et $b\in\R^*$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 874] +Soient $\mathbb{K}=\R$ ou $\C$, $E$ un $\mathbb{K}$-espace vectoriel de dimension finie et $f\in\mc{L}(E)$. On suppose que $f^2$ est un projecteur. Donner une condition nécessaire et suffisante portant sur $f$ pour que $f$ soit diagonalisable. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 875] +Soient $a\in\C$ et $u:P\in\C[X]\mapsto(X-a)P'$. + - Montrer que $u$ est lineaire. + - Trouver les valeurs propres de $u$. + - Trouver les $P$ dans $\C[X]$ tels que $P'$ divise $P$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 876] +Soient $E=\C_n[X]$, $\alpha\in\C$ et $f\,:P\in E\mapsto P-\alpha(X-\alpha)P'$. + - Montrer que $f\in\mc{L}(E)$ et donner sa matrice dans la base canonique. + - Montrer que $f$ est diagonalisable. + - à quelle condition sur $\alpha$, l'endomorphisme $f$ est-il inversible? + - Montrer, pour tout $k\in\N:E=\op{Ker}(f^k)\oplus\op{Im}(f^k)$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 877] +Soit $A\in\M_n(\C)$ telle que $\op{rg}(A)=2$, $\op{tr}(A)=0$ et $A^n\neq 0_n$. Montrer que $A$ est diagonalisable. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 878] +Soit $A=$ $\begin{pmatrix}1&0&0\\ 1&2&1\\ 2&-2&-1\end{pmatrix}$. + - Donner le spectre de $A$ et ses espaces propres. La matrice $A$ est-elle diagonalisable? + - Montrer qu'il existe $P\in\op{GL}_3(\R)$ tel que $A=PTP^{-1}$ avec $T=$ $\begin{pmatrix}0&0&-3\\ 0&1&4\\ 0&0&-1\end{pmatrix}$. + - Trouver l'ensemble des matrices $M\in\M_3(\R)$ telles que $MT=TM$. Quelle est sa structure? sa dimension? + - Trouver l'ensemble des matrices $M\in\M_3(\R)$ qui commutent avec $A$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 879] +Soit $A=\begin{pmatrix}-1&1&1\\ 0&5&-14\\ 0&-3&-8\end{pmatrix}$. + - La matrice $A$ est-elle diagonalisable? + - Soit $n\in\N$. Montrer qu'il existe un unique $(\alpha_n,\beta_n,\gamma_n)\in\R^3$ et un unique $Q_n\in\R[X]$ tels que $X^n=(X+1)^2(X+2)Q_n(X)+\alpha_n(X+2)+\beta_n(X+1)(X+2)+\gamma_n(X+ 1)^2$. + - Déterminer $A^n$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 880] +Soit $u$ l'application définie par : $\forall P\in\C[X]$, $\forall z\in\C$, $u(P)(z)=e^{-z}\sum_{k=0}^{+\i}\frac{P(n)}{n!}z^n$. + - Montrer que $u$ est un endomorphisme de $\C[X]$. + - Trouver les valeurs propres de $u$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 881] +Soit $M\in\M_2(\Z)$ telle qu'il existe $n\in\N^*$ vérifiant $M^n=I_2$. Prouver que $M^{12}=I_2$. Ind. Montrer que $M$ est $\C$-diagonalisable et considérer sa trace. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 882] +Soit $E$ un espace vectoriel de dimension finie $n\geq 1$. On dit que $f\in\mc{L}(E)$ est cyclique lorsqu'il existe $x\in E$ tel que $(x,f(x),\ldots,f^{n-1}(x))$ est une base de $E$. + - On suppose $f^{n-1}\neq 0$ et $f$ nilpotent. Montrer que $f$ est cyclique. + - On suppose que $f$ admet $n$ valeurs propres distinctes. Montrer que $f$ est cyclique. + - On suppose $f$ diagonalisable. Déterminer une condition nécessaire et suffisante pour que $f$ soit cyclique. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 883] +Soient $E$ un $\mathbb{K}$ espace vectoriel de dimension finie $n\geq 2$ et $u\in\mc{L}(E)$. On dit que $u$ est cyclique s'il existe $x_0$ tel que $(x_0,u(x_0),\ldots,u^{n-1}(x_0))$ soit une base de $E$. + +Soient $E=\op{Vect}(1,\cos,\sin)$ dans $\mc C^{\i}(\R,\R)$ et $u$ la derivation. Montrer que $u$ est un endomorphisme cyclique non diagonalisable. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 884] +Soit $A\in\M_n(\R)$ telle que $A^3-A-I_n=0$. Montrer que $\det A\gt 0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 885] +Soit $A\in\M_n(\R)$ telle que la suite $(A^k)_{k\in\N}$ converge vers une matrice $B$. + - Montrer que $B^2=B$. + - On suppose desormais que $A$ est diagonalisable avec $p$ valeurs propres. + +En considérant une division euclidienne, montrer que : $\forall k\in\N,\ A^k\in\R_{p-1}[A]$. + - Décrire $B$ à l'aide des éléments propres de $A$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 886] + - Soit $P\in\C[X]$ un polynôme non constant. Montrer que, pour tout $n\in\N^*$, il existe $M\in\M_n(\C)$ telle que $P(M)=0$. + - Soit $Q\in\R[X]$ un polynôme de degre $2$. Montrer qu'il existe $M\in\M_2(\R)$ telle que $Q(M)=0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 887] +Soit $C\in\M_n(\C)$ une matrice de rang $r$. + - Démontrer le theoreme du rang pour les endomorphismes de $\C^n$. + - Montrer qu'il existe $P,Q\in\op{GL}_n(\R)$ telles que $C=PJ_rQ$ ou $J_r=\begin{pmatrix}I_r&0\\ 0&0\end{pmatrix}$. + - Soient $A,B\in\M_n(\C)$ telles que $AC=CB$. Montrer que $A$ et $B$ possedent $r$ valeurs propres communes en tenant compte des multiplicités. + - Que peut-on dire dans - quand $r=n$? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 888] +Soit $A\in\M_n(\mathbb{K})$. Soit $f_A\in\mc{L}(\M_n(\mathbb{K}))$ définie par $\forall M\in\M_n(\mathbb{K}),\,f_A(M)=AM$. + - Pour $P\in\mathbb{K}[X]$, déterminer $P(f_A)$. + - Montrer que $A$ est diagonalisable si et seulement si $f_A$ est diagonalisable. + - Trouver le lien entre $\chi_A$ et $\chi_{f_A}$. + - Donner le lien entre les éléments propres de $A$ et ceux de $f_A$. Retrouver le résultat de la question -. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 889] +Soit $E_N$ l'ensemble des suites à valeurs complexes $N$-periodiques. + - Montrer que $E_N$ est un espace vectoriel de dimension finie et en déterminer sa dimension. Soit $T\,:E_N\ra E_N$ définie par $\forall u\in E_N,\,(T(u))_n=u_{n+1}$. + - Montrer que $T$ est un endomorphisme de $E_N$. + - Déterminer les éléments propres de $T$ de deux facons différentes, en revenant à la définition et matriciellement. + - L'endomorphisme $T$ est-il diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 890] +Soit $E=\R_3[X]$. pour $P,Q\in E$, on note $\Phi(P,Q)=\sum_{k=0}^3(P(k)+P(1))(Q(k)+Q(1))$. + +Pour tout $i\in\db{0,3}$, on note $L_i(t)=\prod_{0\leq k\leq 3\atop k\neq i}\frac{t-k}{i-k}$. + - Calculer $L_i(j)$ pour tous $i,j\in\db{0,3}$. En déduire que $(L_0,L_1,L_2,L_3)$ est une base de $E$. + - Montrer que $\Phi$ est un produit scalaire sur $E$. + - Trouver une base orthonormée de $E$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 891] +Soient $E=\R_4[X]$, $F$ le sous-espace vectoriel de $E$ forme des polynômes pairs, $G$ le sous-espace vectoriel de $E$ forme des polynômes impairs. + +Pour $P,Q\in E$, on note $\Phi(P,Q)=\sum_{k=0}^4\big(P(k)+(-1)^kP(-k)\big)\big(Q(k)+(-1)^kQ (-k)\big)$. + - Montrer que $\Phi$ est un produit scalaire sur $E$. + - Montrer que $E=F\stackrel{{\perp}}{{\oplus}}G$. + - Déterminer une base orthonormée de $E$ adaptée à $E=F\stackrel{{\perp}}{{\oplus}}G$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 892] +Soit $E$ un espace euclidien de dimension $3$. On considére une isométrie indirecte $f$. Montrere que $f$ se décompose en une rotation d'axe $\Delta$ et une reflexion de plan $\Delta^{\perp}$. Cette décomposition est-elle unique? La rotation et la reflexion commutent-elles? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 893] +On munit $\M_n(\R)$ de sa structure euclidienne canonique. + +Soit $F=\{M\in\M_n(\R),\op{tr}(M)=0\}$. + - Montrere que $F$ est un espace vectoriel et donner sa dimension. + - Pour $A\in\M_n(\R)$, donner $d(A,F)$ en fonction notamment de $\op{tr}(A)$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 894] +Soit $(E,\langle\,\ \rangle)$ un espace prehilbertien réel et $F$ un sous-espace vectoriel de $E$. + - Montrere que $F\subset(F^{\perp})^{\perp}$. + - On munit $E=\R[X]$ du produit scalaire donne par $\colon\langle P,Q\rangle=\int_0^1P(t)Q(t)dt$. Soit + + $F=\{P\in E,\ P(1)=P'(1)=0\}$. Déterminer $F^{\perp}$ et $(F^{\perp})^{\perp}$. + - Pour $E$ prehilbertien, donner une condition suffisante sur $F$ pour que $F=(F^{\perp})^{\perp}$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 895] +Soit $(E,\langle\,\ \rangle)$ un espace euclidien. Pour $x_1,\ldots,x_p$ dans $E$, on note $G(x_1,\ldots,x_p)$ la matrice de coefficient $G_{i,j}=\langle x_i,x_j\rangle$. + - Montrere que $\colon\ G$ est inversible si et seulement si $(x_1,\ldots,x_p)$ est libre. + - Montrere que $\op{rg}(G)=\op{rg}(x_1,\ldots,x_p)$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 896] +Soit $(E,\langle\,\ \rangle)$ un espace euclidien et $F$ une partie fermée, non vide et convexe de $E$. + +Pour $x\in E$ on pose $d(x)=\inf_{f\in F}\|x-f\|$ et $\Gamma(x)=\{f\in F,\ \|x-f\|=d(x,F)\}$. + - Caractériser l'ensemble des $x$ tels que $d(x)=0$. + - Montrere que $d$ est 1-lipschitzienne. En déduire que $\Gamma(x)$ est non vide. + - En utilisant une identite relative à la norme, montrer que : + + $\forall(f,f')\in\Gamma(x)^2,\ f\neq f'\Rightarrow\left\|\frac {1}{2}(f+f')-x\right\|^2\lt d(x)^2$. + - Montrere que $\Gamma(x)$ est reduit à un seul élément, que l'on notera $p(x)$. + - Montrere que $p(x)$ est caractérise par $\colon\forall y\in F,\ \langle x-p(x),y-p(x)\rangle\ \leq 0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 897] +On munit $\R^3$ de sa structure euclidienne canonique. Soit $u$ l'endomorphisme de $\R^3$ dont la matrice dans la base canonique est $\frac{1}{3}\begin{pmatrix}2&2&-1\\ -1&2&2\\ 2&-1&2\end{pmatrix}$. Déterminer sa nature et ses valeurs propres. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 898] + - Que peut-on dire du spectre d'une matrice orthogonale? + - Que peut-on dire de la matrice $A=\frac{1}{7}\begin{pmatrix}-2&6&-3\\ 6&3&2\\ -3&2&6\end{pmatrix}$? Que décrit-elle? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 899] +Soient $p$ et $q$ deux projecteurs orthogonaux d'un espace euclidien. + - Montrere que $u=p-q$ est diagonalisable et que $\op{Sp}(u)\subset[-1,1]$. + - Déterminer $\op{Ker}(u+\op{id})$ et $\op{Ker}(u-\op{id})$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 900] +Soient $(E,\langle\,\ \rangle)$ un espace euclidien, $\alpha$ un réel et $a$ un vecteur de $E$ unitaire. + +On définit $f_{\alpha}:x\mapsto x+\alpha\,\langle x,a\rangle\,a$. + - Montrer que $f_{\alpha}$ est un endomorphisme de $E$. + - Soient $\alpha,\beta$ dans $\R$. Calculer $f_{\alpha}\circ f_{\beta}$. Pour quels $\alpha$, $f_{\alpha}$ est-il bijectif? + - Trouver les valeurs et les vecteurs propres de $f_{\alpha}$. + - Pour quels $\alpha$, $f_{\alpha}$ est-il une isométrie vectorielle? + - Pour quels $\alpha$, $f_{\alpha}$ est-il auto-adjoint? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 901] +Soient $E$ un espace euclidien et $u\in\mc{L}(E)$. Montrer qu'il existe une base orthonormée $(e_1,\ldots,e_n)$ telle que la famille $(u(e_1),\ldots,u(e_n))$ soit orthogonale. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 902] +Déterminer l'ensemble des matrices $M\in\M_n(\R)$ telles que $M^TMM^T=I_n$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 903] +Soit $A\in\mathrm{GL}_n(\R)$ telle que $A^2+A^T=I_n$. + - Montrer que $A^4-2A^2+A=0$. + - Montrer que $1$ n'est pas valeur propre de $A$. + - Montrer que $A$ est diagonalisable dans $\M_n(\R)$ et déterminer l'expression des $A$ possibles. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 904] +On munit $\R_n[X]$ du produit scalaire $\langle P,Q\rangle=\int_0^1PQ$. On pose pour tout $P\in\R_n[X]$ + + $(u(P))\,(x)=\int_0^1(x+t)^nP(t)dt$. + - Montrer que $u$ est un endomorphisme auto-adjoint de $\R_n[X]$. Qu'en deduit-on? + - Montrer que $u$ est un isomorphisme. + +Soit $(P_0,\ldots,P_n)$ une base orthonormée de vecteurs propres de $u$ associes aux valeurs propres $\lambda_0,\ldots,\lambda_n$. + - Montrer que, pour tout $(x,y)\in\R^2$, $(x+y)^n=\sum_{k=0}^n\lambda_kP_k(x)P_k(y)$. + - En déduire que $\mathrm{tr}(u)=\frac{2^n}{n+1}$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 905] +Soit $S=(s_{i,j})_{1\leq i,j\leq n}\in\mc{S}_n(\R)$. On pose $D=\mathrm{diag}(s_{1,1},\ldots,s_{n,n})$. On suppose $S$ et $D$ semblables. Montrer que $S=D$. Ind. Considérer la trace de $S^2$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 906] +Soit $A\in\mc{S}_n(\R)$. On dit que $A\in\mc{S}_n^{++}(\R)$ lorsque, pour toute matrice $X\in\M_{n,1}(\R)$ non nulle, $X^TAX\gt 0$. + - Déterminer une condition nécessaire et suffisante pour que $A\in\mc{S}_n^{++}(\R)$. + - Soit $A\in\mc{S}_n^{++}(\R):A=\begin{pmatrix}B&C\\ C^T&D\end{pmatrix}$. Montrer que $\det(B)\gt 0$, puis montrer que $\det(A)\leq\det(B)\det(D)$. +#+end_exercice + + +** Analyse + +#+begin_exercice [Mines PSI 2024 # 907] +Soient $E=\mc C^0([0,1],\R)$ et $\phi\in E$. On note, pour $f\in E$, $N_{\phi}(f)=\|f\phi\|_{\i}$. + - Montrer que $N_{\phi}$ est une norme si et seulement si $\phi^{-1}(\{0\})$ est d'interieur vide. - Montrer que $N_{\phi}$ et $\|\ \|_{\i}$ sont équivalentes si et seulement si $\phi^{-1}(\{0\})$ est vide. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 908] +Soient $E$ un $\R$ espace vectoriel, $N_1$ et $N_2$ deux normes sur $E$. + - Soit $(u_n)$ une suite qui converge dans $(E,N_1)$. On suppose que $N_1$ et $N_2$ sont équivalentes. Montrer que $(u_n)$ converge dans $(E,N_2)$. + - On suppose qu'une suite $(u_n)$ converge dans $(E,N_1)$ si et seulement si $(u_n)$ converge dans $(E,N_2)$. Montrer que $N_1$ et $N_2$ sont équivalentes. + - On prend $E=\R[X]$ et, pour $a\in\R$, $N_a(P)=|P(a)|+\int_0^1|P'(t)|\dt$. Montrer que, si $a,b\in[0,1]$, $N_a$ et $N_b$ sont équivalentes. + - Soit, pour $n\in\N$, $P_n=\dfrac{X^n}{2^n}$. Trouver les valeurs de $a$ telles que $(P_n)$ converge pour $N_a$ et déterminer alors la limite. + - En déduire que $N_a$ et $N_b$ ne sont pas équivalentes si $0\leq a\lt b$ et $b\gt 1$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 909] +Soit $(E,\|\ \|)$ un espace vectoriel norme. Une suite $(u_n)\in E^{\N}$ est de Cauchy si + + $\forall\eps\gt 0$, $\exists N\in\N$, $\forall n,m\geq N$, $\|u_n-u_m\|\leq\eps$. + - Montrer que toute suite convergente est de Cauchy. + - Dans $E=\R[X]$ muni de la norme $\left\|\sum a_kX^k\right\|=\max|a_k|$, montrer que la suite $(P_n)$ + +de terme general $P_n=1+\sum_{k=1}^n\dfrac{X^k}{k}$ est de Cauchy sans être convergente. + - Montrer que toute suite de Cauchy est bornée. + - Montrer que, si $(u_n)$ est de Cauchy et possede une suite extraite convergente, alors $(u_n)$ est convergente. + - On admet le theoreme de Bolzano-Weierstrass dans $\R$. Montrer que si $E$ est de dimension finie, alors la suite $(u_n)$ est convergente si et seulement si elle est de Cauchy. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 910] +Soit $E$ l'ensemble des applications lipschitzienne de $[0,1]$ dans $\R$. Pour $f\in E$, on note $K(f)=\inf\{k\in\R^+,\,f$ est $k-$lishtzienne}. + - Montrer que $E$ est un espace vectoriel. + - Montrer que, pour tout $f\in E$, $f$ est $K(f)$-lipschitzienne. + - Montrer que toute fonction polynomiale $P$ appartient à $E$ et déterminer $K(P)$. + - L'application $f\mapsto K(f)$ est-elle une norme sur $E$? + - Prouver que $\forall f\in E$, $\|f\|_{\i}\leq\inf_{x\in[0,1]}|f(x)|+K(f)$. + - L'application $f\mapsto\dfrac{K(f)}{\|f\|_{\i}}$ est-elle bornée sur $E\setminus\{0\}$? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 911] +Soit $f\,\colon\,(x,y)\in(\R^{+*})^2\mapsto x^2+y^2+\dfrac{3}{xy}$. La fonction $f$ est-elle prolongeable par continuité en $(0,0)$? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 912] +Soit $a\in\R$. Pour tout $n\in\N^*$, on définit $A_n=\begin{pmatrix}1&a/n\\ -a/n&1\end{pmatrix}$. + - Soient $\alpha\in\R$ et, pour tout $n\in\N^*$, $z_n=\Big(1+i\dfrac{\alpha}{n}\Big)^n$. Montrer que $z_n\ra e^{i\alpha}$. - Diagonaliser $A_n$ dans $\C$. + - Déterminer $\lim A_n^n$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 913] +Soit $(x_n)_{n\in\N^*}$ une suite de réels positifs et, pour $n\geq 1$, $y_n=\sqrt{x_1+\sqrt{x_2+\cdots+\sqrt{x_n}}}$. + - Étudier la convergence de la suite $(y_n)$ lorsque la suite $(x_n)$ est constante. + - Étudier la convergence de la suite $(y_n)$ lorsque $x_n=a\,b^{2^n}$ avec $a\gt 0$ et $b\gt 0$. + - Montrer que la suite $(y_n)$ converge si et seulement si la suite $\left(x_n^{1/2^n}\right)$ est bornée. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 914] +Pour $n\geq 2$, on s'interesse à l'équation $e^x-x^n=0$. + - Montrer que cette équation admet exactement deux solutions positives $u_n$ et $v_n$, avec $u_n\lt v_n$. + - Montrer que $(u_n)$ tend vers une limite $\ell$. + - Trouver un équivalent de $u_n-\ell$. + - Montrer que la suite $(v_n)$ diverge. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 915] +On définit la suite $(u_n)_{n\in\N}$ par : $u_{3n}=\frac{2}{\ln(n+3)}$ et $u_{3n+1}=u_{3n+2}=\frac{-1}{\ln(n+3)}$. + - Montrer que la série $\sum u_n$ est convergente et calculer sa somme. + - Soit $(a_n)_{n\in\N}$ une suite réelle telle que la série $\sum a_n$ converge. A-t-on nécessairement la convergence de la série $\sum a_n^2$? + - Montrer, pour tout entier $p\geq 2$, la divergence de la série $\sum u_n^p$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 916] +On donne $H_n=\sum_{k=1}^n\frac{1}{k}=\ln n+\gamma+o(1)$. + - On pose $u_k=\frac{(-1)^k}{k}$. Étudier la convergence et la somme de $\sum_{k\geq 1}u_k$. + - On donne $\sigma$ bijection de $\N^*$ avec + +\begin{tabular}{|c|c c c c c c c c c c c c|} $k$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $\ldots$ \\ \hline $\sigma(k)$ & $1$ & $3$ & $2$ & $5$ & $7$ & $4$ & $9$ & $11$ & $6$ & $13$ & $15$ & $\ldots$ \\ \end{tabular} + +Donner $\sigma(k)$. + - Déterminer la somme de la série $\sum_{k\geq 1}u_{\sigma(k)}$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 917] +Soit $(u_n)_{n\geq 1}$ une suite définie par $u_1\gt 0$ et, pour tout $n\in\N^*$, $u_{n+1}=\frac{u_n}{n}+\frac{1}{n^2}$. + - Étudier la convergence de la suite $(u_n)_{n\geq 1}$. + - Étudier la convergence de la série $\sum u_n$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 918] +Soit $f$ : $\R^{+*}\ra\R^{+*}$. + - à quelle condition nécessaire la série $\sum\frac{(-1)^k}{f(k)}$ est-elle convergente? Cette condition est-elle suffisante? On suppose par la suite que cette condition est vérifiée. - On suppose de plus que $f$ est croissante à partir d'un certain rang. + +On pose $u_n=\sum_{k=n}^{+\i}\frac{(-1)^k}{f(k)}$. Déterminer le signe de $u_n$ et la limite de la suite $(u_n)$. + - On suppose egalement que, pour tout $k$ assez grand, $\frac{1}{f(k)}+\frac{1}{f(k+2)}\geq\frac{2}{f(k+1)}$. + +Déterminer la nature de la série $\sum u_n$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 919] +Soit $f\colon\R^+\ra\R$ continue et surjective. Montr er que tout $y\in\R$ admet une infinite d'antecedents par $f$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 920] +Soit $f$ une application continue de $\R$ dans $\R$ telle que $f\circ f=2f-\mathrm{id}$. + - Montr er que $f$ est une bijection strictement croissante de $\R$ dans $\R$. + - On pose $f_0=f$ et, pour $n\in\N$, $f_{n+1}=f\circ f_n$. Montr er que $\left(\frac{1}{n}f_n\right)$ admet une limite, que l'on precisera. + - Déterminer $f$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 921] +Soient $f$ et $g$ deux fonctions continues sur $[a,b]$ ou $g$ est à valeurs dans $[0,1]$ et $f$ decroissante. On pose $c=\int_a^bg$. Montr er que $\int_{b-c}^bf\leq\int_a^bfg\leq\int_a^{a+c}f$. + +Ind. On pourra introduire une fonction d'une variable bien choisie. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 922] +Trouver les fonctions $f\in\mc C^1(\R,\R)$ telles que $\forall x\in\R,\,f(x)+\int_0^x(x-t)f(t)\dt=1$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 923] +Soit $\theta\in\R\setminus 2\pi\Z$. + - Soit $n\in\N^*$. Montr er que $\sum_{k=1}^n\frac{e^{ik\theta}}{k}=\int_0^1e^{i\theta}\frac{1-( te^{i\theta})^n}{1-te^{i\theta}}\,d\theta$. + - En déduire que $\sum\frac{e^{ik\theta}}{k}$ converge et que $\sum_{k=1}^{+\i}\frac{e^{ik\theta}}{k}=\int_0^1\frac{e^{i\theta}}{1- te^{i\theta}}\,d\theta$. + - En déduire que $\sum_{k=1}^{+\i}\frac{\sin(k\theta)}{k}=\frac{\pi-\theta}{2}$. + - Déterminer de même $\sum_{k=1}^{+\i}\frac{\cos(k\theta)}{k}$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 924] +Calculer $\int_0^{+\i}\lfloor x\rfloor e^{-x}dx$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 925] +Soit, pour $n\in\N$ et $x\in\R$, $I_n(x)=\int_0^{\pi}\frac{\cos(nt)-\cos(nx)}{\cos(t)-\cos(x)}dt$. + - Montr er que $I_n(x)$ est bien définie. + - Calculer $I_{n+1}(x)+I_{n-1}(x)$ et trouver une relation de récurrence. