From 61331c0db7126da80851df49d5a8793211c3a91c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?S=C3=A9bastien=20Miquel?= Date: Mon, 30 Dec 2024 15:44:37 +0100 Subject: [PATCH] Miscs --- Exercices 2022.org | 3 +- Exercices 2024.org | 1014 ++++++++++++++++++++++++------------ Exercices 2024.pdf | Bin 1155901 -> 1155199 bytes Exercices XENS MP 2024.pdf | Bin 632730 -> 715337 bytes 4 files changed, 692 insertions(+), 325 deletions(-) diff --git a/Exercices 2022.org b/Exercices 2022.org index ce114e4..c6be66b 100644 --- a/Exercices 2022.org +++ b/Exercices 2022.org @@ -1,7 +1,7 @@ #+title: Exercices 2022 #+author: Sébastien Miquel #+date: 25-02-2023 -# Time-stamp: <14-07-24 10:44> +# Time-stamp: <21-12-24 17:17> * Meta :noexport: @@ -513,6 +513,7 @@ Soit $p\geq 1$, $\K$ un sous-corps de $\C$ et $A\in\M_p(\K)$. On dit que $A$ est Pour $A$ unipotente, on pose $\ln A = \sum_{n= 1}^{+\i} \frac{(-1)^{n-1}}{n} (A-I_p)^n$. 1. Justifier la définition de $\ln A$. Montrer que $\exp$ réalise une bijection de l'ensemble des matrices nilpotentes sur les matrices unipotentes. 2. Montrer que les matrices unipotentes sont toutes puissantes. + 3. Déterminer les matrices toutes-puissantes de $\M_n(\C)$. #+END_exercice # 44 diff --git a/Exercices 2024.org b/Exercices 2024.org index f0bd924..41b461d 100644 --- a/Exercices 2024.org +++ b/Exercices 2024.org @@ -1,8 +1,8 @@ -# -*- org-export-switch: "all"; -*- +# -*- org-export-switch: "xens mp"; -*- #+title: Exercices 2024 #+author: Sébastien Miquel #+date: 20-11-2024 -# Time-stamp: <02-12-24 19:14> +# Time-stamp: <30-12-24 15:43> * Meta :noexport: @@ -14,7 +14,7 @@ #+RESULTS: | ? | ! | todo | unexed | -| 1 | 0 | 3 | 1198 | +| 2 | 3 | 10 | 1116 | ** Options @@ -24,8 +24,8 @@ *** All -#+OPTIONS: toc:t -#+export_file_name: Exercices 2024 +# #+OPTIONS: toc:t +# #+export_file_name: Exercices 2024 *** XENS @@ -34,10 +34,10 @@ *** XENS MP -# #+select_tags: xens -# #+exclude_tags: autre -# #+exclude_types: proof -# #+export_file_name: Exercices XENS MP 2024 +#+select_tags: xens +#+exclude_tags: autre +#+exclude_types: proof +#+export_file_name: Exercices XENS MP 2024 *** Centrale @@ -179,17 +179,17 @@ On considère l'équation $2^a + 3^b = 5^c$, où $(a,b,c)\in\N^3$. # Clarifier, mettre une suite, sommes de Gauss -#+BEGIN_exercice [ENS MP 2024 # 7] :todo: +#+BEGIN_exercice [ENS MP 2024 # 7] 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 $\sum_{k=0}^{p-1}\chi(k)$. + - <> 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 $N\in\N$. + - Déduire de [[ref]] 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 $S_k(N)=\sum_{n=0}^N\xi^{-kn}$. @@ -217,7 +217,7 @@ Soit $p$ un nombre premier impair. - - 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 $G_a = \chi(a) G_1$. 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 @@ -249,7 +249,7 @@ Pour $f,g\in A$, on pose $(f*g)(n)=\sum_{d\mid n}f(d)\,g(n/d)$ pour tout $n\in\N # 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 ? + - <> 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 numéro 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 @@ -587,9 +587,9 @@ Si $\la$ est valeur propre de $B$, alors $A^2 X + \la^2 X = \la AX$, donc $\vect En prenant la transposée, on a de même que si $\la$ est valeur propre de $A$, $\la e^{\pm \frac{i\pi}{3}}$ est valeur propre de $B$. Donc si $\la$ valeur propre de $A/B$ les $e^{\pm 2i\frac{\pi}{3}}\la$ le sont aussi. -On a $(A-B)^2 = - BA$, et $A^3 + B^2 A = ABA \ssi A^3 = (A-B) BA = - (A-B)^3$. Les matrices $A$ et $U = B-A$ vérifient $A^3 = U^3$. En particulier, si $\la$ est une valeur propre de $A^3$, l'espace caractéristique $F_{\la}$ associé est stable par $A$ et $U$, donc par $A$ et $B$. Sur cet espace, $A$ a comme valeurs propres des racines $3$-èmes de $\la$, idem pour $U-A$. D'après ce qui précède, les seuls valeurs propres de $B$ possibles sur cet espaces sont les $e^{\pm i\frac{\pi}{3}}$ fois celles de $A$. De plus, $A$ et $B$ ont chacun le même nombre de racines complexes conjuguées. +On a $(A-B)^2 = - BA$, et $A^3 + B^2 À = ABÀ \ssi A^3 = (A-B) BÀ = - (A-B)^3$. Les matrices $A$ et $U = B-A$ vérifient $A^3 = U^3$. En particulier, si $\la$ est une valeur propre de $A^3$, l'espace caractéristique $F_{\la}$ associé est stable par $A$ et $U$, donc par $A$ et $B$. Sur cet espace, $A$ a comme valeurs propres des racines $3$-èmes de $\la$, idem pour $U-A$. D'après ce qui précède, les seuls valeurs propres de $B$ possibles sur cet espaces sont les $e^{\pm i\frac{\pi}{3}}$ fois celles de $A$. De plus, $A$ et $B$ ont chacun le même nombre de racines complexes conjuguées. -tragnagna, ça marche pas. +ragnagna, ça marche pas. #+END_proof @@ -658,7 +658,7 @@ Reste l'étude de matrices $M = \begin{pmatrix}1 & a \\ 0 & 1\end{pmatrix}$, ave À quelle condition sur la matrice $A$, la comatrice de $A$ est-elle diagonalisable? #+end_exercice #+BEGIN_proof -Si $A$ inversible, $\op{Com} A = A^{-1}$. Si $\rg A\leq n-2$. Si $\rg A = n-1$, $\rg \op{Com} A = 1$, elle est diagonalisable si et seulement si son image n'est pas dans son noyau, mais son image est le noyau de $A$, et son noyau est l'image de $A$. +Si $A$ inversible, $\op{Com} À = A^{-1}$. Si $\rg A\leq n-2$. Si $\rg À = n-1$, $\rg \op{Com} À = 1$, elle est diagonalisable si et seulement si son image n'est pas dans son noyau, mais son image est le noyau de $A$, et son noyau est l'image de $A$. #+END_proof @@ -671,13 +671,23 @@ Pour $i\in{\N}$ et $A\in{\cal M}_n({\C})$, on note $c_i(A)$ le coefficient numé #+end_exercice +# ID:7730 #+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. + - s Calculer $S^2$. + - Donner une expression simple de $|\det(S)|$. + - 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. + - s On suppose que $n$ est impair. Déterminer le spectre de $S$ et la multiplicité de chacune de ses valeurs propres. #+end_exercice +#+BEGIN_proof + - On trouve pour $S^2$ une matrice dont les coefficients non nuls sont $(1,1)