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 926] + - Justifier que $I=\int_0^{+\i}\bigg{[}\frac{1}{\sqrt{x}}\bigg{]}{\rm d}x$ converge. +#+end_exercice + + - Calculer explicitement $I$ en admettant que $\sum_{k=1}^{+\i}\frac{1}{k^2}=\frac{\pi^2}{6}$. +#+begin_exercice [Mines PSI 2024 # 927] + +#+end_exercice + +Soit $f\,\colon\,\R\,\ra\R^{+*}$ continue par morceaux telle que $\frac{f(x+1)}{f(x)}\underset{x\ra+\i}{\longrightarrow}\ell\in[0,1[$. Étudier l'intégrabilité de $f$ sur $\R^+$. +#+begin_exercice [Mines PSI 2024 # 928] +On définit $f$ sur $\R^+$ par $f(x)=2x^7+x$. + - Montrer que $f$ realise une bijection de $\R^+$ sur $\R^+$. + - La fonction $F\,\colon\,x\in\R^+\mapsto\sin(2x^7+x)$ est-elle intégrable en $+\i$? +#+end_exercice + + - L'intégrale $\int_0^{+\i}F(x)\,{\rm d}x$ est-elle convergente? +#+begin_exercice [Mines PSI 2024 # 929] +Soit $f\,\colon\,\R\,\ra\R$ continue et $T$-periodique. On se propose de prouver l'existence d'un unique $\lambda\in\R$ tel que l'intégrale $\int_1^{+\i}\frac{\lambda-f(t)}{t}\,{\rm d}t$ converge. + - Étudier le cas particulier ou $f=\sin$. +#+end_exercice + + - Traiter le cas general. +#+begin_exercice [Mines PSI 2024 # 930] + +#+end_exercice + +Soit $f\,\colon\,x\mapsto\,\int_x^{+\i}e^{-t^2}{\rm d}t$. Déterminer la limite puis un équivalent de $f(x)$ quand $x$ tend vers $+\i$. +#+begin_exercice [Mines PSI 2024 # 931] +Soit $f\,:[0,1]\ra\R$ une fonction de classe $\mc C^1$ telle que $f(0)=f(1)=0$. + - Soient $I_1=\int_0^1f(x)f'(x)\,{\rm cotan}(\pi x)\,{\rm d}x$ et $I_2=\int_0^1f^2(x)(1+{\rm cotan}^2(\pi x))\,{\rm d}x$. Montrer que $I_1$ et $I_2$ sont convergentes et exprimer $I_1$ en fonction de $I_2$. +#+end_exercice + + - En déduire que $\int_0^1f^2\leq\frac{1}{\pi^2}\int_0^1(f^{ '})^2$. +#+begin_exercice [Mines PSI 2024 # 932] +Soit la suite de fonctions définies par $f_n\,:x\mapsto\frac{x^n}{n!}e^{-x}$. + - Étudier la convergence simple de la suite $(f_n)$. + - Étudier la convergence uniforme de la suite $(f_n)$. +#+end_exercice + + - Calculer $\int_0^{+\i}f_n$ puis sa limite lorsque $n$ tend vers $+\i$. Est-ce coherent avec les theoremes du cours? +#+begin_exercice [Mines PSI 2024 # 933] +Étudier la convergence simple et la convergence uniforme de la suite de fonctions $(f_n)_{n\geq 0}$ définie sur $\R^{+*}$ par $\forall x\gt 0,\,f_0(x)=x$ et $\forall n\in\N,\,f_{n+1}(x)=\frac{1}{2}\left(f_n(x)+\frac{x}{f_n( x)}\right)$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 934] +Soit $f:x\mapsto\sum_{n=2}^{+\i}\frac{xe^{-nx}}{\ln(n)}$. + - Trouver les domaines de définition/continuité/dérivabilité de $f$. + - Trouver la limite de $f$ en $+\i$ puis un équivalent. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 935] +Pour $a\in\R$, on considére la suite de fonctions définie par $f_0=1$ et, pour $n\in\N^*$, $f_n:x\mapsto e^{-n^a}e^{inx}$. + - Pour quelles valeurs de $a$, la série $\sum f_n$ converge-t-elle simplement sur $\R$? + +On suppose cette condition remplie dans la suite. On pose $S=\sum_{n=0}^{+\i}f_n$. + - Montr'er que $S$ est de classe $\mc C^{\i}$ sur $\R$ et calculer $f^{(k)}(0)$ pour $k\in\N$. + - En utilisant le theoreme de Fubini, montrer que $S$ est développable en série entiere au voisinage de $0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 936] + - Déterminer le rayon de convergence $R$ de la série entiere $\sum\frac{x^n}{n^2}$. + - Montr'er que pour tout $x\in[0,R[,\sum_{n\geq 1}\frac{x^n}{n^2}=x\int_0^1\frac{\ln(t)}{xt-1} \dt$. + - Que se passe-t-il pour $x=1$? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 937] +Soit $f\ :x\mapsto\sum_{n\geq 0}\binom{2n}{n}x^n$. + - Déterminer le rayon de convergence de $f$. + - Quel est le domaine de définition de $f$? La fonction $f$ est-elle dérivable? Si oui, déterminer sa derivée. + - Déterminer une équation différentielle d'ordre $1$ vérifiée par $f$. + - Que vaut $\sum_{n\geq 0}\binom{2n}{n}\frac{(-1)^n}{4^n}$? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 938] +Soient $f:x\mapsto\sum_{n=0}^{+\i}\frac{x^n}{(n+1)!}$ et $F:x\mapsto\int_0^xe^{-t}f(t)\dt$. + - Déterminer le rayon de convergence de $f$ et exprimer $f$ à l'aide de fonctions usuelles. + - Montr'er que $F$ est définie et dérivable sur $\R$. Que vaut $F'$? + - Montr'er que $F$ est développable en série entiere et déterminer ce développement. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 939] +On cherche à déterminer le cardinal $m_n$ de l'ensemble $M_n$ forme des $n$-uplets $(a - {1\leq i\leq n}$ + +tels que : (i) $\forall i,\,a_i\in\{-1,0,1\}$, (ii) $\sum_{i=1}^na_i=0$, (iii) $\forall p\in\db{1,n},\,\sum_{i=1}^pa_i\geq 0$. + +On pose $m_0=1$. + - Calculer $m_1$, $m_2$ et $m_3$. + - Soit $n\in\N^*$. Soit $(a - {1\leq i\leq n}\in M_n$ tel que $a_1=1$.Montrer qu'il existe $r\in[0,n-2]$, $(b_1,\ldots,b_r)\in M_r$, $(c_1,\ldots,c_{n-r-2})\in M_{n-r-2}$ tels que $(a_1,\ldots,a_n)=(1,b_1,\ldots,b_r,-1,c_1,\ldots,c_{n-r-2})$ et justifier l'unicité de cette décomposition. + - En déduire une formule de récurrence sur les $m_1,\ldots,m_n$. + - Soit $:x\mapsto\sum_{n=0}^{+\i}m_nx^n$. Montrer que le rayon de convergence de cette série entiere est $\gt 0$ et déterminer $f$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 940] +Soit $g\,\colon\,x\mapsto\frac{1}{\cos x}$. + - Montrer que $g$ est développable en série entiere au voisinage de $0$. + - Donner un encadrement du rayon de convergence. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 941] +Soit $f:x\mapsto\sum_{n=0}^{+\i}\frac{(-1)^n}{(2^nn!)^2}x^{2n}$. + - Trouver l'ensemble de définition de $f$. + - Trouver une équation différentielle vérifiée par $f$. + - Calculer $\int_0^{+\i}f(t)e^{-xt}dt$ pour $x\in]1,+\i[$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 942] +On pose $f(x,s)=\sum_{n=1}^{+\i}\frac{x^n}{n^s}$. + - Calculer $f(x,0)$ et $f(x,1)$ lorsque cela est possible. + - Donner le rayon de convergence de $x\mapsto f(x,s)$. + - Déterminer l'ensemble de définition de $x\mapsto f(x,s)$, en discutant selon les valeurs de $s$. + - Déterminer une relation entre $f(x,s)$ et $f(x,s-1)$. En déduire $f(x,-1)$ et $f(x,-2)$. + - Soit $p\in\N$. Déterminer un équivalent de $f(x,-p)$ lorsque $x\ra 1^-$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 943] +On définit la suite $(u_n)$ par : $u_0=1$ et, pour $n\in\N^*$, $u_n=\sqrt{n+u_{n-1}}$. + - Montrer que, pour tout $n\in\N$, on a : $\sqrt{n}\leq u_n\leq 2\sqrt{n+1}$. + - Montrer que $u_n\sim\sqrt{n}$ et déterminer la limite de $(u_n-\sqrt{n})$. + - Donner le rayon de convergence $R$ de la série entiere $\sum u_nx^n$. + - Calculer $\lim_{x\ra R^-}\sum_{n=0}^{+\i}u_nx^n$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 944] +Soit $a_n\,=\,\int_0^1\left(\frac{1+t^2}{2}\right)^ndt$. On pose $f\,\colon\,x\mapsto\sum_{n=0}^{+\i}a_nx^n$ et l'on note $R$ le rayon de convergence de cette série entiere. + - Montrer que : $\forall n\in\N,\ \frac{1}{2^n}\leq a_n\leq 1\,$. En déduire un encadrement de $R$. + - Montrer que $\forall n\in\N,\ (2n+3)a_{n+1}=1+(n+1)a_n$. + - En déduire que : $\forall x\in]-R,R[,\ (2x-x^2)f'(x)+(1-x)f(x)=\sum_{n=0}^{+\i}x^n$. - Trouver ainsi une expression de $f(x)$ pour $x\in]-1,1[$. + - Trouver une autre expression de $f(x)$ en montrant que : + +$$\forall x\in]-1,1[,\ f(x)=\int_0^1\sum_{n=0}^{+\i}\left(\frac{(1+t^2)x }{2}\right)^ndt=\int_0^1\frac{1}{1-\frac{(1+t^2)x}{2}} dt\text{ et en calculant cette intégrale.}$$ +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 945] +Soit $\alpha\in\R^{+*}$_: + - Montrer que $\int_0^{+\i}\sin(t)\,e^{-\alpha t}\dt\text{ et }\int_0^{+\i}|\sin(t)|\,e^{-\alpha t}\dt$ convergent et déterminer leur valeur. + - Montrer que $\int_0^{+\i}\frac{\sin(t)}{\mathrm{sh}(t)}\dt$ converge. + - Montrer que $\int_0^{+\i}\frac{\sin(t)}{\mathrm{sh}(t)}\dt=\sum_{n \geq 0}\frac{2}{1+(2n+1)^2}$. + - Adapter les questions précédentes pour déterminer $\int_0^{+\i}\frac{\sin(t)}{\mathrm{ch}(t)}\dt$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 946] +Soient $u_n=\int_1^{+\i}e^{-x^n}\dx\text{ et }I=\int_1^{+\i}\frac{e^{-t}}{t}\dt$. + - Montrer que $u_n$ est bien défini pour tout $n\geq 1$. + - Montrer que $I$ est bien définie. + - Déterminer la nature de $\sum u_n$._Ind._ Effectuer un changement de variable. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 947] +Soient $f\in\mc C^0([0,1],\R)$ et, pour $n\in\N$, $I_n=\int_0^1f(t^n)dt$. Limite de $(I_n)\,$? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 948] + - Soient $a$ et $b$ deux réels $\gt 0$. Montrer que $\int_0^1\frac{t^{a-1}}{1+t^b}\dt=\sum_{n=0}^{+\i}\frac{(-1)^n}{a+bn}$. + - Calculer $\sum_{n=0}^{+\i}\frac{(-1)^n}{1+3n}\text{ et }\sum_{n=0}^{+ \i}\frac{(-1)^n}{1+4n}$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 949] +Soit $f:x\mapsto\int_0^{+\i}\frac{\arctan(tx)}{t(1+t^2)}\dt$. + - Montrer que $\colon\forall u\in\R$, $|\arctan(u)|\leq|u|$. + - Montrer que $f$ est de classe $\mc C^1$ sur $\R$. + - Déterminer le développement en éléments simples de $t\mapsto\frac{1}{1+x^2t^2(1+t^2)}\text{ pour }|x|\neq 1$. + - Montrer que $f(x)=\frac{\pi}{2(x+1)}$ pour $x\gt 0$. En déduire la valeur de $f$ sur $\R$. + - Déterminer $\int_0^{+\i}\left(\frac{\arctan(t)}{t}\right)^2\dt$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 950] +Soit $f\,\colon\,\alpha\mapsto\int_0^{+\i}\frac{dt}{t^{\alpha}(t+1)}$. + - Déterminer le domaine de définition $D$ de $f$. + - Montr e que $f$ est continue sur $D$. + - Montr e que la courbe representative de $f$ admet la droite $x=1/2$ pour axe de symétrie. + - Justifier l'existence d'une borne inférieure pour $f$; la déterminer. + - Déterminer un équivalent de $f$ en $0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 951] +Soit $f:x\mapsto\int_0^{+\i}\arctan(xt)\,e^{-t}\dt$. + - Montr e que $f$ est définie et de classe $\mc C^1$ sur $\R$. + - On définit la suite $(u_n)$ par $u_0\in\R^{+*}$ et $\forall n\in\N$, $u_{n+1}=f(u_n)$. Montr e que la suite $(u_n)$ possede une limite et la déterminer. + - Trouver un équivalent de $u_n$ en $+\i$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 952] +Soit $f:x\mapsto\int_0^{+\i}\frac{1}{x+e^t}dt$. + - Montr e que $f$ est définie au moins sur un intervalle de la forme $]-\alpha,\alpha[$ avec $\alpha\gt 0$. + - Montr e que $f$ est développable en série entiere au voisinage de $0$. + - Calculer ce développement et en déduire une expression $f(x)$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 953] +Soit $f:x\mapsto\int_0^xe^{-t^2}dt$. + - Montr e que $f$ est de classe $\mc C^1$ sur $\R$ et donner $f'$. + - Soit $g:x\mapsto e^{x^2}f(x)$. Montr e que $g$ est solution de $(E):y'-2xy=1$ avec $y(0)=0$. + - Déterminer les solutions de $(E)$ développables en série entiere et preciser le rayon. + - La fonction $g$ est-elle développable en série entiere? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 954] +Soit $\Gamma:x\mapsto\int_0^{+\i}t^{x-1}e^{-t}\dt$. + - Montr e que $\Gamma$ est définie sur $]0,+\i[$ et qu'elle est de classe $\mc C^2$. Montr e plus que $\Gamma(x)\gt 0$ pour tout $x\gt 0$. + - Étudier la convexite de $\Gamma$ et celle de $\ln\circ\Gamma$. + - Pour tout $x\gt 0$, etablir $\colon\lim\limits_{n\ra+\i}\int_0^nt^{x-1}(1-t/n)^n\dt= \Gamma(x)$. + - Exprimer $\int_0^nt^{x-1}(1-t/n)^n\dt$ en fonction de $\int_0^1u^{x-1}(1-u)^n\,du$. + - Montr e que la suite de fonctions $f_n:x\in\R^{+*}\mapsto\frac{n^xn!}{x(x+1)\ldots(x+n)}$ converge simplement vers $\Gamma$. Ind. Procéed par intégrations par parties successives. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 955] +On admet que $\int_0^{+\i}e^{-t^2}dt=\frac{\sqrt{\pi}}{2}$. On pose $f:x\mapsto\int_0^{+\i}\cos(2xt)e^{-t^2}dt$. + - Montr e que $F$ est définie et de classe $\mc C^1$ sur $\R$. + - Trouver une relation entre $f$ et $f'$. - En déduire une expression simple de $f(x)$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 956] +Soit $F\colon\ x\mapsto\int_0^{+\i}\frac{t^3}{\sqrt{1+t^4}}e^{-xt}{\rm d}t\,$. + - Déterminer le domaine de définition $I$ de $F$. + +Montrer que $F$ est de classe $\mc C^1$ sur $I$ et donner son sens de variation. + - Déterminer les limites de $F$ aux bornes de $I$. + - Calculer $G(x)=\int_0^{+\i}t^3e^{-xt}{\rm d}t$ pour $x\gt 0$. + - Montrer que $F(x)\underset{x\ra+\i}{\sim}\frac{6}{x^4}$. + +_Ind._ On pourra étudier $\mid F-G\mid$ et utiliser la relation de Chasles. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 957] +On pose $f:x\mapsto\int_0^1\frac{{\rm e}^{-(t^2+1)x^2}}{t^2+1}{\rm d}t$ et $g:x\mapsto\int_0^x{\rm e}^{-t^2}{\rm d}t$. + - Montrer que $f$ est définie sur $\R$ et qu'elle est paire. Que vaut $f(0)$? + - Montrer que $f$ est de classe $\mc C^1$ sur $\R$ et donner l'expression de $f'(x)$. + - Montrer que $g$ est définie et de classe $\mc C^1$ sur $\R$. + - à l'aide d'un changement de variable affine, montrer que : $\forall x\in\R$, $f'(x)=-2g'(x)g(x)$. + - Montrer que : $\forall x\in\R$, $f(x)=\frac{\pi}{4}-g(x)^2$. + - En déduire la limite de $g$ en $+\i$ puis conclure que $\int\limits_0^{+\i}{\rm e}^{-t^2}{\rm d}t=\frac{\sqrt{\pi}}{2}$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 958] +Soit $f\colon\R\ra\R$ une fonction de classe $\mc C^{\i}$ telle que $f(0)=0$. Soit $g:x\mapsto\frac{f(x)}{x}$. à l'aide de la formule de Taylor avec reste intégral, montrer que $g$ se prolonge en une fonction de classe $\mc C^{\i}$ sur $\R$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 959] +Soit $(E)$ l'équation différentielle : $x^2y'(x)+y(x)=x^2$. + - Montrer que $(E)$ n'admet pas de solution développable en série entiere. + - Résoudre l'équation différentielle sur $]0,+\i[$. + - Montrer qu'il existe une unique solution tendant vers $0$ en $0^+$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 960] +Soit $f:x\mapsto\int_0^{+\i}\frac{e^{-t}e^{-x/t}}{\sqrt{t}}{\rm d}t$. + - Montrer que $f$ est définie sur $\R^+$. + - Montrer que $f$ est de classe $\mc C^2$ et solution de l'équation différentielle $2xy''+y'-2y=0$. + - Résoudre l'équation en posant $y(x)=z(\sqrt{x})$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 961] +On s'interesse aux solutions $f:x\mapsto\sum_{n\geq 0}a_nx^n$ de l'équation différentielle + + $(E):x^2y''+4xy'+(2-x^2)y=1$. + - Montrer que $a_0=1/2,\,a_1=0$ et $\forall n\geq 2,\,a_n=\frac{a_{n-2}}{(n+1)(n+2)}$. - En déduire l'unicité de $f$. + - Déterminer les $a_n$, le rayon de convergence de $f$ puis exprimer $f$ à l'aide de fonctions usuelles. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 962] +On note $(E)$ l'équation différentielle $x(1-x)y''+(1-3x)y'-y=0$. + - Déterminer les solutions de $(E)$ non nulles développables en série entiere. Preciser le rayon de convergence. + - Déterminer l'ensemble des solutions de $(E)$ sur un intervalle raisonnable. + - Les raccorder entre elles. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 963] +On note $(E)$ l'équation différentielle $x^2y''-2xy'+2y=2(1+x)$. + - Trouver les solutions de l'équation homogène associée de la forme $x\mapsto x^{\alpha}$, ou $\alpha\in\R$. + - Trouver une solution particuliere de $(E)$, d'abord sur $]0,+\i[$, puis sur $]-\i,0[$. + +Ind. On la cherchera sous la forme $x\alpha(x)+x^2\beta(x)$, ou $\alpha$ et $\beta$ sont des fonctions de classe $\mc C^1$ + +telles que $x\alpha'(x)+x^2\beta'(x)=0$. + - L'équation $(E)$ admet-elle des solutions sur $\R$? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 964] +Pour $(a,b,c)\in\R^3$, on définit $f_{a,b,c}:t\in\R\mapsto\left(\begin{array}{c}be^t+ce^{-t}\\ 2a-be^t\\ a+ce^{-t}\end{array}\right)\in\R^3$. + +Soit $F=\left\{f_{a,b,c},\ (a,b,c)\in\R^3\right\}$. + - Montr e que $F$ est un espace vectoriel, en donner la dimension et une base. + - Trouver $M\in\M_3(\R)$ telle que $\colon\forall f\in F,\forall t\in\R,\ f'(t)=Mf(t)$. + - La matrice $M$ est-elle inversible? + - Quelles sont les valeurs propres de $M$? Pouvait-on s'y attendre? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 965] + - Soit $\alpha\in\R$. à l'aide d'un changement de variables classique, résoudre l'équation $x\frac{\partial f}{\partial x}(x,y)+y\frac{\partial f}{\partial y}(x,y)=\alpha f (x,y)$ d'inconnue $f\in\mc C^1(\R^{+*}\times\R^{+*},\R)$. + - Résoudre $x\frac{\partial f}{\partial x}(x,y)+y\frac{\partial f}{\partial y}(x,y)=\sqrt{ x^2+y^2}f(x,y)$ d'inconnue + + $f\in\mc C^1(\R^{+*}\times\R^{+*},\R)$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 966] +Soit $J:x\mapsto\int_0^{\pi}\cos(x\sin(\theta))\,d\theta$. + - Montr e que $J$ est bien définie et de classe $\mc C^2$ sur $\R$. + - Montr e que $J$ est développable en série entiere et déterminer le rayon de convergence. + - Montr e que $xJ''(x)+J'(x)+J(x)=0$. + - Soit $(x,y)\mapsto\phi(x,y)=J\left(\sqrt{x^2+y^2}\right)$. Montr e que $\phi$ est de classe $\mc C^2$ sur $\R^2\setminus\{(0,0)\}$ et que $\Delta\phi+\phi=0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 967] +On pose $f(x,y)=\frac{1}{1-y^2}\ln\left(\frac{x+y}{1+xy}\right)$. On note $\Omega$ l'ensemble de définition de $f$. + - Representer $\Omega$ et montrer que c'est un ouvert. + - Monter que $f$ est de classe $\mc C^1$ sur $\Omega$. - Comparer $f(1/x,y)$ et $f(x,y)$. Donner une interpretation geometrique pour $x\gt 0$ et $y\in]0,1[$. + - Montrer que $f$ vérifie $:2yf+(1-x^2)\frac{\partial f}{\partial x}-(1-y^2)\frac{\partial f}{ \partial y}=0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 968] + - Résoudre $(1-t^2)y''-2ty'=0$ sur $I=]-1,1[$. + - Soit $f$ de classe $\mc C^2$ sur $I$ à valeurs dans $\R$. On pose $g(x,y)=f\left(\frac{\cos(2x)}{\mathrm{ch}(2y)}\right)$. + +Déterminer l'ensemble des fonctions $f$ telles que $g$ soit non constante et de laplacien nul, c'est-a-dire telles que $\frac{\partial^2g}{\partial x^2}(x,y)+\frac{\partial^2g}{ \partial y^2}(x,y)=0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 969] +On munit $\R^n$ de sa structure euclidienne canonique. Soit $\rho:x\mapsto\|x\|^2$. + - Montrer que $\rho\in\mc C^2(\R^n,\R)$. + - Soient $g\in\mc C^2(\R^{+*},\R)$ et $f\ \colon\R^n\setminus\{0\}\ra\R$ définie par $x\mapsto f(x)=g(\|x\|^2)$. + +Déterminer les fonctions $g$ vérifiant $\Delta f=0$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 970] +Soit $D=\{(x,y)\in\R^2\,;\,x\geq 0,\,y\geq 0,\ x+y\leq 1\}$. Soient $a,b,c$ des réels $\gt 0$ et $f\ :D\ra\R$ la fonction définie par $(x,y)\mapsto x^ay^b(1-x-y)^c$. Montrer l'existence d'extrema locaux pour $f$ et les déterminer. +#+end_exercice + + +** Probabilités + +#+begin_exercice [Mines PSI 2024 # 971] +On considére une classe de PSI constituée de $N$ eleves, dont $n$ provenant de PCSI et $N-n$ de MPSI. On envoie successivement au tableau des eleves choisis au hasard. Un eleve peut passer plusieurs fois au tableau. + - Quelle est la probabilité qu'au cours des $n$ premiers passages, il n'y ait que des eleves de PCSI? + - Quelle est la probabilité qu'au cours des $n+5$ premiers passages, il y ait $n$ eleves de PCSI? + - Soit $i\in\N^*$. On note $X_i$ la variable aléatoire qui compte le nombre de tirages nécessaires pour faire passer $i$ eleves de PCSI distincts au tableau. Déterminer la loi de $X_i$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 972] +On considére initialement une urne contenant une boule blanche et une boule rouge. On tire une boule, on note sa couleur, on la remet dans l'urne et on rajoute deux boules de la même couleur que celle trée. On repete indéfiniment le processus. + - Calculer la probabilité de ne tirer que des boules rouges lors des $n$ premiers tirages? + - Calculer la probabilité de tirer indéfiniment uniquement des boules rouges? + - Calculer la probabilité de tirer une boule blanche au 42-ieme tirage. + - Le résultat de la question - reste-t-il vrai si on rajoute 3 boules (au lieu de 2)? 4 boules? +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 973] + - Calculer $\int_0^1x^p(1-x)^qdx$ avec $p,q\in\N$. + - On dispose de $p$ unres contenant chacune $p$ boules. Pour $i\in\db{1,p}$, l'urne $i$ contient $i$ boules noires et $p-i$ blanches. On choisit une des urnes aléatoirement et on en tire successivement des boules avec remise. On note $A_{n,p}$ l'evenement : on tire $2n$ boules et on a autant de boules noires que de boules blanches. + - Exprimer $P(A_{n,p})$ sous forme d'une somme. + - Déterminer la limite de $\mathbf{P}(A_{n,p})$ quand $n$ tend vers $+\i$. + - Déterminer la limite de $\mathbf{P}(A_{n,p})$ quand $p$ tend vers $+\i$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 974] +On considére des lancers indépendants avec la probabilité $p\in]0,1[$ d'avoir pile. On pose par convention $T_0=0$ et pour $r\in\N^*$, $T_r$ est la variable aléatoire qui compte le nombre de lancers nécessaires pour avoir $r$ piles. On pose $Z_r=T_r-T_{r-1}$ pour $r\in\N^*$. + - Déterminer la loi de $Z_r$. + - Déterminer la fonction generatrice de $T_r$. + - Pour tout $x\in]0,1[$, calculer $\sum_{k=r}^{+\i}\binom{k}{r}x^{k-r}$ et en déduire la loi de $T_r$. + - Calculer $\mathbf{E}(T_r)$ de deux facons différentes. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 975] +Soient $s\gt 1$ et $\zeta(s)=\sum_{k=1}^{+\i}\frac{1}{k^s}$. Soit $X$ une variable aléatoire à valeurs dans $\N^*$ telle que $\forall n\in\N^*,\mathbf{P}(X=n)=\frac{1}{\zeta(s)} \frac{1}{n^s}$. + - Soit $n\in\N^*$. Calculer $\mathbf{P}(n$ divise $X)$. + - Soit $p$ un nombre premier et $v_p(k)=\max\{i\in\N,p^i$ divise $k\}$ pour tout $k\in\N^*$. + +Déterminer la loi de $v_p(X)$ puis son esperance. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 976] +On effectue des lancers avec une piece dont la probabilité de donner pile est $p\in]0,1[$. On lance la piece jusqu'a obtenir pile pour la deuxieme fois. On note $X$ le nombre de faces obtenues au cours de l'experience. + - Donner la loi de $X$. + - Montrer que $\mathbf{E}(X)\lt +\i$ et la calculer. + - On prend une urne et, si $X=n$, on pose $n+1$ boules numerotées de $0$ à $n$ dans l'une. Donner la loi de $Y$ ou $Y$ est le numero de la boule tirée dans l'urne. Calculer ensuite l'esprance de $Y$ ainsi que sa variance. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 977] + - Soit $S:t\mapsto\sum_{n=0}^{+\i}\frac{n^2+n+1}{n!}t^n$. Déterminer le rayon de convergence et donner une expression de $S$. + - Soit $X$ une variable aléatoire à valeurs dans $\N$ de fonction generatrice $G_X=\lambda S$. Déterminer $\lambda$ et la loi de $X$. + - Calculer $\mathbf{E}(X)$ et $\mathbf{V}(X)$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 978] +Soient $n\in\N$ et $p\in]0,1[$. On considére une variable aléatoire $X$ telle que $X(\Omega)\subset\N$ et $\forall k\in\N,\mathbf{P}(X=k)=a\binom{n+k}{k}p^k$. + - Quelle est la valeur de $a$? + - Déterminer $\mathbf{E}(X)$ et $\mathbf{V}(X)$ si elles existent. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 979] +Soit $(X_k)$ une suite de variables aléatoires i.i.d. suivant la loi de Bernoulli de paramêtre $2/3$. On pose $A_k=(X_{2k-1}X_{2k}=0)$, $B_p=\bigcap_{k=0}^pA_k$. + +Soit $T=\min\{k\geq 2,X_{k-1}=X_k=1\}\in\N\cup\{+\i\}$. + - Montrer que $\mathbf{P}\left(\bigcap_{k=0}^{+\i}A_k\right)=1$ et en déduire que $\mathbf{P}(T\in\N)=1$. + - Etablir une relation de récurrence lineaire d'ordre deux vérifiée par $(\mathbf{P}(T=n))$. + - Calculer l'esperance de $T$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 980] +Soient $X$ et $Y$ deux variables aléatoires indépendantes de lois respectives $\mc{G}(p)$ et $\mc{G}(q)$, ou $p$ et $q$ sont éléments de $]0,1[$. On pose $U=\dfrac{X}{Y}$. + - Donner la loi de $U$. + - Calculer l'esperance de $U$. + - Si $p=q$, montrer que $\mathbf{E}(U)\gt 1$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 981] +Soient $X_1,\ldots,X_n$ des variables aléatoires i.i.d. de loi $\mc{B}(p)$. On note $U$ la matrice ligne $\begin{pmatrix}X_1&\cdots&X_n\end{pmatrix}$ et $M=U^TU$. + - Déterminer les lois de $\op{rg}(M)$ et $\op{Tr}(M)$. + - Déterminer la probabilité que $M$ soit une matrice de projecteur. + - Dans cette question, on prend $n=2$. On note $V$ la matrice ligne $\begin{pmatrix}1&1\end{pmatrix}$ et $X=VMV^T$. + +Déterminer l'esperance et la variance de $X$. +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 982] +Soient $a,b\gt 0$, $X,Y,Z$ des variables aléatoires indépendantes telles que $X\sim\mc{P}(a)$, $Y\sim\mc{P}(b)$, $\mathbf{P}(Z=1)=1-p$ et $\mathbf{P}(Z=-1)=p$. + +Quelle est la probabilité que la matrice $A=\begin{pmatrix}X&Y\\ YZ&X\end{pmatrix}$ soit diagonalisable +#+end_exercice + + +#+begin_exercice [Mines PSI 2024 # 983] +Soit $(X_n)_{n\geq 1}$ une suite de variables aléatoires i.i.d. suivant la loi uniforme sur $\{-1,1\}$. On définit $(S_n)_{n\geq 0}$ par $S_0=0$ et $\forall n\in\N^*$, $S_n=S_{n-1}+X_n$. + - Déterminer la loi de $\dfrac{S_n+n}{2}$. En déduire $\mathbf{E}(S_n)$ et $\mathbf{V}(S_n)$. + - On pose $A_n=|S_n|$. + - Déterminer $A_n(\Omega)$. + - Pour tout $n\in\N$, etablir : $\mathbf{E}(A_{n+1})=\mathbf{E}(A_n)+\mathbf{P}(S_n=0)$. + +Ind. Exprimer $\mathbf{E}(A_{n+1})$ et appliquer la formule des probabilités totales à $X_{n+1}$. + - En déduire pour tout $n\in\N^*$ : $\mathbf{E}(A_{2n})=\mathbf{E}(A_{2n-1})=\sum_{k=0}^{n-1}\begin{pmatrix}2k\\ k\end{pmatrix}\begin{pmatrix}1\\ 4\end{pmatrix}^k$. +#+end_exercice + + +* Mines - Ponts - PC :autre: + +** Algèbre + +#+begin_exercice [Mines PC 2024 # 984] +Soient $A$ un ensemble de réels de cardinal $n\geq 2$ et $B=\{a+a',\;(a,a')\in A^2\}$. + - Montrer que $2n-1\leq\op{Card}B\leq\dfrac{n(n+1)}{2}$. + - Donner des exemples de parties pour lesquelles les bornes sont atteintes. + - Generaliser à $B_k=\{a_1+a_2+\cdots+a_k\;;\;a_1,...,a_k\in A\}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 985] +Trouver tous les polynômes $P\in\C[X]$ tels que $(X+4)P(X)=XP(X+1)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 986] +Déterminer les polynômes réels $P$ vérifiant $P(X)P(X+1)=P(X^2)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 987] + - Soit $P\in\Z[X]$ unitaire. Montrer que ses racines rationnelles sont dans $\Z$. + - Pour $n\in\N^*$, montrer qu'il existe un polynôme unitaire $P_n\in\Z[X]$ de degre $n$ tel que, pour tout $\theta\in\R$, on ait $P_n(2\cos\theta)=2\cos(n\theta)$. + - Montrer que $\cos(\pi\Q)\cap\Q=\biggl{\{}-1,-\dfrac{1}{2},0,\dfrac{1}{2},1 \biggr{\}}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 988] +Soit $P\in\R[X]$ unitaire de degre $n$. Calculer $\sum_{k=0}^n\dfrac{P(k)}{\prod_{i\neq k}(k-i)}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 989] +Soit $n\in\N$, $n\geq 2$. On note $(*)\;(1+iX)^{2n+1}-(1-iX)^{2n+1}=2iXQ_n\,(X)$. + - Montrer qu'il existe un unique $Q_n\in\R\,[X]$ vérifiant $(*)$. Donner le degre et le coefficient dominant de $Q_n$. + - Déterminer les racines de $Q_n$. + - Calculer $\prod_{k=0}^{n-1}\bigg(4+\tan^2\bigg(\dfrac{k\pi}{2n+1}\bigg) \bigg)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 990] +Soient $n\in\N^*$ et $P\in\R\,[X]$ tel que $\forall x\in\R$, $P(x)\geq 0$. On pose $Q=P+P'+\cdots+P^{(n)}$. + - Montrer que $Q$ est minore sur $\R$. + - Montrer que $Q$ est positif sur $\R$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 991] +L'union de deux sous-espaces vectoriels est-elle un sous-espace vectoriel? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 992] +Soit $n\in\N^*$. Trouver toutes les matrices $A\in\M_2(\C)$ telles que $A^n=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 993] +Soit $n\in\N^*$. Soit $E=\{S_1,\ldots,S_k\}$ l'ensemble des parties non vides de $\{1,\ldots,n\}$. Soit $A\in\M_k(\R)$ définie par $a_{i,j}=\left\{\begin{array}{cc}1&\text{si }S_i\cap S_j\neq\emptyset\\ 0&\text{si }S_i\cap S_j=\emptyset\end{array}.$.Déterminer le rang de $A$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 994] + - Soient $A,B\in\M_{n,p}(\R)$. Montrer que $|\mathrm{rg}A-\mathrm{rg}B|\leq\mathrm{rg}(A+B)\leq\mathrm{rg}A+ \mathrm{rg}B$. + - Soit $(v_1,...,v_k)\in(\R^n)^k$ tel que $\sum_{i=1}^kv_i(v_i)^T=I_n$. Montrer que $k\geq n$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 995] +Soit $(A,B)\in\M_n\left(\R\right)^2$ telles que $:A^2=A$, $B^2=B$ et $AB=BA$. Montrer que $\det\left(A-B\right)\in\left\{-1,0,1\right\}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 996] + - Four $A\in\M_n(\R)$, on définit $f_A:M\mapsto\op{tr}(AM)$. Montrer que l'application $f\colon\M_n(\R)\ra\mc{L}(\M_n(\R),\R),\ A\mapsto f_A$ est un isomorphisme. + - Soit $g\in\mc{L}(\M_n(\R),\R)$ telle que $\forall(A,B)\in\M_n(\R)^2,g(AB)=g(BA)$. Montrer que $g$ est proportionnelle à la trace. + - Soit $h$ un endomorphisme de $\M_n(\R)$ tel que $\forall(A,B)\in\M_n(\R)^2,\ h(AB)=h(BA)$. Montrer que $h$ préserve la trace. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 997] +Trouver $\dim(\op{Vect}(A))$ dans les deux cas suivants : + - $A=\{M\in\M_2(\C)$, $M^n=\op{Diag}(1,2)\}$ avec $n\geq 2$, + - $A=\{M\in\M_2(\C)$, $M^2=I_2\}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 998] +Si $A\in\M_n(\R)$, on note $S(A)$ l'ensemble des matrices semblables à $A$. Déterminer les matrices $A$ telles que $S(A)$ est fini. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 999] +Soit $\mathfrak{S}_n$ l'ensemble des permutations de $\db{1,n}$. + - Soit $\sigma\in\mathfrak{S}_n$. Montrer que $\phi_{\sigma}:s\mapsto s\circ\sigma$ est une permutation de $\mathfrak{S}_n$. + - Soient $E$ un $\mathbb{K}$-espace vectoriel de dimension $n\geq 2$, $(e_1,\ldots,e_n)$ une base de $E$. Pour $\sigma\in\mathfrak{S}_n$, on note $f_{\sigma}$ l'endomorphisme de $E$ défini par $\forall i\in\db{1,n}\ f_{\sigma}\left(e_i\right)=e_{\sigma(i)}$. On pose $p_n=\frac{1}{n!}\underset{\sigma\in\mathfrak{S}_n}{\sum}\ f_{\sigma}$. Montrer que $p_n$ est un projecteur et expliciter son image et son noyau. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1000] +Soient $n\geq 2$, $E=\R_n\left[X\right]$ et $\phi:P\in E\mapsto P-P'$. + - Montrer que $\phi$ est bijectif de deux manieres différentes. + - Soit $Q$ l'antecedent de $P$ par $\phi$. On suppose que $Q\geq 0$. Montrer que $P\geq 0$. Exprimer $P$ en fonction de $Q$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1001] +Soit $A\in\M_{3,2}\left(\R\right)$ et $B\in\M_{2,3}\left(\R\right)$ telles que $AB=\left(\begin{array}{rrr}0&-1&-1\\ -1&0&-1\\ 1&1&2\end{array}\right)$. + +Vérifier que $\left(AB\right)^2=AB$. Déterminer $\op{rg}\left(A\right)$, $\op{rg}\left(B\right)$. Montrer que $BA=I_2$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1002] +Soient $E$ un $\R$-espace vectoriel, $\phi$ une forme lineaire sur $E$ et $f\in\mc{L}\left(E\right)$. + - Montrer que $\op{Ker}\left(\phi\right)$ est stable par $f$ si et seulement s'il existe $\lambda\in\R$ tel que $\phi\circ f=\lambda\phi$. + - Soit $\mc{B}$ une base de $E$. On pose $L=\op{Mat}_{\mc{B}}\left(\phi\right)$ et $A=\op{Mat}_{\mc{B}}\left(f\right)$. Montrer que $\op{Ker}\left(\phi\right)$ est stable par $f$ si et seulement s'il existe $\lambda\in\R$ tel que $A^TL^T=\lambda L^T$. + - Trouver toutes les droites stables par l'endomorphisme dont la matrice dans la base canonique de $\R^3$ est $\left(\begin{array}{rrr}1&1&0\\ 0&1&0\\ 0&0&2\end{array}\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1003] +Soient $n\geq 2$, $A,B\in\M_n(\mathbb{K})$. On suppose $ABAB=0$. A-t-on $BABA=0$? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1004] +Soit $f\in\mc{L}\left(E\right)$ telle que $f^2=-4\op{id}$ ou $E$ est un $\R$-espace vectoriel de dimension $n$. - Déterminer le noyau et l'image de $f$. L'endomorphisme $f$ est-il inversible? Si c'est le cas, déterminer $f^{-1}$. - Montr er que $n$ est nécessairement pair. - Pour $x\neq 0$, montrer que $\left(x,f\left(x\right)\right)$ est une famille libre. - On suppose maintenant que $n=4$. Montrer qu'il existe une base de $E$ dans laquelle la + +$$\text{matrice de }f\text{ est }\left(\begin{array}{cccc}0&-4&0&0\\ 1&0&0&0\\ 0&0&0&-4\\ 0&0&1&0\end{array}\right)\text{.}$$ +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1005] +Soit $A\in\M_n\left(\R\right)$. R $\acute{\text{e}}$soudre $X+X^T=\op{tr}\left(X\right)A$ d'inconnue $X\in\M_n\left(\R\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1006] + - Soit $A\in\M_n\left(\C\right)$ telle que, pour tout $X\in\M_{n,1}\left(\C\right)$, $\left(X,AX\right)$ est l $\acute{\text{e}}$e. Que dire de $A$? + - Montrer que toute matrice $A\in\M_n\left(\C\right)$ de trace nulle est semblable à une matrice de diagonale nulle. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1007] +Soit $E$ un espace vectoriel de dimension finie. Soit $u$ un endomorphisme nilpotent tel que tout sous-espace de $E$ stable par $u$ admet un supplementaire stable par $u$. Montrer que $u$ est l'endomorphisme nul. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1008] +Soient $E,F,G$ trois $\mathbb{K}$-espaces vectoriels de dimension finie, $u\in\mc{L}(E,F),v=\mc{L}(F,G)$ et $w=v\circ u$. Montrer que $w$ est un isomorphisme si et seulement si les trois conditions suivantes sont realisées - $u$ est injective, - $v$ est surjective, - $F=\op{Im}u\oplus\op{Ker}v$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1009] +Soient $E$ un $\mathbb{K}$-espace vectoriel de dimension finie et $u,v\in\mc{L}(E)$ tels que $\op{rg}(u)=\op{rg}(v)$ et $u^2\circ v=u$. + - Montrer que $v\circ u\circ v=v$. + - Montrer que $u\circ v$ est un projecteur + - Montrer que $u\circ v\circ u=u$ puis que $v^2\circ u=v$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1010] +Soit $E$ un $\mathbb{K}$-espace vectoriel de dimension finie. + - Que dire de la trace d'un projecteur de $E$? Montrer que, pour $p$ projecteur de $E$, $\op{Im}(p)$ et $\op{Ker}(p)$ sont supplementaires dans $E$. + - Soient $p,q$ deux projecteurs de $E$. Montrer que $p+q$ est un projecteur si et seulement si $p\circ q=q\circ p=0$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1011] +Soient $E,F,G$ trois $\mathbb{K}$-espaces vectoriels de dimensions finies. Soient $f\in\mc{L}(E,F)$ et $g\in\mc{L}(F,G)$. Montrer que $\op{rg}(g\circ f)\geq\op{rg}(f)+\op{rg}(g)-\dim(F)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1012] +Soient $E,F,G$ trois $\mathbb{K}$-espaces vectoriels de dimension finie. Soient $u\in\mc{L}(E,F)$, $v\in\mc{L}(F,G)$. Soit $w=v\circ u$. Montrer que $w$ est un isomorphisme si et seulement si $u$ est injectif, $v$ est surjectif et $\op{Im}u\oplus\op{Ker}v=F$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1013] +Soient $E$ un $\mathbb{K}$-espace vectoriel de dimension finie et $u,v\in\mc{L}(E)$. + - Montrer que $\op{rg}(v)\leq\op{rg}(u\circ v)+\dim(\op{ Ker}u)$. - On suppose que $u$ est nilpotent d'indice $p$. Montrer que $\big(\dim(\op{Ker}u^k)\big)_{k\in\N}$ est strictement croissante puis stationnaire. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1014] + - Existe-t-il deux matrices $A,B\in\M_n(\R)$ telles que $AB-BA=I_n$? + - Soit $A\in\M_n(\R)$ une matrice non nulle de trace nulle. Montrer qu'il existe $u\in\R^n$ telle que la famille $(u,Au)$ soit libre. + - Soit $A\in\M_n(\R)$ de trace nulle. Montrer que $A$ est semblable à une matrice dont tous les coefficients diagonaux sont nuls. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1015] +Soient $A$ et $B$ dans $\M_n(\C)$ telles que $\op{rg}(AB-BA)=1$. + +Montrer que $A(\op{Ker}(B))\subset\op{Ker}(B))$ ou $A(\op{Im}(B))\subset\op{Im}(B))$ +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1016] +Soient $E$ un $\mathbb{K}$-espace vectoriel de dimension $p$ et $f_1,\ldots,f_p$ des formes lineaires sur $E$. Prouver l'équivalence des trois assertions suivantes : + - $(f_1,\ldots,f_p)$ est libre, + - $u:x\in E\mapsto(f_1(x),\ldots,f_p(x))\in\mathbb{K}^p$ est surjective, + - il existe $x_1,\ldots,x_p\in E$ tels que $\det(f_i(x_j))_{1\leq i,j\leq p}\neq 0$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1017] +Donner une condition nécessaire et suffisante sur $(a_1,\ldots,a_n)\in\C^n$ pour que la matrice $\begin{pmatrix}0&\cdots&0&a_1\\ \vdots&&\vdots&\vdots\\ 0&\ldots&0&a_{n-1}\\ a_1&\ldots&a_{n-1}&a_n\end{pmatrix}$ soit diagonalisable. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1018] +Soient $(a_1,\ldots,a_n,b_1,\ldots,b_n)\in\R^{2n}$ et $M=\begin{pmatrix}0&\cdots&0&b_1\\ \vdots&&\vdots&\vdots\\ 0&\ldots&0&b_n\\ a_1&\ldots&a_n&0\end{pmatrix}$. Donner une condition nécessaire et suffisante pour que $M$ soit diagonalisable. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1019] +Soit $\alpha\in\C$. La matrice $M=\begin{pmatrix}1&\alpha&0\\ \alpha&0&1\\ 0&1&-1\end{pmatrix}$ est-elle diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1020] +Redémontrer qu'une matrice diagonalisable à un polynôme annulateur scindé à racines simples. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1021] +Soit $A=\begin{pmatrix}0&0&1\\ 1&0&1\\ 0&1&0\end{pmatrix}$. + - Monter que $A$ est diagonalisable sur $\C$ et qu'elle admet une unique valeur propre réelle strictement positive $a$. + - Montrer que $\sum_{\lambda\in\op{Sp}(A)}\lambda^n$ est un entier pour tout $n\in\N$. + - Déterminer la nature de la série $\sum_{\lambda\in\op{Sp}(A)}\lambda^n$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1022] +Soit $E$ un $\C$-espace vectoriel de dimension finie. + - Soit $f\in\mc{L}(E)$ nilpotent. Montrer qu'il existe une base de $E$ dans laquelle la matrice de $f$ est triangulaire supérieure avec des 0 sur la diagonale. + - Soient $v$ et $w$ dans $\mc{L}(E)$ tels que $v$ est diagonalisable, $w$ est nilpotent et $v\circ w=w\circ v$. Montrer que $v+w$ et $v$ ont le même polynôme caractéristique. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1023] +Soit $E$ un $\R$-espace vectoriel de dimension finie $n\geq 2$. Soit $u\in\mc{L}(E)$ de spectre vide. + - Montrer qu'il existe $P\in\R[X]$ de degre $2$ tel que $\op{Ker}P(u)\neq\{0\}$. + - Montrer qu'il existe un sous-espace vectoriel de $E$ de dimension 2 et stable par $u$. + - En déduire que tout endomorphisme de $E$ admet un sous-espace vectoriel stable de dimension 1 ou 2. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1024] +Soit $f\in\mc{L}(E)$, ou $E$ un $\C$-espace vectoriel de dimension $n\geq 2$. Montrer que $f$ est diagonalisable si et seulement si $f^2$ est diagonalisable et $\op{Ker}f=\op{Ker}f^2$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1025] +Soient $f,g$ deux endomorphismes d'un $\R$-espace vectoriel $E$ de dimension finie tels que $f\circ g=f+g$. + - Montrer que $\op{Im}f=\op{Im}g$ et que $\op{Ker}f=\op{Ker}g$. + - On suppose de plus que $f$ est diagonalisable. Montrer que $f\circ g$ est diagonalisable. + - Montrer qu'aucune valeur propre de $f\circ g$ n'appartient à $]0,4\,[$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1026] +Soit $A\in\M_3(\C)$. Montrer que $A$ est semblable à $-A$ si et seulement si $\op{tr}(A)=0$ et $\det(A)=0$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1027] +Déterminer toutes les matrices $A\in\M_4(\R)$ telles que $A^2=\op{diag}(1,2,-1,-1)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1028] +Soient $A\in\M_n(\C)$ et $B=\begin{pmatrix}0&A\\ A&0\end{pmatrix}\in\M_{2n}(\C)$. + - Exprimer le rang de $B$ en fonction du rang de $A$. + - Étudier la diagonalisabilité de $B$ en fonction de celle de $A$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1029] +Soient $A\in\M_n(\C)$ et $B=\begin{pmatrix}0&I_n\\ A&0\end{pmatrix}$. + - Trouver une relation entre les valeurs propres de $A$ et celles de $B$ ainsi qu'entre les sous-espaces propres de $A$ et ceux de $B$. + - Déterminer les dimensions des sous-espaces propres de $B$ en fonction des dimensions des sous-espaces propres de $A$. + - Trouver une condition nécessaire et suffisante sur $A$ pour que $B$ soit diagonalisable. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1030] +Soient $A,B\in\M_n\left(\C\right)$ telles que $AB=BA$. Peut-on trigonaliser $A$ et $B$ dans une même base? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1031] +Soient $\left(\alpha_i\right)_{1\leq i\leq n}\in\R^n$ et $\left(\beta_i\right)_{1\leq i\leq n}\in\R^n$. On pose $A=\left(\alpha_i\beta_j\right)_{1\leq i,j\leq n}$. + - Quel est le rang de $A$? + - Montrer que $A^2=\op{tr}\left(A\right)A$. - Soit $M\in\M_n\left(\R\right)$ telle que $\op{rg}\left(M\right)=1$. Montrer qu'il existe $\left(X,Y\right)\in\left(\R^n\right)^2$ telles que $M=X^TY$. + - Trouver toutes les matrices de $\M_3\left(\R\right)$ telles que $M^2=0_3$. + - à quelle condition la matrice $A$ est-elle diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1032] +Soient $A=\left(\begin{array}{ccc}0&0&1\\ 1&0&0\\ 0&1&0\end{array}\right)$ et $M\in\M_3\left(\R\right)$ telle que $M^3=I_3$ et $M\neq I_3$. + - La matrice $A$ est-elle diagonalisable dans $\M_3\left(\C\right)$? dans $\M_3\left(\R\right)$? Donner ses valeurs propres. + - La matrice $M$ est-elle diagonalisable dans $\M_3\left(\C\right)$? Montrer que $\op{Sp}_{\C}\left(M\right)\subset\left\{1,j,j^2\right\}$ et que les multiplicités de $j$ et $j^2$ sont les memes. Donner le spectre de $M$. + - Montrer que $A$ et $M$ sont semblables dans $\M_3\left(\C\right)$, puis dans $\M_3\left(\R\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1033] +Soient $M\in\M_n\left(\C\right)$, $\left(A,B\right)\in\M_n\left(\C\right)^2$ et $\left(\lambda,\mu\right)\in\left(\C\ast\right)^2$ tels que $\lambda\neq\mu$. On suppose : $I_n=A+B,\ M=\lambda A+\mu B,\ M^2=\lambda^2A+\mu^2B$. + - Montrer que $M$ est inversible et déterminer $M^{-1}$. + - Montrer que $A$ et $B$ sont des projecteurs. + - La matrice $M$ est-elle diagonalisable? Si oui, trouver $\op{Sp}\left(M\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1034] +Soient $A$ et $B$ dans $\M_n(\R)$ telles que $AB-BA=A$. + +On note $\Psi:M\in\M_n(\R)\mapsto MB-BM$. + - Montrer que $\Psi$ est un endomorphisme de $\M_n(\R)$ et que, pour tout $k\in\N$, $\Psi(A^k)=kA^k$. + +Calculer $\op{tr}(A)$. + - Montrer que si $A$ n'est pas nilpotente alors $\Psi$ à une infinite de valeurs propres. Conclure +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1035] +Soient $n\geq 2$ et $A\in\M_n(\R)$ telle que $\op{Tr}(A)\neq 0$. + - On considére $\Phi\colon\M_n(\R)\ra\M_n(\R)$ définie par $\Phi:M\mapsto\op{Tr}(A)M-\op{Tr}(M)A$. + - Trouver $\op{Ker}\Phi$ et $\op{Im}\Phi$. + - Déterminer les éléments propres de $\Phi$. + - Déterminer la trace, le déterminant et le polynôme caractéristique de $\Phi$. + - On considére $\Psi\colon\M_n(\R)\ra\M_n(\R)$ définie par $\Psi:M\mapsto\op{Tr}(A)M+\op{Tr}(M)A$. + - Trouver les éléments propres de $\Psi$. + - Montrer que $\Psi$ est bijective et déterminer sa réciproque. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1036] +Soit $A\in\M_n(\R)$. Soit $f_A\in\mc{L}(\M_n(\R))$ défini par $f_A(M)=AM$. Montrer que $A$ et $f_A$ ont les memes valeurs propres. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1037] +Soit $A\in\M_3(\R)$. On cherche le nombre de solutions de l'équation $B^3=A$ dans $\M_3(\R)$. + - Montrer que, si $B$ est solution, alors $AB=BA$. + - Montrer que si $A$ est diagonalisable et à un sous-espace propre de dimension $\geq 2$ alors il y a une infinite de solutions. + - Traiter le cas ou $A$ admet trois valeur propres réelles distinctes. - Traiter le cas ou $A=\begin{pmatrix}r\cos(\theta)&-r\sin(\theta)&0\\ r\sin(\theta)&r\cos(\theta)&0\\ 0&0&\lambda\end{pmatrix}$ avec $r\gt 0$, $\lambda\in\R$ et $\theta\in\R\setminus\pi\Z$. - Cas general? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1038] +On note $D:P\mapsto P'$ l'endomorphisme derivation de $\R[X]$. + - Montrer que, pour tout $n\in\N$, $\R_n[X]$ est stable par $D$ et déterminer la matrice de l'endomorphisme induit par $D$ dans la base canonique de $\R_n[X]$. + - Soit $F$ un sous-espace vectoriel de $\R[X]$ de dimension finie non nulle stable par $D$. + - Montrer qu'il existe un entier $n$ et un polynôme $R$ de degre $n$ tel que $R\in F$ et $F\subset\R_n[X]$. + - Montrer que la famille $(D^j(R))_{0\leq j\leq n}$ est libre. + - En déduire que $F=\R_n[X]$. + - Expliciter tous les sous-espaces vectoriels de $\R[X]$ stables par $D$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1039] +On note $E=\mc C^0(\R^+,\R)$. Soit $\Phi$ l'application qui à $f\in E$ associe la fonction $\Phi(f)$ définie par : $\Phi(f)(0)=f(0)$ et $\forall x\in]0,+\i[$, $\Phi(f)(x)=\frac{1}{x}\int_0^xf(t)\dt$. + - Montrer que $\Phi$ est un endomorphisme de $E$. + - Déterminer les valeurs propres de $\Phi$ et les espaces propres associes. + - Soit $n\in\N$. Montrer que $\Phi$ stabilise $\R_n[X]$. L'endomorphisme induit par $\Phi$ sur $\R_n[X]$ est-il diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1040] +Soit $A\in\M_2(\R)$. On suppose qu'il existe $n\in\N^*$ tel que $A^{2^n}=I_2$. Montrer que $A^2=I_2$ ou qu'il existe $k\in\N^*$ tel que $A^{2^k}=-I_2$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1041] +Soit $n\in\N^*$. Soit $E$ un sous-espace vectoriel de $\M_n(\R)$ ne contenant que des matrices diagonalisables. + - Montrer que $\dim(E)\leq\frac{n(n+1)}{2}$. + - Lorsque $\mathbb{K}=\R$, quelle est la dimension maximale de $E$? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1042] +Soit $A\in\M_n(\R)$ telle que $A^2$ soit triangulaire supérieure avec des coefficients diagonaux egaux à $1,2,\ldots,n$. Montrer que $A$ est triangulaire supérieure. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1043] +Soit $A\in\M_n(\R)$. On suppose que la suite $(A^k)_{k\in\N}$ admet une limite $B\in\M_n(\R)$. + - Montrer que $B^2=B$, $BA=AB$. Déterminer $\mathrm{Ker}(B)$ et $\mathrm{Im}(B)$. + - Montrer que $\mathrm{Sp}(A)\subset\{z\in\C\,,\;|z|\lt 1\}\cup\{1\}$. Montrer que si $1$ n'est pas valeur propre de $A$ alors $B=0$. + - Montrer que la multiplicité de 1 dans le polynôme caractéristique de $A$ est egale à la dimension de $\mathrm{Ker}(A-I_n)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1044] +Soient $a,b$ deux réels et $n$ un entier. + +Montrer que $\Phi:P\in\R_n[X]\mapsto(X-a)(X-b)P'-nP$ est un endomorphisme et déterminer ses éléments propres. L'endomorphisme $\Phi$ est-il diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1045] + - Soient $A\in\M_n(\R)$ diagonalisable et $B=I_n+A+A^3$. Montr are $A$ est un polynôme en $B$. + - Le résultat de - subsiste-t-il lorsque $A$ est complexe? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1046] + - Soit $A\in\M_3(\R)$ non trigonalisable. Montr are $A$ est $\C$-diagonalisable. + - Soit $A\in\M_4(\R)$. Montr are $\mathfrak{l}$'une des conditions suivantes est realisées : + - $A$ est $\R$-trigonalisable ; - $A$ est $\C$-diagonalisable ; + - $A$ est $\R$-semblable à une matrice de la forme $\begin{pmatrix}B&C\\ 0&B\end{pmatrix}$ avec $B,C\in\M_2(\R)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1047] +Soient $n\in\N^*$ et $A\in\M_n(\R)$ telle que $A^{n-1}\neq 0$ et $A^n=0$. Soit $L$ l'ensemble $L=\{M\in\M_n(\R),\ AM=MA\}$. + - Montr are qu'il existe $x_0\in\R^n$ tel que la famille $(x_0,Ax_0,A^2x_0,\ldots,A^{n-1}x_0)$ soit une base de $\R^n$. + - En déduire que la famille $(I_n,A,A^2,\ldots,A^{n-1})$ est une base de $L$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1048] +Soit $E$ un $\C$-espace vectoriel de dimension finie. Soit $u$ un automorphisme de $E$ tel que, pour tout $x\in E$, l'ensemble $\{u^k(x)\ ;\ k\in\N\}$ est fini. + - Montr are qu'il existe $N\in\N^*$ tel que $u^N=\mathrm{id}$. + - L'endomorphisme $u$ est-il diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1049] +Soit $(u_1,...,u_p)$ une famille de vecteurs de $\R^n$ telle que $\forall i\neq j$, $\langle u_i,u_j\rangle\lt 0$. + +Montr are que toute sous-famille de $(u_1,...,u_p)$ de cardinal $(p-1)$ est libre. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1050] +Soit $A\in\M_n(\R)$ une matrice nilpotente non nulle. + - Montr are qu'il existe $V\in\R^n$ tel que $AV\neq 0$ et $A^2V=0$. + - On note $\langle\,\ \rangle$ le produit scalaire usuel sur $\R^n$. + +Déterminer l'ensemble $\{\langle AX,X\rangle\ ;\ X\in\R^n\}$. + - Trouver les matrices $B\in\M_n(\R)$ telles que $\{\langle BX,X\rangle\ ;\ X\in\R^n\}=\{0\}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1051] +Soit $E=\R_n[X]$. Soient $a_0\lt a_1\lt \cdots\lt a_n$ des réels. Pour $P,Q\in E$, on pose $\langle P,Q\rangle=\sum_{k=0}^nP(a_k)Q(a_k)$. + - Montr are que $\langle\,\ \rangle$ est un produit scalaire sur $E$. + - Trouver une base orthonormée de $E$ pour ce produit scalaire. + - Soit $H$ l'ensemble des $Q\in E$ tels que $\sum_{k=0}^nQ(a_k)=0$. Montr are que $H$ est un sous-espace vectoriel de $E$ et preciser sa dimension. + - Pour $P\in E$, déterminer $d(P,H)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1052] +Soient $a,b\in\R$ et $A=\begin{pmatrix}a^2&ab&ab&b^2\\ ab&a^2&b^2&ab\\ ab&b^2&a^2&ab\\ b^2&ab&ab&a^2\end{pmatrix}$. Preciser le spectre et les sous-espaces propres. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1053] + - Montrer que $\phi:P\mapsto(X^2-1)P''+2XP'$ définit un endomorphisme de $\R_n[X]$ qui est symétrique pour le produit scalaire $\left\langle P,Q\right\rangle=\int_{-1}^+P(t)Q(t)\dt$. + - Déterminer les valeurs propres de $\phi$. + - Montrer qu'il existe une unique base orthonormée de vecteurs propres $\left(P_0,\ldots,P_n\right)$ telle que, pour tout $k\in\N$, $\deg P_k=k$ et $\left\langle P_k,X^k\right\rangle\gt 0$. + - On pose $Q_k(X)=(-1)^kP_k(-X)$. Montrer que $\left(Q_0,\ldots,Q_n\right)$ vérifie les propriétés de -. + +Que peut-on en déduire? + - Montrer que $P_n$ est scindé à racines simples sur $]-1,1[$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1054] +Soient $a,b\in\R$ et $\Phi_{a,b}$ l'endomorphisme de $\M_n(\R)$ défini par $\Phi_{a,b}:M\mapsto aM+bM^T$. + - Trouver les valeurs propres et les sous-espaces propres de $\Phi_{a,b}$. + - Déterminer $\mathrm{Tr}(\Phi_{a,b})$ puis son polynôme caractéristique. + - à quelle condition $\Phi_{a,b}$ est-il un automorphisme? Déterminer alors $\Phi_{a,b}^{-1}$. + - L'endomorphisme $\Phi_{a,b}$ est-il autoadjoint? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1055] +Soit $A\in\M_n\left(\C\right)$ telle que $A^2+A^T=I_n$ et $\mathrm{tr}\left(A\right)=0$. + - Montrer que toute valeur propre de $A$ vérifie $\lambda^4-2\lambda^2+\lambda=0$ et que $A$ est diagonalisable. + - Montrer que $n$ est multiple de 4. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1056] +Soient $\left(E,\left\langle\ \,\ \ \right\rangle\right)$ un espace euclidien, $a$ et $b$ deux vecteurs libres de $E$ et $f\colon x\in E\mapsto\left\langle a,x\right\rangle a+\left\langle b,x\right\rangle b$. + - Déterminer le noyau et l'image de $f$. + - Déterminer les éléments propres de $f$. L'endomorphisme $f$ est-il diagonalisable? Aurait-on pu le prevoir sans étudier les éléments propres? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1057] +Soit $A\in\M_n\left(\R\right)$ telle que, pour tout $X\in\M_{n,1}\left(\R\right)$, $X^TAX=0$. Montrer que $\det\left(A\right)\geq 0$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1058] +Soit $A\in\mc{S}_n^{++}\left(\R\right)$ de spectre $0\lt \lambda_1\leq\lambda_2\leq\cdots\leq\lambda_n$. Soit $X\in\M_{n,1}\left(\R\right)$. + - Montrer que $\left\|X\right\|^4\leq\left\langle AX,X\right\rangle\left\langle A^{-1 }X,X\right\rangle$. + - Montrer que $\left\langle AX,X\right\rangle\left\langle A^{-1}X,X\right\rangle \leq\dfrac{\left(\lambda_1+\lambda_n\right)^2}{4\lambda_1\lambda_{ n}}\left\|X\right\|^4$. + - Montrer qu'il existe une base orthonormale $\left(P_0,\ldots,P_n\right)$ de $E$ telle que, pour tout $k\in\left[\![0,n]\!\right]$, $\deg\left(P_k\right)=k$ et $\left\langle P_k,X^k\right\rangle\gt 0$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1059] +Pour $t\in\R$, on pose $M\left(t\right)=\left(\begin{array}{ccc}1&1&1\\ 1&1&0\\ 1&0&t\end{array}\right)$. On note $\alpha\left(t\right)\leq\beta\left(t\right)\leq\gamma\left(t\right)$ les valeurs propres de $M\left(t\right)$. + - Montrer que $\alpha\left(t\right)\lt 0\lt \beta\left(t\right)\lt 2\lt \gamma\left(t\right)$. + - Montrer que, lorsque $t\ra+\i$, $\alpha\left(t\right)\ra 0$, $\beta\left(t\right)\ra 2$ et que $\gamma\left(t\right)=t+O\left(\dfrac{1}{t}\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1060] +Soit $M\in{\cal M}_n({\R})$. Montrer que $M$ est antisymétrique si et seulement si pour toute $P\in{\cal O}_n({\R})$, la matrice $P^{-1}MP$ est à diagonale nulle. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1061] +Soit $M=(m_{i,j})_{1\leq i,j\leq n}\in{\cal O}_n({\R})$. Montrer : + +$$\sum_{i,j}m_{i,j}^2=n,\ \ \left|\sum_{i,j}m_{i,j}\right|\leq n,\ \ \ n \leq\sum_{i,j}|m_{i,j}|\leq n\ln(n).$$ +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1062] +Soient $E$ un espace euclidien et $p,q$ deux projecteurs orthogonaux. On considére $h=p\circ q$. + - Montrer que ${\rm Im}(q)$ et ${\rm Ker}(p)$ sont stables par $h$. + - Montrer que $p$ et $q$ sont autoadjoints. + - On pose $F={\rm Im}(q)+{\rm Ker}(p)$. Montrer que $E=F\oplus F^{\perp}$. En déduire que $h$ est diagonalisable. + - Montrer que le spectre de $h$ est contenu dans le segment $[0,1]$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1063] +Soient $n\geq 2$, $A\in{\cal S}_n^{++}({\R})$ et $B\in{\cal S}_n^+({\R})$. + - Montrer qu'il existe une matrice $C$ telle que $C^2=A^{-1}$. + - Montrer, en posant $D=CBC$, que $(\det(I_n+D))^{\frac{1}{n}}\geq 1+(\det D)^{\frac{1}{n}}$. + - En déduire que $(\det(A+B))^{\frac{1}{n}}\geq(\det A)^{\frac{1}{n}}+(\det B)^{\frac{1}{n}}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1064] +Soit $A\in{\rm GL}_n({\R})$. Montrer qu'il existe $O\in{\cal O}_n({\R})$ et $S\in{\cal S}_n^{++}({\R})$ telles que $A=OS$. Étudier l'unicité d'une telle décomposition. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1065] +Soient $A,B\in{\cal S}_n({\R})$ telles que $ABA=B$ et $BAB=A$. + - Montrer que $A^2=B^2$. + - On suppose que $A$ est inversible. Montrer que $A$ et $B$ sont des symétries orthogonales qui commutent. + - On ne suppose plus que $A$ est inversible. Montrer que ${\rm Im}\,A={\rm Im}\,B$ et ${\rm Ker}\,A={\rm Ker}\,B$. +#+end_exercice + + +** Analyse + +#+begin_exercice [Mines PC 2024 # 1066] +Les parties $E=\left\{(x,y)\in{\R}^2\,\ x^2(x-1)(x-3)+y^2(y^2-4)=0\right\}$ et + + $F=\left\{(x,y)\in{\R}^2\,\ 2x^2-y(y-1)=0\right\}$ sont elles fermées? bornées? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1067] +Soit $E$ un espace euclidien. Soit $u\in{\cal S}^{++}(E)$. Montrer qu'il existe $m\gt 0$ et un ouvert $\Omega$ dense dans $E$ tels que $\forall x\in\Omega$, $\frac{\left\|u^{k+1}(x)\right\|}{\left\|u^k(x)\right\|}\xrightarrow[k\ra+ \i]{}m$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1068] + - Soit $A\in{\cal M}_2({\C})$. Montrer que $\left\{Q(A)\ ;\ Q\in{\C}[X]\right\}$ est un ferme de ${\cal M}_2({\C})$. + - Soient $B\in{\cal M}_n({\C})$ et $Q\in{\C}[X]$ non constant. On suppose que $B$ à $n$ valeurs propres distinctes. Montrer qu'il existe $A\in{\cal M}_n({\C})$ telle que $B=Q(A)$. + - Soit $Q\in{\C}[X]$ non constant. Montrer que $\left\{Q(A)\ ;\ A\in{\cal M}_2({\C})\right\}$ est une partie dense de ${\cal M}_2({\C})$. Cet ensemble est-il ferme? borne? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1069] +Soient $\left(a_n\right)_{n\geq 0}$ et $\left(b_n\right)_{n\geq 0}$ deux suites réelles convergeant vers $a$ et $b$ respectivement. Montrer que $\dfrac{1}{n+1}\sum_{k=0}^na_kb_{n-k}\xrightarrow[n\ra+\i]{}ab$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1070] +Soit $(u_n)_{n\geq 1}$ une suite réelle définie par $u_1\in\R$ et $\forall n\geq 1$, $u_{n+1}=nu_n-1$. Montrer que $u_1=e-1$ si et seulement si il existe $a\in\R$ vérifiant $u_n=O(n^a)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1071] +Pour $n$ et $p$ dans $\N^*$, on pose $u_{n,p}=\dfrac{1}{p^n}\left(\sqrt[n]{1+\dfrac{1}{p}}+\sqrt[n]{1+\dfrac{2}{ p}}+\cdots+\sqrt[n]{1+\dfrac{p}{p}}\right)^n$. + - Calculer $\lim\limits_{n\ra+\i}\lim\limits_{p\ra+\i}u_{n,p}$. + - Calculer $\lim\limits_{p\ra+\i}\lim\limits_{n\ra+\i}u_{n,p}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1072] +On pose $S_n(t)=\sum_{k=0}^n\dfrac{(-1)^k}{(2k+1)!}t^{2k+1}$ et $x_n=\min\{t\gt 0,\ S_n(t)=0\}$. + - Montrer que $x_n$ est bien défini pour tout $n\in\N^*$. + - Étudier les variations et la convergence de $(x_n)_{n\in\N^*}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1073] +Soit $\left(x_n\right)_{n\geq 0}$ une suite réelle telle que $x_0\gt 1$ et, pour tout $n\in\N$, $x_{n+1}=x_n+x_n^{-1}$. Montrer que $x_n\sim\sqrt{2n}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1074] +Pour tout $n\in\N^*$, on note $x_n$ la solution de $e^x=n-x$. Limite, équivalent et développement asymptotique à deux termes de $x_n$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1075] +Soit $\alpha\in\R$. Nature de la série de terme general $u_n=n^{\alpha}\prod_{k=1}^n\left(1+\dfrac{(-1)^{k-1}}{\sqrt{k}}\right)?$ +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1076] +Nature de la série de terme general $u_n=\dfrac{(-1)^n}{\sum_{k=1}^n\frac{1}{\sqrt{k}}+(-1)^n}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1077] +Pour $n\in\N^*$, on pose $x_n=\sum_{k=1}^n\dfrac{1}{k},H_n=\sum_{k=1}^n\dfrac{1}{k},u_n= \sum_{k=1}^n\dfrac{(-1)^k\ln k}{k},v_n=\sum_{k=1}^n\dfrac{\ln k}{k}$ et $w_n=\sum_{k=1}^n\dfrac{\ln(2k)}{k}$. + - Montrer que $(x_n)$ converge vers un réel $\ell$ à déterminer. Montrer que $x_n=\ell+\mc{O}\left(\dfrac{1}{n}\right)$. + - Exprimer $u_{2n}$ en fonction de $v_{2n}$ et $w_n$. + - Montrer que $H_n=\ln(n)+\gamma+o(1)$. + - Etablir la convergence de $(u_n)$ et preciser sa limite. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1078] +Soit $\left(a_n\right)_{n\in\N^*}$ la suite définie par $a_1=1$ et, pour tout $n\geq 2$, $a_n=2a_{\lfloor n/2\rfloor}$. Montrer que $\sum{\dfrac{1}{a_n^2}}$ converge. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1079] +Soit $f\in\mc C^1(\R,\R^{+*})$ telle que $f'\leq 0$ et $f(0)=1$. On pose $a_0=1$ et, pour $n\in\N$, $a_{n+1}=a_nf(a_n)$. Montrer que $(a_n)_{n\in\N}$ decroit et tend vers 0. Étudier la nature de la série $\sum a_n$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1080] +Soit $\alpha\in\R$. On pose, pour $n\in\N$, $u_n=\int_{n\pi}^{(n+1)\pi}\frac{\sin t}{t^{\alpha}}dt$ et $v_n=u_{2n}+u_{2n+1}$. Déterminer la nature de $\sum u_n$ et $\sum v_n$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1081] +Pour $n\in\N^*$, on pose $R_n^{(0)}=\frac{(-1)^n}{n}$, $R_n^{(1)}=\sum_{k=n}^{+\i}\frac{(-1)^k}{k}$ et pour $\ell\in\N^*$, + + $R_n^{(\ell)}=\sum_{k=n}^{+\i}R_k^{(\ell-1)}$. Justifier l'existence et étudier le signe de $R_n^{(\ell)}$. Ind. Calculer $\int_0^1t^k\dt$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1082] +Soit $f$ une fonction continue et injective de $\R$ dans $\R$. + +En considérant $g_x:t\mapsto f(x+t)-f(x)$ montrer que $f$ est strictement monotone. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1083] +Déterminer les applications $f\colon\R\ra\R$ telles que l'image de tout segment est un segment de même longueur. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1084] +Soit $f\colon\R^p\ra\R^n$ telle que $\forall(x,y)\in(\R^p)^2$, $f(x+y)=f(x)+f(y)$. Montrer que $f$ est continue si et seulement si $f$ est lineaire. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1085] +Trouver toutes les fonctions $f\colon\R\ra\R$ dérivables en 0 telles que : + + $\forall x\in\R,f(2x)=2f(x)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1086] +Soit $f\in\mc C^0(\R,\R)$ telle que $(*)\colon\forall\left(x,y\right)\in\R^2$, $f\left(x+y\right)f\left(x-y\right)=\left(f\left(x\right)f\left(y\right)\right) ^2$. + - Donner toutes les valeurs que peut prendre $f\left(0\right)$. + - Montrer que, pour tout $x_0\in\R$ tel que $f\left(x_0\right)=0$, on a $f\left(\frac{x_0}{2^n}\right)=0$. En déduire que si $f$ s'annule en un point, $f$ est identiquement nulle. + - Trouver toutes les fonctions continues vérifiant $(*)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1087] +Soient $f$, $g$ deux fonctions continues de $[0,1]$ dans $[0,1]$ telles que $f\circ g=g\circ f$. Montrer qu'il existe $x\in[0,1]$ tel que $f\left(x\right)=g\left(x\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1088] +Soit $f:[0,1]\ra\R$ dérivable et non nulle pour laquelle il existe $M\gt 0$ tel que $\forall x\in[0,1]$, $f'\left(x\right)\leq Mf\left(x\right)$. Montrer que $f$ ne s'annule pas. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1089] +Montrer que $x\mapsto\cos\left(x\right)$ admet un unique point fixe. Montrer qu'il n'existe pas de fonction $f$ dérivable telle que $\cos=f\circ f$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1090] +Soit $f$ une fonction telle que, pour $0\lt x\lt 1$, $f\left(x\right)=\frac{1}{\sqrt{x}}\ln\left(\frac{1+\sqrt{x}}{1-\sqrt{x}}\right)$. Trouver $g\in\mc C^{\i}\left(]-\i,1[\right)$ telle que $g\mid_{0,1}[=f$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1091] +Soit $f\in\mc C^1(\left[a,b\right],\R)$ telle que $f'(a)=f'(b)=0$. Montrer qu'il existe $x\in\left]a,b\right[$ tel que $f'(x)=\dfrac{f(x)-f(a)}{x-a}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1092] +Soit $f\colon\R\ra\R$ dérivable telle que $f^2+\left(1+f'\right)^2\leq 1$. Montrer que $f=0$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1093] +Soit $f\colon\R\ra\R$ une fonction de classe $\mc C^{n+1}$ telle que $f(0)=0$. Pour $x\gt 0$, on pose $g(x)=\dfrac{f(x)}{x}$. Déterminer, pour $k\in\left\{0,1,\ldots,n\right\}$, $\lim\limits_{x\ra 0}g^{(k)}(x)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1094] +Soit $x\in\R$. Montrer qu'il existe un unique $a\in\R$ tel que $\int_x^a\exp\left(t^2\right)dt=1$. On définit alors $x\mapsto a\left(x\right)$. Montrer que $a$ est $\mc C^{\i}$. Montrer que le graphe de $a$ est symétrique par rapport à la droite d'équation $y=-x$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1095] +Trouver un équivalent simple en $0$ de $f:x\mapsto\int_{x^2}^{x^3}\dfrac{e^t}{\arcsin t}\dt$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1096] +Calculer $\int_0^{\pi/4}\ln\left(1+\tan(x)\right)dx$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1097] +Soit $f\in\mc C^1(\left[0,1\right],\R)$. Pour $n\in\N^*$, on pose $U_n=\int_0^1f(x)\dx-\dfrac{1}{n}\sum_{k=0}^{n-1}f \Big(\dfrac{k}{n}\Big)$. Déterminer la limite de $\left(nU_n\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1098] +Soit $f\colon\left[0,1\right]\ra\R$ continue. On suppose que $\int_0^1f(x)x^ndx=0$ pour $0\leq k\leq n$. Montrer que $f$ s'annule au moins $n+1$ fois sur $]0,1[$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1099] +Soit $x$ un nombre complexe de module différent de 1. Calculer $I=\int_0^{2\pi}\dfrac{dt}{x-e^{it}}:$ + + - en utilisant la décomposition en éléments simples de la fraction rationnelle $\dfrac{nX^{n-1}}{X^n-1}$, + + - par une autre methode. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1100] +Soient $(a,b)\in\R^2$ avec $a\lt b$, et $f,g\in\mc C^0(\left[a,b\right],\R^{+*})$. On pose $m=\inf\limits_{\left[a,b\right]}\dfrac{f}{g}$ et + + $M=\sup\limits_{\left[a,b\right]}\dfrac{f}{g}\cdot$ Montrer que $\int_a^bf^2\int_a^bg^2\leq\dfrac{\left(M+m\right)^2}{4 Mm}\left(\int_a^bfg\right)^2$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1101] +Soient $c\in\R$, $u$ et $v$ deux fonctions continues sur $\R^+$ à valeurs respectivement dans $\R$ et dans $\R^+$ telles que $\forall x\in\R^+$, $u\left(x\right)\leq c+\int_0^xv\left(t\right)u\left(t\right) dt$. + +Montrer que $\forall x\in\R^+$, $u\left(x\right)\leq c\exp\left(\int_0^xv\left(t\right) dt\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1102] +Montrer qu'il existe $(A,B)\in\R^2$ tel que, pour tout $f\in\mc C^1\left(\R,\R\right)$ $2\pi$-periodique, on ait $\sup_{\R}|f|\leq A{\int_0^{2\pi}|f|+B{\int_0^{2\pi}|f'|.}}$ L'inegalite subsiste-elle si on enleve une hypothese. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1103] +On considére une fonction $f:[a,b]\mapsto\R$ de classe $\mc C^1$. On suppose qu'on dispose de $x_0\in]a,b[$, $y_0\gt f(x_0)$ et qu'un cercle $C$ de centre $(x_0,y_0)$ passant par $(x_0,f(x_0))$ est au-dessus du graphe de $f$. Montrer que $f'(x_0)=0$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1104] +Soit $M:t\in\R\mapsto\begin{pmatrix}2e^{-t}&(t-1)^2\\ 1&0\end{pmatrix}$. + - Montrer que l'application $N:A=(a_{i,j})\in\M_2(\R)\mapsto\sup_{1\leq i,j \leq 2}|a_{i,j}|$ est une norme. + +Déterminer $\phi(t)=N(M(t))$ et tracer le graphe de $\phi$. La fonction $\phi$ est-elle de classe $\mc C^1$? + - Déterminer la primitive $\Phi$ de $\phi$ telle que $\Phi(0)=0$. $\Phi$ est-elle $\mc C^1$? + - Soit $F$ la primitive de $M$ telle que $F(0)=0$. Prouver $\forall t\geq 0,N(F(t))\leq\Phi(t)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1105] +Nature de l'intégrale ${\int_0^{+\i}\frac{\sin\left(x\right)}{\sqrt{x}+\sin\left(x\right)}}$d $x$? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1106] +Pour $\alpha\gt 0$ déterminer la nature de ${\int_0^{+\i}\left(1+\ln(\op{sh}x^{\alpha})-2 \op{sh}(\ln(x^{^{\alpha}}+1))\right)dx}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1107] +Nature de ${\int_1^{+\i}\frac{\ln|1-x|\cos\left(\ln\left(x\right)\right)}{x^{ \alpha}\left(1+x\right)}dx}$ et ${\int_0^1\frac{\ln|1-x|\cos\left(\ln\left(x\right)\right)}{x^{ \alpha}\left(1+x\right)}dx}$? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1108] +Étudier la convergence de l'intégrale ${\int_0^{+\i}\left|\sin x\right|^x\dx}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1109] +Existence et calcul des intégrales ${I=\int_0^{+\i}\frac{x}{\op{sh}x}\dx}$ et ${J=\int_0^{+\i}\frac{x}{\op{ch}x}\dx}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1110] +On considére ${E=\{f\in\mc C^2([0,1],\R),\ f(0)=f(1)=0\}}$. Soit $f\in E$. + - Montrer que ${I(f)=\int_0^1\frac{\cos(\pi t)}{\sin(\pi t)}f'(t)f(t)\dt}$ est bien définie, et que + + ${I(f)=\frac{\pi}{2}\int_0^1\frac{f(t)^2}{\sin(\pi t)^2}\dt}$ + - En considérant ${\int_0^1\left(\pi\frac{\cos(\pi t)}{\sin(\pi t)}f(t)-f'(t) \right)^2\dt}$, montrer que + + ${\int_0^1f'(t)^2\dt}\geq\pi^2\int_0^1f(t)^ {2}\dt$. + - Déterminer les fonctions $f$ pour lesquelles il y a egalite dans -. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1111] +Soit $p\in\N$. Montrer que la fonction $t\mapsto e^{-\left(t-p\pi\right)^2}\sin(t)$ est intégrable sur $\R$ et que son intégrale est nulle. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1112] +Existence et calcul de $\int_0^{+\i}e^{-t}\left(\ln(t)-\frac{1}{t}+\frac{1}{1-e^{-t}}\right)\, dt$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1113] +Soit $f\colon\R^+\ra\R$ continue, positive, decroissante et telle que $\int_0^{+\i}f(t)\dt$ converge. + +Montrer que $tf(t)\underset{t\ra+\i}{\longrightarrow}0$. Ind. Considérer $\int_t^{2t}f(x)\dx$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1114] +Soit $f\colon\R^+\ra\R$ de classe $\mc C^1$. On suppose que $\int_0^{+\i}f'(t)^2\dt$ et $\int_0^{+\i}t^2f(t)^2\dt$ convergent. Montrer que $\int_0^{+\i}f(t)^2\dt$ converge et que + +$$\int_0^{+\i}f(t)^2\dt\leq\left(\int_0^{+\i}f^{ '}(t)^2\dt\right)^{1/2}\left(\int_0^{+\i}t^2f(t)^2 \dt\right)^{1/2}.$$ +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1115] +Pour $n\in\N^*$, on pose $A_n(x)=\sum_{k=1}^n\frac{x^k}{k}$. + - Montrer que, pour tout $y\geq 0$, il existe un unique $x\geq 0$ tel que $A_n(x)=y$. On pose $f_n(y)=x$. + - Étudier la monotonie de $(f_n)_{n\in\N^*}$ et montrer que la suite converge simplement vers une fonction $f$. + - Montrer que $\forall x\geq 0$, $0\leq f(x)\lt 1$. + - Montrer que $\forall x\geq 0$, $f(x)=1-e^{-x}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1116] +Soit $f:x\mapsto\sum_{n=2}^{+\i}\left(\frac{1}{n-x}-\frac{1}{n+x}\right)$. + - Montrer que $f$ est bien définie sur $[\,0\,;1\,]$. + - Montrer que $f$ est continue et intégrable sur $[\,0\,;1\,]$. + - Calculer $\int_0^1f(x)\dx$. + - Montrer que $f$ est dérivable. Est-elle de classe $\mc C^k$? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1117] +Soit $f:x\mapsto\sum_{n=1}^{+\i}\frac{1}{n+n^2x^2}$. + - Déterminer le domaine de définition et de continuité de $f$. + - Déterminer la limite de $f$ et un équivalent en $+\i$. + - Déterminer la limite de $f$ et un équivalent en $0$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1118] +Soit $F:x\mapsto\sum_{n=1}^{+\i}\mathrm{e}^{-n^2x^2}$. Déterminer les limites et équivalents de $F$ en $0$ et en $+\i$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1119] +Soit $f:x\mapsto\dfrac{1}{x^2}{\sum_{n=1}^{+\i}\dfrac{1}{\left(n-x \right)^2}}+{\sum_{n=1}^{+\i}\dfrac{1}{\left(n+x\right)^2}}$. + +On note $(*)$ la propriété $\colon\forall x\in\R\setminus\Z$, $g\left(\dfrac{x}{2}\right)+g\left(\dfrac{x+1}{2}\right)=4g\left(x\right)$. + - Montrere que $f$ est continue sur $\R\setminus\Z$ et 1-periodique. + - Montrere que $f$ vérifie $(*)$. + - Montrere que, si $g$ est continue sur $\R$, 1-periodique et vérifie $(*)$ alors $g$ est nulle. + - Montrere que, pour tout $x\in\R\setminus\Z$, $f\left(x\right)=\dfrac{\pi^2}{\sin^2\left(\pi x\right)}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1120] +Preciser le domaine de définition de $f:x\mapsto\sum_{n\geq 0}e^{-n}e^{in^2x}$. Montrere que l'application $f$ est de classe $\mc C^{\i}$ sur $\R$. Est-elle développable en série entiere? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1121] +Étudier la convergence uniforme de la série de fonctions $\sum e^{-x}\dfrac{x^k}{k!}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1122] +Soit $\alpha\gt 0$. Pour $n\in\N^*$ et $x\gt 0$, on pose $u_n(x)=x^{\alpha}e^{-n^2x}$ puis $f_{\alpha}(x)=\sum_{n=1}^{+\i}u_n(x)$. + - Montrere que $f_{\alpha}$ est bien définie sur $\R^{+*}$. + - Trouver les $\alpha$ pour lesquels la série $\sum u_n$ converge normalement sur $\R^{+*}$. + - Trouver la limite puis un équivalent de $f_{\alpha}(x)$ lorsque $x\ra+\i$. + - Trouver la limite puis un équivalent de $f_{\alpha}(x)$ lorsque $x\ra 0^+$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1123] +Soit $(a_n)_{n\in\N}$ une suite réelle telle que $\forall n\geq 2$, $a_n=a_{n-1}+(n-1)a_{n-2}$. + +Trouver $f$ de classe $\mc C^{\i}$ au voisinage de 0 telle que $\forall n\in\N$, $f^{(n)}(0)=a_n$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1124] +Soit $f:x\in\,]-1,1[\,\mapsto\sum_{n=1}^{+\i}\dfrac{(-1)^n}{x+n}$. + - Montrere que $f$ est de classe $\mc C^{\i}$. + - Montrere que $f$ est développable en série entiere. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1125] +On s'interesse à la série entiere suivante $:f(x)=\sum_{n=1}^{+\i}u_nx^n$ avec $u_n=\int_1^{+\i}e^{-t^n}\dt$. + - Déterminer la limite de la suite $(u_n)$. + - Déterminer le domaine de convergence de la série entiere. + - Déterminer la limite de $f$ à la borne de droite du domaine de convergence. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1126] +Soit $N$ un entier qui n'est pas un carre parfait. On pose $a=\sqrt{N}$. + - Montrere qu'il existe une suite d'entiers $(p_n)_{n\in\N}$ telle que $na-p_n\in\left[-\dfrac{1}{2},\dfrac{1}{2}\right]$. + - Montrere qu'il existe une constante $c\gt 0$ tels que $\forall n\in\N^*$, $\sin(na\pi)\gt cn^{-1}$. - En déduire le rayon de convergence de $f(x)=\sum_{n=0}^{+\i}\frac{x^n}{\sin(n\pi\sqrt{2})}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1127] +On pose $b_0=1$ et, pour $n\in\N$, $b_{n+1}=-\frac{1}{n+2}\sum_{k=0}^n\binom{n+2}{k}b_k$. + - Montrer que, pour tout $n$, $|b_n|\leq n!$. + - Pour $|z|\lt 1$, montrer que $\sum_{k=0}^{+\i}\frac{b_k}{k!}z^k=\frac{z}{e^z-1}$. + - Montrer que $x\mapsto\op{cotan}(x)-\frac{1}{x}$ est développable en série entiere. + - Quel est le lien entre les deux dernieres questions? On pourra poser $z=2i\pi x$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1128] +Soit $S:x\mapsto\sum_{n=0}^{+\i}\frac{x^n}{\binom{2n}{n}}$. + - Déterminer le rayon de convergence $R$ de $S$. Montrer que $S$ est solution de l'équation différentielle $x(x-4)y'+(x+2)y=2$. + - En déduire $S(x)$ pour tout $x\in]0,R[$. + - Calculer $\sum_{n=0}^{+\i}\frac{1}{\binom{2n}{n}}$ +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1129] +Montrer qu'il existe une fonction $\phi$ développable en série entiere en 0 vérifiant au voisinage de 0 : $\phi'\left(x\right)=x+\phi^2\left(x\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1130] +Pour $n\in\N^*$, on pose $I_n=\int_0^{\frac{\pi}{2}}\frac{\sin(2n+1)t}{\sin t}dt$ et $J_n=\int_0^{\frac{\pi}{2}}\frac{\sin((2n+1)t)}{t}dt$. + - Que dire de $I_n$? + - Montrer que $\left(I_n\right)$ et $\left(J_n\right)$ convergent vers la même limite. Trouver cette limite. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1131] +Pour $n\in\N$, on pose $I_n=\int_0^{+\i}\frac{dt}{\left(1+t^2\right)\sqrt[n]{1+t^{ n}}}$. Montrer que chaque intégrale $I_n$ est convergente puis déterminer la limite de la suite $\left(I_n\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1132] +Pour $n\geq 2$, on pose $I_n=\int_1^{+\i}\frac{dt}{1+t+\cdots+t^n}$. + +Justifier que $I_n$ existe puis déterminer un équivalent de $I_n$ quand $n\ra+\i$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1133] +Pour $n\in\N$ et $x\in[0,1]$, on pose $f_n(x)=\frac{2^nx}{1+n2^nx^2}$. + - Étudier la convergence simple de la suite $\left(f_n\right)_{n\in\N}$. + - Pour $n\in\N$, on pose $I_n=\int_0^1f_n(x)\dx$. Calculer $I_n$ et $\lim_{n\ra+\i}I_n$. + - Étudier la convergence uniforme de la suite $\left(f_n\right)_{n\in\N}$ sur $[0,1]$. + - Donner un développement asymptotique à deux termes de $I_n$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1134] +Soient $a\gt -1$ et $b\gt 0$. On définit les suites $(I_n)$ et $(J_n)$ par $J_n=\int_0^{+\i}x^ae^{-nx}\dx$ et $I_n=\int_0^{+\i}\frac{x^ae^{-nx}}{\sqrt{1+x^b}}\dx$. + - Étudier l'existence de $J_n$ et en déduire celle de $I_n$. + - Déterminer la limite de $(J_n)$. + - Exprimer $J_n$ à l'aide de la fonction $\Gamma:x\mapsto\int_0^{+\i}t^{x-1}e^{-t}\dt$ et retrouver ainsi la limite de la suite. + - Déterminer un équivalent de $I_n$ à l'aide de $J_n$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1135] +Montrer : $\int_0^1\frac{dx}{1+x^p}=\sum_{k=0}^{+\i} \frac{{(-1)}^k}{1+kp}$. Calculer $\sum_{k=0}^{+\i}\frac{{(-1)}^k}{1+k}$, $\sum_{k=0}^{+\i}\frac{{(-1)}^k}{1+2k}$, $\sum_{k=0}^{+\i}\frac{{(-1)}^k}{1+3k}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1136] +Pour tout $n\in\N$, on pose $I_n=\int_0^1\ln{(1+t^n)}\dt$. + - Déterminer la limite de $(I_n)$. + - Justifier l'existence de $J=\int_0^1\frac{\ln{(1+u)}}{u}du$. + - Montrer que $I_n\sim\frac{J}{n}$. + - Montrer que $J=\sum_{n=1}^{+\i}\frac{{(-1)}^{n-1}}{n^2}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1137] + - Montrer que $I=\int_0^{+\i}\frac{\ln(x)}{x^2-1}\dx$ est convergente. + - On pose $J=\int_0^1\frac{\ln(x)}{x^2-1}\dx$. Montrer que $I=2J$. + - Exprimer $J$ à l'aide de la somme d'une série. + - On donne $\sum_{n=1}^{+\i}\frac{1}{n^2}=\frac{\pi^2}{6}$. Calculer $J$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1138] +On considére $J=\int_0^1\ln(t)\ln(1-t)\dt$. + +Montrer que $J$ est bien définie et que $J=\sum_{n=0}^{+\i}\frac{1}{n(n+1)^2}$. En déduire la valeur de $J$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1139] +Soit $F:x\mapsto\int_0^{+\i}\frac{\mathrm{sh}\,t}{t}e^{-xt}dt$. + - Déterminer le domaine de définition et la limite en $+\i$ de $F$. + - Donner une expression simple de $F(x)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1140] +Étudier $x\mapsto\int_0^{+\i}\frac{1-\cos(xt)}{t^2}dt$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1141] +Soit $F:x\mapsto\int_0^{+\i}\frac{\arctan(xt)}{t(1+t^2)}dt$. + - Montr are $F$ est définie sur $\R$ et impaire. + - Montr are $F$ est dérivable et calculer $F'$. + - En déduire la valeur de $F(x)$ pour tout $x\in\R$. + - En déduire la valeur de $\int_0^{+\i}\frac{\arctan(t^2)}{t^2}dt$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1142] +Soit $F:x\mapsto\int_0^{+\i}\frac{e^{-at}-a^{-bt}}{t}\cos(xt)\dt$, ou $0\lt a\lt b$. + - Montr are que $f$ est définie sur $\R$ et de classe $\mc C^1$. + - Montr are qu'il existe une constante $C$ telle que $\forall x\in\R$, $F(x)=\frac{1}{2}\ln\left(\frac{x^2+b^2}{x^2+a^2}\right)+C$. + - Déterminer $\lim\limits_{x\ra+\i}F(x)$ et conclure quant à la constante $C$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1143] +Soit $f\colon\R\ra\R$ continue et bornée. Soit $g:x\in\R\mapsto-\frac{1}{2}\int_{-\i}^{+\i}f(t)\,e^{-|x-t|}\, dt$. + - Montr are $g$ est définie sur $\R$ et bornée. + - Montr are que $g$ est de classe $\mc C^2$ et vérifie l'équation différentielle $(*):y''-y=f(x)$. + - Soit $h\colon\R\ra\R$ de classe $\mc C^2$ et bornée sur $\R$ vérifiant l'équation $(*)$. A-t-on $g=h\,$? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1144] +Soit $F:x\mapsto\int_0^{+\i}\frac{\sin\left(xt\right)}{t}\mathrm{e}^{-t} dt$. Trouver le domaine de définition de $F$ et exprimer $F$ sans le signe intégral. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1145] +Soit $F:x\mapsto\int_1^{+\i}\frac{t-\lfloor t\rfloor}{t^{x+1}}dt$. + - Déterminer le domaine de définition de $F$. + - Montr are $F$. + - Pour $x\geq 1$, donner l'expression de $F\left(x\right)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1146] +Pour tout $x\gt 0$, on pose $f\left(x\right)=\int_0^1\ln\left(t\right)\ln\left(1-t^x\right)dt$. + - La fonction $f$ est-elle bien définie? + - Écrire $f$ comme la somme d'une série. + - Déterminer la limite de $f\left(x\right)$ quand $x$ tend vers 0. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1147] +Pour $x\gt 0$, on pose $F:x\mapsto\int_0^{+\i}\frac{\mathrm{e}^{-xt}}{\sqrt{t+t^2}}dt$. + - Calculer $F'\left(x\right)$. + - Calculer $\lim\limits_{x\ra+\i}F\left(x\right)$, puis déterminer un équivalent de $F$ en $+\i$. - Montrer que $\underset{x\ra 0}{\lim}F\left(x\right)=+\i$, puis déterminer un équivalent de $F$ en 0. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1148] +Soit $f\in\mc C^0([0,1],\R^{+*})$. Pour $x\gt 0$, on pose $N_f(x)=\left(\int_0^1f(t)^x\dt\right)^{1/x}$. + - Montrer que $N_f$ est de classe $\mc C^{\i}$ sur $\R^{+*}$. + - Déterminer la limite de $N_f(x)$ lorsque $x\ra+\i$. + - Déterminer la limite $\dfrac{1}{x}\left(\int_0^1f(t)^x\dt-1\right)$ lorsque $x\ra 0^+$. + - Déterminer la limite de $N_f(x)$ lorsque $x\ra 0$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1149] +Soit $f$ une fonction continue de $[a,b]\times[c,d]$ dans $\R$. + +Montrer que $\int_a^b\left(\int_c^df(x,y)dy\right)dx=\int_c^{ d}\left(\int_a^bf(x,y)\dx\right)dy$. + +Ind. Considérer $g:t\mapsto\int_a^b\left(\int_c^tf(x,y)dy\right) dx$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1150] +Soit $(E):x^2y''+4xy'+2y=\ln\left(1+x\right)$. + - Trouver les solutions de $(E)$ développables en série entiere et déterminer leur rayon de convergence. + - Écrire ces fonctions à l'aide des fonctions usuelles. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1151] +Soit $A\in\M_n(\R)$ telle que $\mathrm{Tr}(A)\gt 0$. Soit $x\colon\R\ra\R^n$ une fonction de classe $\mc C^1$ telle que : (i) pour tout $t\in\R$, on a $x'(t)=Ax(t)$, (ii) pour tout $i\in\db{1,n}$, on a $\lim_{t\ra+\i}x_i(t)=0$. + +Montrer qu'il existe une forme lineaire $\ell\colon\R^n\ra\R$ non nulle telle que $\forall t\in\R,\,\ell(x(t))=0$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1152] +On définit $E=\mc C^0(\left[0,1\right],\R)$ et $F=\mc C^{\i}(\left[0,1\right],\R)$. + +Soit $n\in\N^*$. Pour $u\in E$ et $\left(\lambda_1,\ldots,\lambda_n\right)\in\R^n$, on considére le systeme d'équations différentielles $(L)\colon\forall i\in\db{1,n}$, $\forall t\in\left[\,0,1\right],x'_i(t)=\lambda_ix_i(t)+u(t)$. + - Résoudre le systeme $(L)$. + - Pour $i\in\db{1,n\rrbracket$, on note $\phi_i(u)$ la valeur en $t=1$ de la solution de la $i$-eme équation de $(L)$ qui s'annule en $t=0$. On note $\Phi(u)=(\phi_1(u),\ldots,\phi_n(u))$. Montrer que, pour tout $i\in\llbracket 1,n}$, $\phi_i\in\mc{L}(E,\R)$ et que $\Phi\in\mc{L}(E,\R^n)$. + - Pour $i\in\db{1,n}$, on définit un élément de $F$ en posant $f_i:s\mapsto e^{\lambda_i(1-s)}$. Montrer que la famille $(\phi_1,\ldots,\phi_n)$ est libre si et seulement si la famille $(f_1,\ldots,f_n)$ est libre. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1153] +Soit $f\colon\R^2\ra\R$ de classe $\mc C^2$ telle que $\dfrac{\partial^2f}{\partial x\partial y}=0$. Déterminer $f$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1154] +Déterminer les extrema de $f(x,y)=x\ln y-y\ln x$ pour $(x,y)\in(\R^{+*})^2$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1155] +Trouver les extrema de $f:(x,y)\mapsto x^4+y^4-2(x-y)^2$. +#+end_exercice + + +** Probabilités + +#+begin_exercice [Mines PC 2024 # 1156] +Une poite contient $n$ boules numerotées de $1$ à $n$. On tire des boules, une à une, avec remise, tant que le numero de la boule tirée est supérieur au précédent. On note $Z$ le nombre de boules tires. Déterminer la loi de $Z$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1157] +Une urne contient deux boules. L'une est blanche et l'autre est soit blanche soit noire avec probabilité $1/2$. On tire successivement deux boules de l'urne sans remise. Quelle est la probabilité de tirer une boule blanche au second tirage sachant qu'on a tire une boule blanche au premier tirage? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1158] +Soient $a\in\N^*$, $n\in\N^*$ et $N=an$. On dispose de $N$ boules indiscernables et $n$ unres numerotées de $1$ à $n$. On depose les $N$ boules dans les unres. On note $T_i$ la variable aléatoire qui vaut $1$ si l'urne $i$ est vide, et $0$ sinon. On note $Y_n$ le nombre d'urnes vides et $S_n=\dfrac{1}{n}Y_n$. + - Donner la loi de $T_i$. Calculer l'esperance et la variance de $T_i$. + - Calculer l'esperance et la variance de $S_n$. Étudier les limites de $(\mathbf{E}(S_n))$ et $(\mathbf{V}(S_n))$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1159] +Une panier contient $r$ pommes rouges et $v$ pommes vertes. On mange les pommes une à une, on s'arrête lorsqu'on a mange toutes les pommes vertes. Déterminer la probabilité d'avoir mange toutes les pommes. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1160] +On repartit $N$ objets dans $N-1$ boites. Probabilité pour qu'aucune boite ne soit vide? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1161] +La durée de vie (en jours) d'une ampoule suit la loi geometrique de paramêtre $\dfrac{1}{2}$. + - Quelle est la durée de vie moyenne de cette ampoule? + - L'ampoule à deja vecu $n$ jours. Quelle est la durée de vie moyenne de cette ampoule à partir du $n$-eme jour? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1162] +On considére deux des et, pour $i\in\db{1,6}$, on note $p_i$ (respectivement $q_i$) la probabilité que le premier de (respectivement le second de) donne le résultat $i$. On note $P$ et $Q$ les fonctions generatrices des deux des. On note $R$ la fonction generatrice de la somme des deux des. + - Donner $R$. + - On suppose d'orenavant que $R$ est egale à la fonction generatrice de la somme de deux des non pipes. + - Quelles sont les racines de $R$? + - Montrer que les deux des ne sont pas pipes. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1163] +Dans un magasin, on a $n$ caisses et $np$ clients. Chaque client choisit une caisse de facon indépendante et avec la même probabilité pour chacune des caisses. On note $X_i$ le nombre de clients à la caisse numero $i$. + - En écrivant $X_i$ comme une somme de variables aléatoires indépendantes, déterminer la loi, l'esperance et la variance de $X_i$. + - Pour $(i,j)\in\db{1\,;\,n}^2$, calculer $\op{Cov}(X_i,X_j)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1164] +Soient $A$ et $B$ deux evenements. Montrer que $|\mathbf{P}(A)\mathbf{P}(B)-\mathbf{P}(A\cap B)|\leq\dfrac{1}{4}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1165] +Soient $X$ et $Y$ deux variables aléatoires à valeurs dans $\N^*$ telles que, pour tout $n\in\N^*$, la loi de $X$ sachant $(Y=n)$ est la loi uniforme sur $\db{1,n}$. + - Montrer que $Y+1-X$ et $X$ ont même loi. + - On suppose $X$ suit la loi geometrique $\mc{G}(p)$. Montrer que $X$ et $Y+1-X$ sont indépendantes. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1166] + - On munit l'ensemble des fonctions $f\colon\db{1,n\rrbracket\ra\llbracket 1,n-1}$ de la loi uniforme. Déterminer la probabilité pour que $f$ soit surjective. + - Même question avec $f\colon\db{1,n\rrbracket\ra\llbracket 1,n-2}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1167] +Soient $U$ une variable aléatoire discrete, $n$ et $\ell$ deux entiers naturels tels que $n\geq\ell+3$ et $\mathbf{P}\left(U\gt n\right)\mathbf{P}\left(U\gt \ell\right)\gt 0$. On pose $Y=\left\lfloor\frac{U}{2}\right\rfloor$ et $Z=\left\lfloor\frac{U+1}{2}\right\rfloor$. + - Montrer que $Y$ et $Z$ ne sont pas indépendantes. + - On suppose que $U\sim\mc{B}\left(n,p\right)$ avec $n\geq 4$ pair. Montrer que $Y$ ne suit pas une loi binomiale. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1168] +Soit $Z$ une variable aléatoire à valeurs dans $\Z$ telle que $|Z|+1\sim\mc{G}(p)$ et telle que $\forall n\in\Z,\mathbf{P}(Z=n)=\mathbf{P}(Z=-n)$. Soit $A=\left(\begin{array}{ccc}0&Z&Z\\ Z&0&1\\ 1&1&0\end{array}\right)$. + - Déterminer la loi du rang de $A$. + - Déterminer la probabilité pour que $A$ soit diagonalisable. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1169] +Soient $X$ et $Y$ deux variables aléatoires indépendantes suivant des lois geometriques de paramêtres $p$ et $q$ respectivement. En notant $M=\left(\begin{array}{cc}X&1\\ 0&Y\end{array}\right)$, donner la probabilité pour $M$ soit diagonalisable. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1170] +Soit $M=(X_{i,j})_{1\leq i,j\leq n}$ une matrice aléatoire réelle ou les $(1+X_{i,j})$ sont i.i.d. de loi $\mc{G}(p)$ avec $p\in]0,1[$. + - Déterminer la probabilité que $M$ soit symétrique. + - Déterminer la probabilité que $M$ soit orthogonale. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1171] +Soit $a$ un réel. On pose $g:t\mapsto\frac{a\,e^t}{2-t}$. + - Montrer qu'il existe une unique valeur de $a$ pour laquelle il existe une variable aléatoire $X$ à valeurs dans $\N$ dont $g$ soit la fonction generatrice. + +On suppose maintenant que $a$ est egal à cette valeur et que $X$ est une variable aléatoire à valeurs dans $\N$ dont $g$ est la fonction generatrice. + - Trouver la probabilité que $X$ soit pair. + - Quelle est la probabilité que la matrice $\left(\begin{array}{ccc}X&X&0\\ -X&-X&0\\ X&X&0\end{array}\right)$ soit diagonalisable? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1172] +Soit $(X,Y)$ un couple de variables aléatoires à valeurs dans $(\N^*)^2$. On suppose que $X\leq Y$, que $\forall i\in\N^*,\mathbf{P}(Y=i)\gt 0$, $\forall 1\leq k\leq i,\mathbf{P}(X=k|Y=i)=\dfrac{1}{i}$. Montrer que $X$ et $Y-X+1$ ont la même loi. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1173] +Soient $n\in\N^*$, $X_1,\ldots,X_n$ des variables aléatoires indépendantes suivant la loi de Bernoulli de paramêtre $p\in]0,1[$. On pose $M=(X_iX - {1\leq i\leq n,1\leq j\leq n}$. + - Déterminer la loi du rang de $M$, de la trace de $M$. + - Quelle est la probabilité que $M$ soit un projecteur? +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1174] +Soit $T$ une variable aléatoire à valeurs dans $\N$ telle que $\forall n$, $\mathbf{P}\left(T\gt n\right)\gt 0$. Pour tout entier naturel $n$, on pose $\theta_n=\mathbf{P}\left(T=n\,|\,T\geq n\right)$. + - Montrer que $\forall n\in\N$, $\theta_n\in[0,1[$. + - Exprimer $\theta_n$ en fonction de $\mathbf{P}\left(T\geq n\right)$. En déduire que $\sum\theta_n$ diverge. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1175] +Soient $n\in\N^*$ et $p\in\,]0,1]$. + - Soit $U$ une variable aléatoire telle que $U\sim\mc{B}\left(n,p\right)$. Déterminer la fonction generatrice de $U$. + - Soient $Y$ et $Z$ deux variables aléatoires dicretes indépendantes telles que $U=Y+Z$ et $U\sim\mc{B}\left(n,p\right)$. Montrer que $Y$ et $Z$ suivent des lois binomiales (pas nécessairement de memes paramêtres). +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1176] + - Soit $r\in\N^*$ et $x\in\,]\,-1\,;1\,[$. Montrer que $\sum_{n=r-1}^{+\i}\binom{n}{r-1}x^{n-r+1}=\dfrac{1}{(1-x)^r}$. + - Soit $(U_n)$ une suite de variables aléatoires indépendantes suivant la loi $\mc{B}(p)$. Soit $X$ le rang du $r$-eme succès. Quelle est la loi de $X$? Déterminer $\mathbf{E}(X)$, $\mathbf{E}(X(X+1))$ et $\mathbf{V}(X)$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1177] +Soit $X$ une variable aléatoire suivant la loi de Poisson de paramêtre $\lambda$. + - Montrer que $\mathbf{P}(X\geq\lambda+1)\leq\lambda$. + - Montrer que $\mathbf{P}\!\left(X\leq\dfrac{\lambda}{3}\right)\leq\dfrac{9}{4\lambda}$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1178] +Soient $X$ et $Y$ deux variables aléatoires discretes à valeurs strictement positives indépendantes et suivant la même loi. Montrer que $\mathbf{E}(X/Y)\geq 1$. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1179] + - Montrer qu'il existe une variable aléatoire à valeurs dans $\N^*$ telle que, pour tout $n\in\N^*$, $\mathbb{P}(Y=n)=\dfrac{1}{n(n+1)}$. + - Si $X\colon\Omega\ra\N^*$ est un variable aléatoire telle que $X(\Omega)=\N^*$, on définit le_taux de defaillance_ de $X$ pour $n\in\N^*$ par $x_n=\mathbb{P}(X=n|X\geq n)$. + - Pour $n\in\N^*$, montrer que $\mathbb{P}(X\geq n)=\prod_{k=1}^{n-1}(1-x_k)$. + - En déduire $\mathbb{P}(X=n)$ en fonction des $x_k$ pour $n\in\N^*$. + - Quelle variable aléatoire admet un taux de defaillance constant à partir du rang $1$? + - Calculer le taux de defaillance de la variable $Y$ introduite à la première question. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1180] +Soit $(p_n)_{n\in\N}$ une suite d'éléments de $]0,1[$ tel que la série $\sum p_n$ converge. Pour tout $n\in\N$, soit $X_n$ une variable aléatoire suivant la loi de Bernoulli de paramêtre $p_n$. On pose $S_n=\sum_{k=0}^nX_k$ et $S=\sum_{k=0}^{+\i}X_k$. + - Soit $k\in\N$. Exprimer l'evenement $(S\geq k)$ à l'aide des evenements $(S_n\geq k)$. En déduire que $S$ est une variable aléatoire. + - Montrer que $S$ est presque-surement finie. + - Montrer que $S$ admet une esperance et la calculer. +#+end_exercice + + +#+begin_exercice [Mines PC 2024 # 1181] +Soient $p\in]0,1[$ et $(X_i)_{i\geq 1}$ une suite de variables aléatoires indépendantes identiquement distribuées suivant la loi geometrique de paramêtre $p$. + +Pour $n\in\N^*$, on pose $M_n=\max\{X_1,\ldots,X_n\}$. + - Montrer que $\mathbf{E}(M_n)=\sum_{k=0}^{+\i}1-(1-q^k)^n$ ou $q=1-p$. + - Soit $f_n:t\mapsto 1-(1-q^t)^n$. Montrer que $f_n$ est intégrable sur $\R^+$ et donner un équivalent de $\int_0^{+\i}f_n(t)\dt$ lorsque $n\ra+\i$. + - En déduire un équivalent de $\mathbf{E}(M_n)$ lorsque $n\ra+\i$. +#+end_exercice + + +* Centrale - MP + +** Algèbre + +#+begin_exercice [Centrale MP 2024 # 1182] +Un entier $n\geq 2$ est un faux premier (FP) s'il n'est pas premier et si, pour tout $a\in\Z$ premier à $n$, $a^{n-1}\equiv 1\;[n]$. + - Montrer que, si $n$ est FP, $n$ est impair. + - On suppose que $n$ s'écrit $\prod_{i=1}^rp_i$ ou $r\geq 2$, les $p_i$ sont des nombres premiers impairs distincts tels que, pour tout $i\in\db{1,r},p_i-1$ divise $n-1$. Montrer que $n$ est FP. + - On admet que, pour tout $p$ premier impair et tout $v\in\N^*$, le groupe multiplicatif $(\Z/p^v\Z)^{\times}$ est cyclique. En déduire la réciproque de la question précédente. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1183] +Soient $G$ un groupe admettant un nombre fini de generateurs, $H$ un groupe fini, $f:G\ra G$ un morphisme de groupes surjectif et $g:G\ra H$ un morphisme de groupes. + - Montrer que l'ensemble des morphismes de groupes de $G$ vers $H$ est fini. + - Soit $a\in\mathrm{Ker}\,f$. Montrer que, pour tout $n\in\N^*$, il existe $b_n\in G$ tel que $f^n(b_n)=a$, puis calculer $g\circ f^m(b_n)$ pour tout $m\gt n$. + - Montrer que $\mathrm{Ker}\,f\subset\mathrm{Ker}\,g$. + - On pose $\Gamma=\{M\in\M_2(\Z),\ \det M=1\}$. Montrer que $\Gamma$ est un groupe, engendre par les matrices $S=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ et $T=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}$. + - Montrer que tout endomorphisme surjectif de $\Gamma$ est bijectif. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1184] +On note $\mathbb{U}$ le groupe des nombres complexes de module $1$. Soit $q$ un entier $\geq 2$ fixe. On pose $H_q=\left\{z\in\C\ ;\ \exists n\in\N,\ z^{q^n}=1\right\}$. + - Montrer que $H_q$ est un sous-groupe dense de $\mathbb{U}$. + - Soit $f$ un endomorphisme du groupe $H_q$, continu en $1$. Montrer que $f$ se prolonge de maniere unique en un endomorphisme continu $\overline{f}$ du groupe $\mathbb{U}$. + - En déduire qu'il existe $m\in\Z$ tel que $f(z)=z^m$ pour tout $z\in H_q$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1185] + - Rappeler, pour tous $P,Q\in\C[X]$, la définition de $P\circ Q$ et preciser le degre de de ce polynôme. + - Montrer que seuls les polynômes de degre 1 possedent un inverse pour la loi $\circ$. + - On pose $P=X^2+\alpha$ avec $\alpha\in\C$. Montrer qu'il existe au plus un polynôme de degre $n$ qui commute avec $P$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1186] + - Soit $I$ un idéal de $\Q[X]$ distinct de $\{0\}$. Montrer qu'il existe un polynôme $\mu\in\Q[X]$ tel que $I=\mu\Q[X]$. + - Soit $\lambda\in\C$. Montrer que $I_{\lambda}=\{P\in\Q[X],\ P(\lambda)=0\}$ est un idéal de $\Q[X]$. + - Soit $P\in\Q[X]$ irreductible. Montrer que les racines complexes de $P$ sont simples. + - Soient $P\in\Q[X]$ et $\lambda\in\C$ racine de $P$ avec multiplicité $m\gt \dfrac{\deg P}{2}$. Montrer que $\lambda\in\Q$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1187] + - Rappeler la définition d'une $\mathbb{K}$-algèbre et d'un endomorphisme de $\mathbb{K}$-algèbre. + - Soit $\phi$ un endomorphisme de la $\mathbb{K}$-algèbre $\mathbb{K}(X)$. Montrer que $\phi(X)\neq 0$. + - Déterminer les endomorphismes de la $\mathbb{K}$-algèbre $\mathbb{K}(X)$. + - Déterminer les automorphismes de la $\mathbb{K}$-algèbre $\mathbb{K}(X)$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1188] +Soit $n\in\N^*$. Pour $A,B\in\M_n(\C)$, on pose $[A,B]=AB-BA$. + +Soit $E=\{[A,B],(A,B)\in\M_n(\C)^2\}$. + - Montrer que $\op{tr}(M)=0$ pour toute matrice $M\in E$. + - Montrer que l'ensemble $E$ est stable par similitude matricielle et par multiplication par un scalaire. + - Montrer qu'une matrice de trace nulle est semblable à une matrice de diagonale nulle. + - Montrer que $E$ est egal à l'ensemble des matrices de trace nulle. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1189] +Soient $E$ un espace vectoriel réel de dimension finie, $G$ un sous-groupe fini de $\op{GL}(E)$, $p=\dfrac{1}{|G|}\sum_{g\in G}g$, $V^G=\{x\in E\ ;\ \forall g\in G,\ g(x)=x\}$. + - Montrer que, si $h\in G$, $g\in G\mapsto h\circ g\in G$ est une bijection de $G$ sur lui-meme, puis que $p$ est un projecteur. + - Montrer que $\text{dim}(V^G)=\dfrac{1}{|G|}\sum\op{tr}(g)$. + - Montrer que tout sous-espace $V$ de $E$ stable par tous les éléments de $G$ admet un supplementaire stable par tous les éléments de $G$. On pourra partir d'un projecteur $q$ de $E$ sur $V$ et considérer $\dfrac{1}{|G|}\sum_{g\in G}g\circ q\circ g^{-1}$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1190] +Soient $E$ un $\R$-espace vectoriel de dimension $n\geq 2$ et $u\in\mc{L}(E)$. + - Calculer, en fonction de $\op{tr}u$ et de $\op{tr}(u^2)$, les coefficients de $X^{n-1}$ et de $X^{n-2}$ du polynôme caractéristique de $u$. + - On suppose $u$ de rang $2$. + - Montrer que l'on peut écrire $\chi_u=X^{n-2}P(X)$, ou $P$ est un polynôme de degre $2$ dont on precisera les coefficients en fonction de $\op{tr}u$ et $\op{tr}(u^2)$. + - à quelle condition l'endomorphisme $u$ est-il trigonalisable? +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1191] +Pour $a\in\Z$, on pose $S_a=\begin{pmatrix}1&a\\ 0&-1\end{pmatrix}$ et $T_a=\begin{pmatrix}1&a\\ 0&1\end{pmatrix}$. + - Donner le lien entre l'inverse d'une matrice carrée inversible et sa comatrice. + - Montrer que $\op{GL}_2(\Z)$ (ensemble des matrices de $\M_2(\Z)$ inversibles et dont l'inverse est à coefficients dans $\Z$) est un groupe et que $S_a,T_a\in\op{GL}_2(\Z)$ pour tout $a\in\Z$. + - Que vaut $T_bS_aT_b^{-1}$ pour $a,b\in\Z$? + - Soit $M\in\M_2(\Z)$ de polynôme caractéristique $X^2-1$. Montrer qu'il existe $P\in\op{GL}_2(\Z)$ tel que $M=PS_0P^{-1}$ ou $M=PS_1P^{-1}$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1192] +Soit $E$ un $\C$-espace vectoriel de dimension finie $n$. + - Cours : lemme des noyaux (avec demonstration). + - Soit $u$ et $v$ deux symétries telles que $u\circ v=-v\circ u$. Montrer que $n$ est pair. + - On pose $n=2p$. Montrer qu'il existe une base $\mc{B}$ de $E$ dans laquelle les matrices de $u$ et $v$ sont respectivement : $\begin{pmatrix}I_p&0\\ 0&-I_p\end{pmatrix}$ et $\begin{pmatrix}0&I_p\\ I_p&0\end{pmatrix}$. + - Quels sont les entiers $k$ pour lesquels il existe des symétries $s_1,\ldots,s_k$ vérifiant : + + $\forall(i,j)\in\db{1,k}^2,\ (i\neq j\implies s_i\circ s_j=-s_j\circ s_i)$? +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1193] + - Montrer que les valeurs propres d'un endomorphisme d'un espace de dimension finie sont les racines de son polynôme caractéristique. + - Montrer que, pour $p,q\in\Q$ avec $p\neq q$, il existe $a,b,c\in\Z$ premiers entre eux dans leur ensemble tels que $p=a/c$ et $q=b/c$. + - Existe-t-il $x,y,z\in\Z$ premiers entre eux tels que $x^2+y^2=3z^2$? + - Existe-t-il $M\in\M_2(\Q)$ symétrique dont $\sqrt{2}$ est valeur propre? +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1194] + - Pour $A\in\M_n(\mathbb{K})$, rappeler la définition des polynômes minimal $\pi_A$ et caractéristique $\chi_A$. + - Donner une condition nécessaire et suffisante sur $\pi_A$ pour que $A$ soit trigonalisable. + - Donner la définition et la dimension du sous-espace caractéristique de $A$ associe à la valeur propre $\lambda$. + - Soient $k\in\N^*$ et $A\in\M_n(\R)$. + - Montrer que, si $\chi_A$ est scindé, alors $\chi_{A^k}$ l'est aussi. + - Trouver un contre-exemple à la réciproque. + +Ind. On pourra examiner le cas des rotations. + - On suppose $\chi_{A^2}$ scindé à racines dans $\R^+$. Montrer que $\chi_A$ est scindé sur $\R$. + - Soit $A\in\M_n(\C)$. On suppose : $\forall X\in\C^n,\ \exists p\in\N^*,\ A^pX=X$. Montrer que $A$ est diagonalisable. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1195] +Soit $A\in{\cal M}_n({\C})$. On note $\chi_A=\sum_{i=0}^na_iX^{n-i}$ et $\lambda_1,\ldots,\lambda_n$ les valeurs propres de $A$. + - Donner et démontrer la décomposition en éléments simples de $P'/P$. + +En déduire que : $\forall x\in{\C}\setminus{\rm Sp}(A),\ \frac{\chi'_A(x)}{ \chi_A(x)}={\rm tr}\big((xI_n-A)^{-1}\big)$. + - Pour tous $j\in\db{0,n}$ et $x\in{\C}$, on pose $B_j=\sum_{i=0}^ja_iA^{j-i}$ puis $Q(x)=\sum_{j=1}^nx^{n-j}B_{j-1}$. + +Montrer que $Q(x)(xI_n-A)=\chi_A(x)I_n$ et ${\rm tr}\big(Q(x)\big)=\chi'_A(x)$. + - Pour tout $k\in\db{0,n-1]\!]$, on pose $S_k=\sum_{j=1}^n\lambda_j^k$. Montrer que : $\forall j\in[\![0,n-1}$, + +$$\sum_{i=0}^ja_iS_{j-i}=(n-j)a_j.$$ +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1196] +Soient $n\in{\N}^*$ et $S_n$ l'ensemble des polynômes unitaires de degre $n$ à coefficients dans ${\Z}$ dont toutes les racines complexes ont un module majore par $1$. + +Soit $P(X)=X^n+a_{n-1}X^{n-1}+\cdots+a_0\in S_n$. + +On note $z_1,...,z_n$ les racines de $P$ eventuellement confondues. + - - Rappeler les relations coefficients-racines pour un polynôme complexe. + - Montrer que $\forall k\in\db{0,n-1}$, $|a_k|\leq\binom{n}{k}$. + - Conclure que $S_n$ est fini. + - Montrer que $P$ est le polynôme caractéristique de la matrice + - - Montrer que $\forall p\in{\N},\ \exists Q_p\in S_n,\ \forall 1\leq i \leq n,Q_p(z_i^p)=0$. + - Conclure que les racines non nulles de $P$ sont de module $1$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1197] +Soient $n\in{\N}^*$ et $p$ un entier premier impair. On note ${\rm GL}_n({\Z})$ l'ensemble des matrices $M\in{\rm GL}_n({\R})$ telles que $M$ et $M^{-1}$ sont à coefficients entiers. + - Rappeler la définition de la comatrice. + - Montrer que ${\rm GL}_n({\Z})$ est l'ensemble des matrices de ${\cal M}_n({\Z})$ dont le déterminant vaut $\pm 1$. + - Soit $M\in{\cal M}_n({\C})$. On suppose qu'il existe $k\in{\N}^*$ tel que $M^k=I_n$ et que $A=\frac{1}{p}(M-I_n)\in{\cal M}_n({\Z})$. Montrer que $M=I_n$. + - Déterminer un majorant des cardinaux des sous-groupes finis de ${\rm GL}_n({\Z})$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1198] + - Si $n\in{\N}^*$, montrer que le groupe ${\rm GL}_n({\Z})$ des inversibles de l'anneau ${\cal M}_n({\Z})$ est l'ensemble des $M\in{\cal M}_n({\Z})$ de déterminant $\pm 1$. - Pour $a\in\Z$, soient $T_a=\left(\begin{array}{cc}1&a\\ 0&1\end{array}\right)$ et $S_a=\left(\begin{array}{cc}1&a\\ 0&-1\end{array}\right)$. Pour $a\in\Z$ et $b\in\Z$, calculer $T_bS_a{T_b}^{-1}$. + - Soit $M\in\M_2(\Z)$ telle que $M^2=I_2$. Montrer qu'il existe $P\in\text{GL}_2(\Z)$ telle que $PMP^{-1}\in\{S_0,S_1\}$. + - Existe-t-il $Q\in\text{GL}_2(\Z)$ telle que $S_1=QS_0Q^{-1}$? +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1199] + - Rappeler le theoreme de Cayley-Hamilton et le prouver dans le cas diagonalisable. Soient $A,B\in\M_n(\C)$ telles que $\op{Sp}(A)\cap\op{Sp}(B)=\emptyset$. + - Montrere que $\chi_A(B)$ et $\chi_B(A)$ sont inversibles. + - Montrere que, pour toute matrice $C\in\M_n(\C)$, il existe une unique matrice $D\in\M_n(\C)$ telle que $AD-DB=C$. + - Soit $C\in\M_n(\C)$. + - Montrere que les matrices $\left(\begin{matrix}A&C\\ 0&B\end{matrix}\right)$ et $\left(\begin{matrix}A&0\\ 0&B\end{matrix}\right)$ sont semblables. + - En déduire une condition nécessaire et suffisante sur les matrices $A$ et $B$ pour que $\left(\begin{matrix}A&C\\ 0&B\end{matrix}\right)$ soit diagonalisable. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1200] + - Rappeler les définitions de morphisme de groupes et d'ordre d'un élément. + +On appelle representation de degre $n$ un morphisme de groupes de $\mc{S}_3$ dans $\text{GL}_n(\C)$. + - Soit $f$ une representation de $\mc{S}_3$. Soit $\sigma$ un élément de $\mc{S}_3$. Montrer que $f(\sigma)$ est diagonalisable. Montrer que l'image de $f$ est entierement déterminée par l'image de la transposition $(1\ 2)$ et du cycle $(1\ 2\ 3)$. + - Donner les representations de degre $1$. + - Donner un exemple de representation de degre $3$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1202] +Soit $M\in\M_n(\R)$ à coefficients positifs et telle que la somme des coefficients sur chaque ligne vaut $1$. + - Montrere que $1$ est valeur propre de $M$ puis que toute valeur propre complexe de $M$ vérifie $|\lambda|\leq 1$. + - On suppose que tous les coefficients diagonaux de $M$ sont strictement positifs. Montrer que $1$ est la seule valeur propre de $M$ de module $1$. + - Montrere que $\op{Ker}(M-I_n)^2=\op{Ker}(M-I_n)$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1203] +Soit $A\in\M_n(\C)$. On designe par $\mu_A$ son polynôme minimal. + - Montrere que tout idéal de $\C[X]$ est de la forme $P\C[X]$, ou $P\in\C[X]$. + - Pour $x\in\M_{n,1}(\C)$ non nul, on note $\mu_{A,x}$ le generateur unitaire de l'idéal annulateur ponctuel $\{P\in\C[X],\ P(A)x=0\}$. Montrer qu'il existe $x\in\M_{n,1}(\C)$ tel que $\mu_{A,x}=\mu_A$. - Soit $A$ une matrice diagonale par blocs dont la diagonale vaut $(A_1,A_2)$ ou $A_1$ et $A_2$ sont des matrices de Frobenius (compagnon) et $\chi_{A_1}\wedge\chi_{A_2}=1$. Montrer que $A$ est semblable à une matrice de Frobenius. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1204] + - Définir l'exponentielle d'une matrice de $\M_n(\C)$. + +Pour $P\in\mathrm{GL}_n(\C)$, montrer que $\exp(P^{-1}AP)=P^{-1}\exp(A)P$. + - Soit $B\in\mathrm{GL}_n(\C)$ diagonalisable. Montrer qu'il existe $A\in\M_n(\C)$ telle que $B=\exp(A)$. + - Montrer qu'il existe $P\in\C[X]$ tel que $B=\exp\bigl(P(B)\bigr)$. + - Soit $M\in\mathrm{GL}_n(\C)$. On suppose que $M=I_n+A$ avec $A$ nilpotente. Montrer qu'il existe $P\in\C[X]$ tel que $M=\exp\bigl(P(M)\bigr)$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1205] + - Justifier la définition de l'exponentielle de matrice. + - Calculer $\exp(A)$ pour $A=\begin{pmatrix}0&t\\ -t&0\end{pmatrix}$ et $t\in\R$. + - Considérons une matrice $A$ diagonalisable. Calculer $\exp(A)$ en utilisant des matrices de passage. Montrer que $\exp(A)\in\mathbb{K}[A]$. + - Dans cette question, on admet l'existence et l'unicité de la décomposition de Jordan-Dunford. En notant $D+N$ la décomposition de Jordan-Dunford de $A$, montrer que la décomposition de $\exp(A)$ est $\exp(D)+\exp(D)(\exp(N)-I_n)$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1206] +Soit $E$ un espace vectoriel euclidien de dimension $n$. + - Montrer que, pour tout hyperplan $H$ de $E$, il existe $a\in E$ tel que $H=\mathrm{Vect}(a)^{\perp}$. + - Soit $(x_0,\ldots,x_n)$ une famille de vecteurs unitaires de $E$ tels que $\langle x_i,x_j\rangle=\alpha$ pour tous $i\neq j$, ou $\alpha$ est un réel strictement negatif fixe. Déterminer $\alpha$. + - Montrer l'existence d'une telle famille. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1207] +Soient $n\in\N\setminus\{0,1\}$ et $(E,\langle\,\ \rangle)$ un espace euclidien de dimension $n$. On considére une base orthonormale $\mc{B}=(e_1,\ldots,e_n)$ et deux familles $(x_1,\ldots,x_n)$ et $(y_1,\ldots,y_n)$ d'éléments de $E$. On note respectivement $A$ et $B$ les matrices representatives des familles précédentes dans la base $\mc{B}$. + - Exprimer les coordonnées et la norme d'un vecteur $x$ de $E$ à l'aide des éléments de $\mc{B}$. + - Explorer les coefficients de $A$, $B$ et $A^TB$ à l'aide de produits scalaires. + - On suppose ici que $(x_1,\ldots,x_n)$ est une base de $E$. D'eduire de la question précédente que l'on peut choisir $y_1,\ldots,y_n$ de telle sorte que $\langle x_i,y_j\rangle=\delta_{i,j}$. Montrer que, ainsi choisis, $(y_1,\ldots,y_n)$ est une base de $E$. + - On suppose ici que : + +(i) $\forall i\in\db{1,n}$, $\|x_i\|=1$ et $\forall i\neq j$, $\langle x_i,x_j\rangle\lt 0$, + +(ii) $\exists v\in E,\ \forall i\in\db{1,n},\ \langle x_i,v\rangle\gt 0$. + +Montrer que $(x_1,\ldots,x_n)$ est une base de $E$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1208] + - Rappeler le procede d'orthonormalisation de Gram-Schmidt. + - Soit $M\in\mathrm{GL}_n(\R)$. Montrer qu'il existe $O\in\mc{O}_n(\R)$ et $T$ triangulaire supérieure telles que $M=OT$. - Soient $M\in\M_n(\R)$, $C_1$,..., $C_n$ ses colonnes. Montrer que $|\mathrm{det}(M)|\leq\prod_{i=1}^n\lVert C_i\rVert$. - On suppose que le résultat précédent est vérifie pour des matrices dans $\M_n(\C)$ avec le produit scalaire hermitien $((x_1,\ldots,x_n),(w_1,\ldots,w_n))\mapsto\sum_{k=1}^nx_i \overline{w_i}$. + +Soit $\overline{\mathbb{D}}=\{z\in\C,\ |z|\leq 1\}$. Trouver le maximum et les points realisant le maximum de $f:(z_1,\ldots,z_n)\in\overline{\mathbb{D}}^n\mapsto\prod_{1\leq i\lt j \leq n}|z_i-z_j|$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1209] +On note $E$ l'ensemble des fonctions réelles, continues et de carre intégrable sur $\R^+$. - - Définir la notion de fonction intégrable sur $[0,+\i[$. + - Montrer que, pour $f,g\in E$, $fg$ est intégrable et en déduire que $E$ est un $\R$-espace vectoriel. + - Pour $(f,g)\in E^2$, on pose $\langle f,g\rangle=\int_0^{+\i}fg$. + - Montrer que $\langle\,\ \rangle$ est un produit scalaire sur $E$. + - Soit $f\in E$ de classe $\mc C^2$ telle que $f''\in E$. Montrer que $f'\in E$. + - Exprimer $\langle f',f'\rangle+\langle f,f''\rangle$, $\langle f,f'\rangle$, $\langle f',f''\rangle$ en fonction de $f(0)$ et $f'(0)$. + - On pose $A=\left(\begin{array}{ccc}\langle f,f\rangle&\langle f',f\rangle& \langle f'',f\rangle\\ \langle f,f'\rangle&\langle f',f'\rangle&\langle f^{ ''},f'\rangle\\ \langle f,f''\rangle&\langle f',f''\rangle& \langle f'',f''\rangle\end{array}\right)$. + +Montrer que $\mathrm{det}(A)\geq 0$ et étudier le cas d'egalite. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1210] +Soit $E$ un espace vectoriel euclidien. + - Montrer que toutes les valeurs propres d'une isométrie vectorielle de $E$ sont de module $1$. + - Soit $u\in\mc{L}(E)$ dont toutes les valeurs propres sont de module $1$ et vérifiant : $\forall x\in E$, $\lVert u(x)\rVert\leq\lVert x\rVert$. Montrer que $\mathrm{Ker}(u-\mathrm{Id}_E)$ et $\mathrm{Im}(u-\mathrm{Id}_E)$ sont en somme directe. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1211] +Pour tout $t\in\left]-1,1\right[$, on note $\omega(t)=\sqrt{\frac{1-t}{1+t}}$. Pour $(P,Q)\in\R_n[X]^2$, on pose $\langle P,Q\rangle=\int_{-1}^1P(t)Q(t)\omega(t)\dt$. + - Montrer que $\langle\,\ \rangle$ est un produit scalaire sur $\R_n[X]$. + - On pose $\phi:P\in\R_n[X]\mapsto(X^2-1)P''+(2X+1)P'$. Montrer que $\phi$ est un endomorphisme autoadjoint de $\R_n[X]$. + - Déterminer ses valeurs propres. + - Montrer qu'il existe une base de vecteurs propres de degres echelonnes. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1212] +Soit $A=(a_{i,j})_{1\leq i,j\leq n}\in\M_n(\R)$ telle que $a_{i,j}=2$ si $i=j$, $a_{i,j}=-1$ si $|i-j|=1$ ou $|i-j|=n-1$, $a_{i,j}=0$ sinon. + - Montrer que les sous-espaces propres d'une matrice symétrique sont orthogonaux. + - Montrer que $A$ est diagonalisable et que ses valeurs propres sont réelles. + - Montrer que le spectre de $A$ est inclus dans $[0,4]$. + - Lister les valeurs propres de $A$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1213] +Soit $n\in\N^*$. On note $\Omega=\{M\in\M_n(\R),\;I_n+M\in\mathrm{GL}_n(\R)\}$. + - Montrer que $(\mc{O}_n(\R),\times)$ est un groupe. + - Montrer que $\mc{A}_n(\R)\subseteq\Omega$. + +On pose $f:M\in\Omega\mapsto(I_n-M)(I_n+M)^{-1}$. + - Montrter que, pour tout $M\in\mc{O}_n(\R)\cap\Omega$, $f(M)\in\mc{A}_n(\R)$ et $f(f(M))=M$. + - Montrter que, pour tout $M\in\M_n(\R)$, il existe une matrice diagonale $J$ à coefficients diagonaux dans $\{-1,1\}$ telle que $\det(M+J)\neq 0$. + +Ind. On pourra faire une récurrence et comparer deux déterminants. + - Soit $M\in\mc{O}_n(\R)$. Montrter qu'il existe une matrice diagonale $J$ à coefficients diagonaux dans $\{-1,1\}$ et $A\in\mc{A}_n(\R)$ telles que $M=Jf(A)$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1214] +Soient $A$ une matrice réelle antisymétrique de taille $n$ et $f$ l'endomorphisme de $\R^n$ canoniquement associe. + - Enoncer le theoreme du rang. + - On suppose $A$ inversible. Montrter que $n$ est pair. + - On suppose $\R^n$ muni de son produit scalaire canonique. Que dire de $f^*$? + - Montrter que $A$ est semblable à une matrice de la forme $A'=\begin{pmatrix}0&0\\ 0&A''\end{pmatrix}$ ou $A''$ est inversible. + - En déduire que le rang de $A$ est pair. + - On suppose $n$ pair et l'on prend une autre matrice antisymétrique $B$ dans $\M_n(\R)$. Montrter que les sous-espaces propres de $AB$ sont de dimension supérieure à $2$. + +Ind. On pourra commencer par le sous-espace propre associe à la valeur propre nulle. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1215] + - Rappeler la définition d'un matrice symétrique définie positive. Caractérisation a l'aide du spectre? + - Montrter que l'exponentielle définit une bijection continue de $\mc{S}_n(\R)$ sur $\mc{S}_n^{++}(\R)$. + - Montrter que sa réciproque est continue. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1216] + - Rappeler la définition d'une matrice définie positive. Caractérisation a l'aide du spectre? + - Soit $A\in\mc{S}_n^{++}(\R)$. + - Montrter qu'il existe $R\in\mc{S}_n^{++}(\R)$ telle que $A=R^2$. + - Montrter son unicité. On la note $\sqrt{A}$. + +Ind. Considérer les sous-espaces propres de l'endomorphisme canoniquement associe à $A$. + - Soient $A$ et $B\in\mc{S}_n^{++}(\R)$. Montrter que l'équation $XA^{-1}X=B$ admet une unique solution dans $\mc{S}_n^+(\R)$ qui est : $A\#B=\sqrt{A}\sqrt{\sqrt{A^{-1}}B\sqrt{A^{-1}}}\sqrt{A}$ (moyenne geometrique de $A$ et $B$). + - Montrter les relations : $A\#A=A$, $A\#B=B\#A$, $(A\#B)^{-1}=A^{-1}\#B^{-1}$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1217] +Soient $E$ et $F$ des espaces vectoriels euclidiens de dimensions respectives $n$ et $m$. + - Soit $u\in\mc{L}(E,F)$. Montrter qu'il existe un unique $u^*\in\mc{L}(F,E)$ tel que $\forall x\in E,\forall y\in F$, $\left\langle u(x),y\right\rangle_F=\left\langle x,u^*(y)\right\rangle_E$. + - Montrter que $u^*u$ est autoadjoint positif. - Soit $M\in\M_{m,n}(\R)$ de rang $r$. Montrer qu'il existe $P\in\mc{O}_m(\R)$, $Q\in\mc{O}_n(\R)$ et $\sigma_1,\ldots,\sigma_r\in\R^{+*}$ tels que $(PMQ)_{i,i}=\sigma_i$ si $i\leq r$, les autres coefficients etant nuls. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1218] +Soit $E$ un espace euclidien. + - Soit $s\in\mc{S}^+(E)$. Montrer qu'il existe un unique $r\in\mc{S}^+(E)$ tel que $s=r^2$. + - Soit $u\in\mc{L}(E)$ tel que $\forall x\in\mathrm{Ker}(u)^{\perp}$, $\left\|u(x)\right\|=\left\|x\right\|$. + - Montrer que $\forall x,y\in\mathrm{Ker}(u)^{\perp}$, $\left\langle u(x),u(y)\right\rangle=\left\langle x,y\right\rangle$. + - Montrer que $u^*u$ est le projecteur orthogonal sur $\mathrm{Ker}(u)^{\perp}$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1219] + - Soit $M\in\mc{S}_n(\R)$. Montrer que $M\in\mc{S}_n^+(\R)$ si et seulement si $\mathrm{sp}(M)\subset\R^+$. + - Soit $M\in\mc{S}_n(\R^+)$ c'est-a-dire symétrique à coefficients positifs. Est-ce que toutes les valeurs propres de $M$ peuvent être strictement negatives? Peut-on trouver $M$ avec une unique valeur propre strictement positive? + - Soient $A\in\mc{S}_n(\R)$ et $X_1,\ldots,X_n$ une base orthonormée de vecteurs propres associes aux valeurs propres $\lambda_1\leq\cdots\leq\lambda_n$. Pour $\alpha\in\R$ on pose $B(\alpha)=\left(\begin{array}{cc}A&\alpha X_n\\ \alpha X_n^T&0\end{array}\right)$. Montrer que $\lambda_1,\ldots,\lambda_{n-1}$ sont valeurs propres de $B(\alpha)$ et exprimer les deux restantes en fonction de $\lambda_n$ et $\alpha$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1220] +Soient $E$ un espace euclidien et $u,v$ dans $\mc{S}(E)$ avec $u\in\mc{S}^{++}(E)$. + - Caractériser spectralement le fait que $u\in\mc{S}^{++}(E)$. + - Montrer qu'il existe un unique $w\in\mc{L}(E)$ tel que $w\circ u+u\circ w=v$ puis que $w^*=w$. + - A-t-on l'équivalence $v\in\mc{S}^{++}(E)\Leftrightarrow w\in\mc{S}^{++}(E)$? +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1221] + - Soit $M\in\mc{S}_d\left(\R\right)$. Montrer que le spectre de $M$ est inclus dans $\R^+$ si et seulement si $\forall x\in\R^d$, $\left\langle Mx,x\right\rangle\geq 0$. + - Soient $M_1,\ldots,M_n\in\M_d\left(\R\right)$ telles que $\sum_{i=1}^nM_i^TM_i=I_d$. On pose, pour $X\in\mc{S}_d(\R)$, + + $\mc{L}\left(X\right)=\sum_{i=1}^nM_i^TXM_i$. Montrer que $\mc{L}$ préserve le caractere symétrique positif. + - Donner $p\in\N$, $\Pi\colon\M_d\left(\R\right)\ra\M_p\left( \R\right)$ morphisme d'algèbre vérifiant $\Pi\left(X^T\right)=\Pi\left(X\right)^T$ et $V\in\M_{p,d}\left(\R\right)$ vérifiant $V^TV=I_d$ tels que $\forall X\in\M_d(\R)$, $\mc{L}\left(X\right)=V^T\Pi\left(X\right)V$. + +Pour $M,N\in\M_d\left(\R\right)$, on note $M\geq N$ si et seulement si $M-N$ est symétrique positive. + - Montrer $\mc{L}\left(X^TX\right)\geq\mc{L}\left(X^T\right)\mc{ L}\left(X\right)$. + - On suppose qu'il existe $\mc{K}$ du même type que $\mc{L}$ tel que $\mc{L}\circ\mc{K}=\mc{K}\circ\mc{L}=\mc{I}$. Montrer que : $\forall X\in\M_d(\R)$, $\mc{L}\left(X^TX\right)=\mc{L}\left(X^T\right)\mc{L}\left(X\right)$. +#+end_exercice + + +** Analyse + +#+begin_exercice [Centrale MP 2024 # 1222] + - Formuler et démontrer le cas d'egalite du theoreme des accroissements finis. On note $\mc{E}$ l'ensemble des polynômes à coefficients dans $\left\{-1,0,1\right\}$ et $A$ l'ensemble des racines réelles des polynômes de $\mc{E}$. + - Montrer que $A\setminus\left\{0\right\}$ est stable par $x\mapsto-x$ et $x\mapsto\dfrac{1}{x}$. + - Montrer que $A\cap\left]2,+\i\right[=\emptyset$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1224] +Soit $A=(a_0,\ldots,a_n)\in\N^{n+1}$ avec $a_0\lt a_1\lt \cdots\lt a_n$. Pour $P\in\R_n[X]$ on pose $\|P\|_A=\max_{0\leq k\leq n}|P(a_k)|$. On pose $d_{n,A}=\inf_{P\in\R_{n-1}[X]}\|X^n-P\|_A$. + - Montrer que $\|\ \|_A$ est une norme sur $\R_n[X]$. + - Soit $P\in\R_{n-1}[X]$. Montrer qu'il existe un unique $(n+1)$-uplet $(b_0,\ldots,b_n)$ de réels tel que $X^n-P=\sum_{k=0}^nb_k\prod_{j\neq k}(X-a_j)$. Montrer que $\sum_{k=0}^nb_k=1$. + - Montrer que, pour tout $k\in\db{0,n}$, $\prod_{j\neq k}|a_k-a_j|\geq\frac{n!}{\binom{n}{k}}$. + - Montrer que $\|X^n-P\|_A\geq\frac{n!}{2^n}$ pour tout $P\in\R_{n-1}[X]$. + - Calculer $d_{n,A}$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1225] +Soient $a\lt b$ des réels fixes. On munit l'espace $E=\mc C^0([a,b],\R)$ de la norme $\|\ \|_{\i}$. On fixe enfin un entier $n\geq 0$ et $f\in E$, et on pose $m=d(f,\R_n[X])$. + - On pose $C=\{g\in\R_n[X]\ ;\ \|f-g\|_{\i}\leq m+1\}$. Montrer que $C$ est compact et non vide. En déduire qu'il existe $p\in\R_n[X]$ tel que $m=\|f-p\|_{\i}$. + - Montrer que l'équation $|f(x)-p(x)|=m$ admet au moins $n+2$ solutions. + - En déduire que $p$ est unique. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1226] +Notons $\mc C$ l'espace des fonctions continues de $[0,1]$ dans $\R$ muni de la norme $\i$. Pour $f\in\mc C$, notons $Af(x)=\int_x^1\frac{f(t)}{\sqrt{t-x}}dt$ si $x\in[0,1[$ et $Af(1)=0$. + - Donner une condition nécessaire et suffisante sur $\alpha\in\R$ pour que l'intégrale $\int_0^1\frac{dt}{t^{\alpha}}$ soit convergente. La démontrer. + - Justifier que, pour tout $f\in\mc C$, $Af$ est correctement définie. + - Montrer que, pour tout $f\in\mc C$, $Af\in\mc C$. + - Montrer que $A$ est un endomorphisme continu de $\mc C$; calculer sa norme subordonnée. + - Étudier la dérivabilité de $Af$ pour une fonction $f:[0,1]\ra\R$ de classe $\mc C^1$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1227] + - Soient $E$ et $E'$ deux espaces vectoriels normes et $u\in\mc{L}(E,E')$. + +Montrer que $u$ est continue sur $E$ si et seulement si elle est continue en $0$. + +On considére desormais l'espace $E=\mc C^1([-1,1],\R)$ muni de la norme infinie. + +Pour $\phi$ forme lineaire sur $E$, on pose $N(\phi)=\sup\{|\phi(f)|,\ f\in S_{\|\ \|_{\i}}(0,1)\}\in[0,+\i]$. + - Calculer $N(\phi)$ avec $\phi:f\mapsto\int_{-1}^0f-\int_0^1f$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1228] +Soit $(\lambda_n)$ une suite de réels positifs strictement croissante telle que $\lambda_0=0$, $\lambda_n\ra+\i$ et la série de terme general $\frac{1}{\lambda_n}$ diverge. Pour $m\in\N^*$ fixe, on pose $Q_0:x\mapsto x^m$ et, pour tout $n$, + +$$Q_{n+1}:x\mapsto(\lambda_{n+1}-m)\,x^{\lambda_{n+1}}\int_x^1Q_n(t)\,t^{-( 1+\lambda_{n+1})}dt.$$ + - Rappeler le theoreme de Weierstrass. + - Montrer que la suite $(Q_n)$ est bornée sur $[0,1]$ et que, pour tout $n$, $\|Q_n\|_{\i}\leq\prod_{j=1}^n\left|1-\frac{m}{\lambda_j}\right|$. + +En déduire que $(Q_n)$ converge uniformément vers la fonction nulle sur $[0,1]$. + - Montrer que, toute fonction continue sur $[0,1]$ est limite uniforme d'une suite de fonctions appartenant à $\mathrm{Vect}\left\{x\mapsto x^{\lambda_n},\ n\in\N\right\}$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1229] + - Enoncer et démontrer le theoreme des bornes atteintes. + +Soit $C$ une partie convexe compacte non vide d'un espace euclidien $E$. + - Soit $x\in E$. + - Montrer l'existence et unicité d'un vecteur $p(x)\in C$ tel que $d(x,C)=\|x-p(x)\|$. + - Soit $y\in C$. Montrer que $y=p(x)$ si et seulement si $\forall c\in C$, $\langle x-p(x),c-p(x)\rangle\leq 0$. + - Montrer que l'application $p$ définie dans ce qui precede est continue. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1230] +Si $A\in\M_n(\C)$, on pose $\rho(A)=\{|\lambda|\ ;\ \lambda\in\text{Sp}(A)\}$. On munit $\C^n$ d'une norme $\|\ \|$ et $\M_n(\C)$ de la norme $\|\ \|$ d'operateur associe. + - L'application $A\mapsto\rho(A)$ est-elle une norme? + - Soit $A\in\M_n(\C)$. Montrer que, pour tout $k\in\N^*$, $\rho(A)\leq\|A^k\|^{1/k}$. + - Montrer que, pour toute norme $N$ sur $\M_n(\C)$, $N(A^k)^{1/k}\ra\rho(A)$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1231] +On note $E$ l'espace vectoriel des fonctions continues de $\R$ dans $\R$ et de limite nulle en $\pm\i$. On munit $E$ de la norme $\|\ \|_{\i}$ et on définit $T(f)$ pour tout $f\in E$ par : + +$$\forall x\in\R,\,T(f)(x)=\frac{1}{2}\int_{\R}\mathrm{e}^{-|x-t| }f(t)dt$$ + - Rappeler le theoreme de Heine. + - Montrer que $f$ est uniformément continue. + - Montrer que $T\in\mc{L}(E)$ puis que $T$ est continu. + - Déterminer sa norme d'operateur. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1232] +Soient $A,B\in\M_p(\mathbb{K})$. Pour $A\in\M_p(\mathbb{K})$, on pose $\|M\|=\max_{1\leq i\leq p}\sum_{j=1}^p|a_{i,j}|$. + - Montrer que $\|\ \|$ est une norme et que $\forall A,B\in\M_p(\mathbb{K})$, $\|AB\|\leq\|A\|\cdot\|B\|$. + - Définir $\exp(A)$ et montrer que $\|\exp(A)\|\leq\exp(\|A\|)$. + - On pose $K=\max(\|A\|,\|B\|)$. Montrer : $\forall n\in\N$, $\|A^n-B^n\|\leq nK^{n-1}\|A-B\|$. + - Trouver $\lim_{n\ra+\i}\left(\mathrm{e}^{A/n}\mathrm{e}^{B/n}\right)^n$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1233] + - Rappeler la définition de la borne inférieure d'une partie non vide de $\R$. - On note $X$ une partie non vide minorée de $\R$. Montrer qu'il existe une suite de $X$ convergent vers la borne inférieure de $X$. Réciproquement, prouver que si une suite de $X$ converge vers un minorant $m$ de $X$, alors $m$ est la borne inférieure de $X$. + - Soit $n\in\N^*$. On pose $S_{\alpha}=\{M\in\mc{S}_n^+(\R),\ \det(M)\geq\alpha\}$ pour $\alpha\gt 0$. On souhaite prouver que, si $A\in\mc{S}_n^+(\R)$, $\inf\limits_{M\in S_{\alpha}}\op{tr}(AM)=n(\alpha\det(A))^{1/n}$. Prouver ce résultat lorsque $A=I_n$ puis lorsque $A\in\mc{S}_n^+(\R)$. + - Est-ce toujours le cas lorsque $\alpha=0$? +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1234] +On note $\mc{A}$ l'ensemble des matrices de $\M_n(\R)$ à coefficients dans $[-1,1]$. + - Montrer la continuité du déterminant sur $\M_n(\R)$. + - Montrer que le déterminant admet un maximum $\alpha$ sur $\mc{A}$. + - Montrer que le maximum est atteint en une matrice inversible $A$ de déterminant strictement positif et à coefficients dans $\{-1,1\}$. + - Montrer que $\alpha\leq n^{n/2}$ avec egalite si et seulement si les colonnes de $A$ sont deux à deux orthogonales pour le produit scalaire euclidien canonique de $\R^n$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1235] + - Montrer que les parties connexes par arcs de $\R$ sont ses convexes. + - Soit $n\in\N^*$. On note $\Gamma_n$ l'ensemble des matrices carrées de taille $n$ à coefficients dans $\{0,1\}$. Justifier l'existence de $a_n=\max\limits_{M\in\Gamma_n}\det(M)$ et étudier son comportement quand $n\ra+\i$. + - On note $C_n$ l'ensemble des matrices carrées de taille $n$ à coefficients dans $[0,1]$. + +Montrer que $\det(C_n)=[-a_n,a_n]$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1236] +Soient $d\in\N^*$ et $(\omega_n)\in\C^{\N^*}$ une suite $d$-periodique. + +Pour $n\in\N^*$ et $\lambda\in\C$, on pose $S_n(\lambda)=\sum\limits_{k=1}^n\frac{\lambda+\omega_k}{k}$. + - Soit $(u_n)\in\C^{\N}$. Montrer que, si $\sum u_n$ converge, alors $u_n\longrightarrow 0$ quand $n\ra+\i$. + +La réciproque est-elle vraie? + - Montrer qu'il existe au plus un complexe $\lambda$ tel que la suite $(S_n(\lambda))_{n\in\N^*}$ converge. + - Montrer l'existence de $\Omega,\alpha\in\C$ tels que $S_{(m+1)d}(0)-S_{md}(0)=\frac{\Omega}{md}+\frac{\alpha}{m^2}+o\left(\frac{1 }{m^2}\right)$ quand $m\ra+\i$. + - Donner une condition nécessaire et suffisante sur $\lambda\in\C$ pour que la suite $(S_n(\lambda))$ converge. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1237] +Soit $(u_n)$ une suite réelle strictement positive telle que $\frac{u_{n+1}}{u_n}=1-\frac{\alpha}{n}+v_n$ ou $\sum|v_n|$ converge. + - Étudier la convergence de $a_{n+1}-a_n$ ou $a_n=\ln(n^{\alpha}u_n)$. En déduire qu'il existe $K\gt 0$ tel que $u_n\sim\frac{K}{n^{\alpha}}$. + - On prend $u_n=n^{-n}n!e^n$. Montrer qu'il existe $K\gt 0$ tel que $u_n\sim K\sqrt{n}$. + - On suppose maintenant que $\frac{u_{n+1}}{u_n}=1-\frac{\alpha}{n}+o\left(\frac{1}{n}\right)$. Montrer que si $\alpha\gt 1$ alors la série $\sum u_n$ converge, et que si $\alpha\lt 1$ alors elle diverge. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1238] +_Pour $n\in\N^*$, on pose $H_n=\sum_{k=1}^n\frac{1}{k}$ et $d_n=\op{card}\{p\in\db{1,n}\;;\;p\mid n\}$._ + +Pour $x\in\R^+$, on définit $f(x)=\sum_{1\leq k\leq x}d_k$. + -_Cours : comparaison série-intégrale. L'utiliser pour montrer l'équivalent $H_n\sim\ln n$._ + -_Déterminer un équivalent de $f$ en $+\i$._ + -_Déterminer le deuxieme terme du développement asymptotique de $f$._ +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1239] + -_Enoncer le theoreme de Rolle._ + -_Soient $a,b\in\R\cup\{\pm\i\}$ tels que $a\lt b$. Montrer que le theoreme reste vrai pour $f\colon\,]a,b[\ra\R$ dérivable et admettant en $a$ et $b$ une même limite finie._ + -_On définit la fonction $f:]-1,1[\ra\R,x\mapsto\exp\left(-\frac{1}{1-x^2}\right)$._ + -_Montrer que $f$ est $\mc C^{\i}$ et que, pour tout $n\in\N$, il existe un polynôme $P_n\in\R[X]$ tel que $f^{(n)}(x)=\frac{P_n(x)}{(1-x^2)^{2n}}f(x)$ pour tout $x\in]-1,1[$._ + -_Quel est le degre de $P_n$?_ + -_Que dire du nombre de zeros de $f^{(n)}$?_ +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1240] +_Soit $A\subset\R^n$. On note $\mc C(A,\R)$ l'ensemble des fonctions continues de $A$ dans $\R$ et $\mc{UC}(A,\R)$ l'ensemble des fonctions uniformément continues de $A$ dans $\R$._ + -_Pour $n=1$ et $A$ un segment, montrer que $f\in\mc C(A,\R)$ si et seulement si $f\in\mc{UC}(A,\R)$._ + -_Montrer que $\mc{UC}(A,\R)$ est stable par composition. Est-il stable par produit?_ + -_Soit $T\in\R^{+*}$ et $f$ une fonction continue et $T$-periodique. Montrer que $f\in\mc{UC}(A,\R)$._ +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1241] +_Soient $E$ l'espace vectoriel des suites réelles et $\Delta$ l'endomorphisme de $E$ défini par : pour $u\in E$ et tout $n\in\N$, $[\Delta(u)]_n=u_{n+1}-u_n$._ + -_Démontrer le theoreme des accroissements finis._ + -_Soit $f\colon\R^+\mapsto\R$ de classe $\mc C^{\i}$. On pose, pour tout $n\in\N$, $u_n=f(n)$. Montrer que, pour tout $n\in\N$ et tout $p\in\N^*$, il existe $x\in]n,n+p[$ tel que $[\Delta^pu]_n=f^{(p)}(x)$._ +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1242] + -_Soient $I$ un intervalle non vide et $f\in\mc C^0(I,\R)$. Montrer que, pour tout $a\in I$, l'application $F_a:x\mapsto\int_a^xf(t)dt$ est dérivable, de derivée $f$._ + +Pour $h\gt 0$, soit $W_h=\bigg{\{}f\in\mc C^0(\R,\R)\;;\;\forall x\in \R,\;\int_{x+h}^{x+2h}f(t)dt=2\int_x^{x+h}f(t)dt \bigg{\}}$._ + -_Montrer que, pour tout $h\gt 0$, $W_h$ est un espace vectoriel de dimension infinie._ + -_Existe-t-il des fonctions non bornées appartenant à $W_h$?_ +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1243] +_On pose $E=\mc C^{\i}(\R,\R)$.__Pour $f\in E$, on définit la suite $(f_n)$ de fonctions de $E$ par $f_0=f$ et $\forall n\in\N$, $\forall x\in\R$, $f_{n+1}(x)=\int_0^xtf_n(t)dt$._ + -_Enoncer le theoreme de derivation terme à terme._ + -_On se place dans le cas ou $f$ est constante. Montrer que la suite $(f_n)$ et la série $\sum f_n$ convergent simplement sur $\R$. Y a-t-il convergence uniforme?_ - On revient au cas general. + - Montrer la convergence simple de la suite $(f_n)$ et de la série $\sum f_n$. + - Montrer que l'application $T:f\in E\mapsto\sum_{n=0}^{+\i}f_n$ est un automorphisme de l'espace vectoriel $E$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1244] +Pour $x\geq 0$ et $n\in\N^*$, on définit $g_n(x)=\dfrac{1}{(1+x)\cdots(1+x^n)}$. + - Étudier la convergence simple de $(g_n)$ sur $\R^+$. + - Étudier la convergence uniforme de $(g_n)$ sur $[1,+\i[$ et sur tout segment. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1245] +Pour $n\in\N^*$, soient $\Omega(n)$ le nombre de facteurs premiers de $n$ comptes avec multiplicité, $\lambda(n)=(-1)^{\Omega(n)}$, $\Lambda(n)=\sum_{d|n}\lambda(d)$. + - Montrer que, si $m$ et $n$ sont deux éléments de $\N^*$ premiers entre eux, $\lambda(mn)=\lambda(m)\lambda(n)$ et $\Lambda(mn)=\Lambda(m)\Lambda(n)$. + - Pour $n\in\N^*$, donner une expression simple de $\Lambda(n)$. + - Montrer que, si $|z|\lt 1$, $\sum_{n=1}^{+\i}\dfrac{\lambda(n)z^n}{1-z^n}=\sum_{n=1}^{+ \i}z^{n^2}$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1246] + - Pour $x\in\R\setminus\Z$, montrer que la suite $\left(\sum_{k=-n}^n\dfrac{1}{x-k}\right)_{n\geq 1}$ converge. On note $f(x)$ sa limite. + - Montrer que $f$ est continue et $1$-periodique sur $\R\setminus\Z$. + - Pour $x\in\R\setminus\Z$, exprimer $f\left(\dfrac{x}{2}\right)+f\left(\dfrac{x+1}{2}\right)$ en fonction de $f(x)$. + - Montrer que, pour tout $x\in\R\setminus\Z$, $f(x)=\pi\op{cotan}(\pi x)$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1247] + - Retrouver le développement en série entiere de la fonction $\arctan$ et montrer que : $\sum_{k=0}^{+\i}\dfrac{(-1)^k}{2k+1}=\dfrac{\pi}{4}$. + - Pour $n\in\N^*$, on pose $S_n=4\sum_{k=0}^n\dfrac{(-1)^k}{2k+1}$. + +Montrer que $\left|\pi-S_n+(-1)^n\left(\dfrac{1}{n}-\dfrac{1}{n^2}\right)\right| \leq\dfrac{1}{n^3}$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1248] + - Soit $\alpha\in\R$. Donner $R\gt 0$ tel que : $\forall x\in\left]-R,R[\,,\ (1+x)^{\alpha}=\sum_{n=0}^{+\i}\binom{\alpha}{n}x^n$. Que vaut $\binom{\alpha}{n}$? - Soit $\beta\gt 0$. Montrer que $p_n=\prod_{k=1}^n\left(1-\frac{\beta}{k}\right)$ tend vers $0$ quand $n$ tend vers l'infini. + - Soit $(\alpha,\alpha')\in\R^2$. Montrer : $\forall n\in\N,\ \left(\begin{matrix}\alpha+{\alpha'}'\\ n\end{matrix}\right)=\sum_{\scriptsize{(p,q)\in\N^2\atop p+q=n}} \left(\begin{matrix}\alpha\\ p\end{matrix}\right)\left(\begin{matrix}\alpha'\\ q\end{matrix}\right)$. + - Soit $0\lt x\lt y$. Montrer que $(x+y)^{\alpha}=\sum_{n=0}^{+\i}\left(\begin{matrix}\alpha\\ n\end{matrix}\right)$ $x^ny^{\alpha-n}$. +#+end_exercice + + - Montrer que $2^{\alpha}=\sum_{n=0}^{+\i}\left(\begin{matrix}\alpha\\ n\end{matrix}\right)$ pour tout $\alpha\gt -1$. +#+begin_exercice [Centrale MP 2024 # 1249] + - Soit $\sum a_nz^n$ une série entiere qui converge sur $]-\alpha,\alpha[$, avec $\alpha\gt 0$. Montrer que sa somme est de classe $\mc C^{\i}$ sur $]-\alpha,\alpha[$. + - Est-ce que toute fonction de classe $\mc C^{\i}$ sur un ouvert contenant $0$ est développable en série entiere au voisinage de $0$? + - Soit $f$ une fonction de classe $\mc C^{\i}$ sur un ouvert contenant $0$. Montrer qu'elle est développable en série entiere au voisinage de $0$ si et seulement si : +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1250] +Soient $R\gt 0$ et $\mc{A}_R$ l'ensemble des fonctions $f\colon\R\ra\R$ développables en série entiere de rayon $\geq R$. + - Montrer que $\mc{A}_R$ est une $\R$-algèbre pour des lois que l'on precisera. + - Déterminer les morphismes d'algèbre de $\R[X]$ dans $\R$. + - Soit $\Phi$ un morphisme d'algèbre de $\R[X]$ dans $\R$. On dit que $\delta$ est une $\Phi$-derivation si $\delta$ est un endomorphisme de $\R[X]$ et si : $\forall P,Q\in\R[X]$, $\delta(PQ)=\Phi(P)\delta(Q)+\Phi(Q)\delta(P)$. Déterminer les $\Phi$-derivations. +#+end_exercice + + - Déterminer les morphismes d'algèbres $\Phi$ de $\mc{A}_R$ dans $\R$, puis les $\Phi$-derivations de $\mc{A}_R$. +#+begin_exercice [Centrale MP 2024 # 1251] + - Montrer que la fonction $f\colon\R\ra\R,x\mapsto\left\{\begin{array}{l}e^{-1/x^2}\ \text{ si }x\neq 0\\ 0\ \text{si }x=0\end{array}$. est de classe $\mc C^{\i}$. + +Est-elle développable en série entiere au voisinage de $0$? + - Soit $f\colon\R\ra\R$ une fonction $\mc C^{\i}$ telle qu'il existe $C,a,\delta\gt 0$ vérifiant $|f^{(n)}(x)|\leq Ca^nn!$ pour tous $n\geq 0$ et $x\in[-\delta,\delta]$. Montrer que $f$ est développable en série entiere au voisinage de $0$. +#+end_exercice + + - Étudier la réciproque. +#+begin_exercice [Centrale MP 2024 # 1252] + +#+end_exercice + +Soit $f$ une fonction développable en série entiere au voisinage de $0$, telle que $f(0)\neq 0$. Montrer que la fonction $\frac{1}{f}$ est développable en série entiere au voisinage de $0$. +#+begin_exercice [Centrale MP 2024 # 1253] +Soit $f:t\in[0,\pi/2[\mapsto-\ln(\cos(t))$. + - Montrer que $f(t)\geq t^2/2$ pour tout $t\in[0,\pi/2[$. + - Soit $\alpha\in\R^+$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1256] +On munit $\M_n(\R)$ de la norme euclidienne canonique. Soient $A\in\mc{S}_n^+(\R)$ et $B\in\M_n(\R)$. On s'interesse à l'équation différentielle $(E):X'=-AX+B$. On suppose que l'ensemble $S=\big{\{}U\in\M_n(\R)\,;\;AU=B\big{\}}$ est non vide. + - Montrer que les valeurs propres de $A$ sont positives. + - Quelles sont les solutions constantes de $(E)$? + - Soient $X$ et $Y$ deux solutions de $(E)$. Montrer que $t\mapsto\|X(t)-Y(t)\|$ est decroissante. En déduire que toute solution est bornée sur $\R^+$. + - Soit $X$ une solution de $(E)$. Montrer que $X(t)$ admet une limite $X_{\i}$ quand $t$ tend vers $+\i$. + - Montrer que $\|X(0)-X_{\i}\|=\inf_{U\in S}\|X(0)-U\|$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1257] + - Montrer que toute série numerique absolument convergente est convergente. + +On définit $s(z)=\frac{e^{iz}-e^{-iz}}{2i}$ pour tout complexe $z$ et $\phi(z)=|s(z)|$. + - Est-ce que $s$ est bornée sur $\C$? Le cas echeant, donner un majorant de $\phi$. + - Memes questions sur $D=\{z\in\C\,;\,|z|\leq 1\}$. + - Montrer que $\phi$ admet deux extrema sur $D$ et trouver les points ou ils sont attentions. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1258] +Soit $f:A\in{\cal M}_n({\R})\mapsto A^TA$. + - Montrer que $f$ est de classe ${\cal C}^1$, et calculer sa différentielle. + - Pour $A\in{\cal M}_n({\R})$, calculer $\dim\op{Ker}\op{d}\!f(A)$ en fonction du rang de $A$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1259] +Soient $A\in{\cal S}^{++}_n({\R})$, $b\in{\R}^n$ et $f$ la fonction de ${\R}^n$ dans ${\R}$ telle que, pour tout $x\in{\R}^n$, + + $f(x)=\frac{1}{2}x^TAx-b^Tx$. + - Justifier que $f$ est de classe ${\cal C}^1$ sur ${\R}^n$. Pour $x\in{\R}^n$, calculer $\nabla f(x)$. + - Montrer que $f(x)\underset{\|x\|\ra+\i}{\longrightarrow}+\i$ et montrer que $f(\omega)=\min\{f(x)\;;\;x\in{\R}^n\}$. + - Soit $\gamma\in{\R}^{+*}$ et $(x_j)_{j\geq 0}$ une suite telle que, pour tout $j\in{\N}$, $x_{j+1}=x_j-\gamma\nabla f(x_j)$. Montrer que, pour $j\in{\N}$, $x_{j+1}-\omega=(I_n-\gamma A)(x_j-\omega)$. + - Montrer que, pour $\gamma$ bien choisi, $(x_j)_{j\geq 0}$ converge vers $\omega$ indépendamment du choix de $x_0$. Comment choisir $\gamma$ pour que la vitesse de convergence soit la meilleure possible? +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1260] + - Soit $G$ un ensemble non vide. Rappeler les conditions sur la loi $*$ pour que $(G,*)$ soit un groupe. + - Rappeler la définition de la différentielle d'une application en un point. Faire le lien avec les derivées partielles dans le cas ${\cal C}^1$. + - Soit $*$ une loi de groupe sur ${\R}$, d'élément neutre note $e$. On suppose que $f:(x,y)\mapsto x*y$ est de classe ${\cal C}^1$ sur ${\R}^2$. Montrer que, pour tout $(x,y)\in{\R}^2$, $\partial_2f(x*y,e)=\partial_2f(x,y)\times\partial_2f(y,e)$. En déduire que, pour tout $y\in{\R}$, $\partial_2f(y,e)\gt 0$. + - Montrer qu'il existe un ${\cal C}^1$-diffeomorphisme $\phi$ de ${\R}$ sur ${\R}$ tel que $\phi(x*y)=\phi(x)+\phi(y)$ pour tout $(x,y)\in{\R}^2$. +#+end_exercice + + +** Geometrie + +#+begin_exercice [Centrale MP 2024 # 1261] + - Soit $f:{\R}^2\ra{\R}$ différentiable. On suppose que $f$ admet un extremum en $a\in{\R}^n$. Rappeler la valeur de $\nabla f(a)$ (avec demonstration). + - Soit $\theta\in[0,\pi]$. Soient $A$ et $B$ du cercle unite de ${\R}^2$ tels que $\widehat{(OA,OB)}(=\theta$. Exprimer l'aire de la_lunule_ constituée des points exterieurs au disque unite et interieurs au disque de diamêtre $[AB]$. + - Soient $A$, $B$ et $C$ trois points du cercle unite tels que les trois angles $(\oa{OA}, \oa{OB})$, $(\oa{OB}, \oa{OC})$ et $(\oa{OC}, \oa{OA})$ soient dans $[0,\pi]$. Maximiser la somme des aires des trois lunules qu'ils définissent. +#+end_exercice + +#+BEGIN_exercice [Centrale MP 2024 # 1262] +Soient $X,Y$ des variables aléatoires indépendantes de même loi géométrique de paramètre $p\in \interval]{0, 1}[$. + - Déterminer la loi de $\min (X,Y)$. + - Montrer que $\min (X,Y)$ et $X-Y$ sont indépendantes. +#+END_exercice + + +#+BEGIN_exercice [Centrale MP 2024 # 1263] + - Soit $u$ un endomorphisme de $\C^n$. Montrer que $u$ est diagonalisable si et seulement si $u$ admet un polynôme annulateur scindé à racines simples. + - Soient $A=\begin{pmatrix}1&-2\\ -2&1\end{pmatrix}$, $\eps_1$, $\eps_2$ deux variables aléatoires indépendantes de loi geometrique de paramêtre $p\in]0,1[$ et $Q=\eps_1X+\eps_2$. Déterminer la probabilité que $Q(A)$ soit inversible. + - Soit $u$ un endomorphisme de $\C^n$ et $Q\in\C[X]$ tel que $Q(u)$ soit diagonalisable et $Q'(u)$ inversible. Montrer que $u$ est diagonalisable. +#+END_exercice + + +#+begin_exercice [Centrale MP 2024 # 1264] +La fonction de repartition d'une variable aléatoire réelle $X$ est $F_X:t\mapsto\mathbf{P}(X\leq t)$. + - Montruer que, pour une variable aléatoire $X$, $F_X$ est croissante de limite $1$ en $+\i$. + +Soient $E$ un ensemble dénombrable de $\R$, $(X_n)$ une suite de variables aléatoires à valeurs dans $E$ et $X$ une variable à valeurs dans $E$. On suppose que, pour tout $x\in E$, $\mathbf{P}(X_n=x)\ra\mathbf{P}(X=x)$. + - Montruer que $\sum_{x\in E}|\mathbf{P}(X_n=x)-\mathbf{P}(X=x)|\ra 0$. + +Ind. Pour $\eps\gt 0$ fixe, considérer une partie finie $I\subset E$ telle que $\mathbf{P}(X\in I)\gt 1-\eps$. + - Montruer que $(F_{X_n})$ converge uniformément vers $F_X$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1265] + - Soient $p$ un réel $\gt 1$ et $q=\dfrac{p}{p-1}$. + - Montruer que $xy\leq\dfrac{x^p}{p}+\dfrac{y^q}{q}$ pour tous $x,y\in\R^+$. + - Soit $(X,Y)$ un couple de variables aléatoires. On suppose que $X\in L^p$ et $Y\in L^q$. Montrer que $XY\in L^1$ et que $\mathbf{E}(|XY|)\leq\mathbf{E}(X^p)^{1/p}\mathbf{E}(Y^q)^{1/q}$. + - Soient maintenant deux réels tels que $1\leq p\lt q$. Montrer que si $X\in L^q$, alors $X\in L^p$ et que $\mathbf{E}(X^p)^{1/p}\leq\mathbf{E}(X^q)^{1/q}$. + - Soient $(\eps_n)_{n\geq 1}$ une suite de variables de Rademacher indépendantes et $p$ un réel $\geq 1$. Montrer que, si $X\in\text{Vect}(\eps_n)_{n\geq 1}$, alors $\mathbf{E}(X^p)^{1/p}\leq C\sqrt{p}\,\mathbf{E}(X^2)^{1/2}$, ou $C$ est une constante absolue. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1266] + - En utilisant une comparaison série-intégrale, dont on rappellera le principe, donner un équivalent de $S_n=\sum_{k=1}^n\dfrac{1}{k}$. + - On dit que $n\in\N^*$ est sans facteur carre s'il n'existe pas de $k\geq 2$ tel que $k^2$ divise $n$. Montrer que pour tout $i\geq 1$, $i$ s'écrit d'une unique maniere sous la forme $i=ma^2$, ou $a\in\N^*$ et $m\in\N^*$ est sans facteur carre. + - Soient $X,Y,Z$ trois variables aléatoires indépendantes suivant la loi uniforme sur $\db{1,n}$. On pose $M=\begin{pmatrix}X&Y\\ Z&X\end{pmatrix}$. Soit $p_n$ la probabilité que $M$ ne soit pas inversible. Montrer que $p_n=O\left(\dfrac{\ln n}{n^2}\right)$. +#+end_exercice + + +#+begin_exercice [Centrale MP 2024 # 1267] +Une suite $(Z_n)_{n\geq 1}$ de variables aléatoires entieres est dite transiente si, pour toute partie bornée $A$ de $\Z$, $\sum_{n=1}^{+\i}\mathbf{P}(Z_n\in A)\lt +\i$. - Soient $\alpha\in\R^{+*}$, $(X_i)_{i\geq 1}$ une suite de variables aléatoires indépendantes telles que, pour tout $i\in\N^*$, $X_i\sim\mc{P}\left(\frac{\alpha}{i}\right)$. Pour $n\in\N^*$, quelle est la loi de $Y_n=\sum_{i=1}^nX_i$? La suite $(Y_n)_{n\geq 1}$ est-elle transiente? + - Soient $p\in]0,1[$ et $(R_i)_{i\geq 1}$ une suite i.i.d. de variables aléatoires telles que, pour tout $i\in\N^*$, $\mathbf{P}(X_i=1)=p,\mathbf{P}(X_i=-1)=1-p$. Pour $n\in\N^*$, soit $S_n=\sum_{i=1}^nX_i$. La suite $(S_n)_{n\geq 1}$ est-elle transiente? +#+end_exercice + +#+begin_exercice [Centrale MP 2024 # 1268] +Soient $p\in]0,1[$ et $q=1-p$. On suppose que $\mu=\frac{\ln 2}{|\ln q|}$ n'est pas un entier. + - Soit $X$ une variable aléatoire suivant la loi geometrique de paramêtre $p$. Montrer qu'il existe un unique entier $m$ tel que $\mathbf{P}(X\geq m)\geq\frac{1}{2}$ et $\mathbf{P}(X\leq m)\geq\frac{1}{2}$. + - Soit $(X_n)_{n\geq 1}$ une suite de variables aléatoires indépendantes suivant la loi geometrique de paramêtre $p$. On pose $Y_n=\mathbf{1}_{X_n\geq m}$ et $S_n=Y_1+\cdots+Y_{2n-1}$ pour $n\geq 1$. Montrer que $\mathbf{P}(S_n\geq n)\underset{n\ra+\i}{\longrightarrow}1$. +#+end_exercice diff --git a/Exercices 2024.pdf b/Exercices 2024.pdf new file mode 100644 index 0000000000000000000000000000000000000000..09b99419f377fed3929dbb11d338b76625d5ec82 GIT binary patch literal 1155901 zcma&NQVUZjIvAu~Y6dxayvx}3dp)Hih=8dkl^A;PL-+aA6JAA1bF$lPOOLjQhkx37;Wp8SE za%Ks9Q*DP0Ze{8@%Z~mYu$~|m3MKchtqzX~3HM4T0RG$?V0e+GQ|7|)C^6UL=j=r> z`FyRE05X^oNx3~T8A_rSc^DQ@4tY8Hv(8Gqid_X_Cd zIwZtUsL|YSP9R~7FwRn=?>HUu7hN*`?u~z0$GAhuItLFzL)`!9a=l9-UM-y?mEt0pwMt0UCorbQu&fzS@;8`9_{N z83y3~%LxNc`6C}|ItS7_eI_shwdu<1YDMfh=!h2Q3iE1cE zohw}O!evhK=!0LOG0Q<4BG>elPxofM3q_Y*YFr+TE5~P~(%D^v6>s8E(VFMJ00XjQ zoOqIV2Zh129Eo~k4r38%r{}2aybwq^tlca;0f!`{G3+2?1PD4JZ;-`*SKxlY06cjI zdfdtsY_B)^+cP{kEYs};F%+00gXC>N1L6gI>Tx5H?y?juRIP%!b@meSdkP4fZ z_>RI9HOw#sm)D7aHQcvB#R?sR#iMPzZ%H9vhrY3MV3HFhvH5bJc4q=yEcrcyzM) zIP#`u8j@?o?LiiW{98taH_-sUQf745nIW*?f=9oYNMHc>gRZEDlF%oU_PYL61*Ki{ zoO@uJSTy=1%t6rVH>t*PJ;Xo6fu;$4LC|#CjruH{g|} 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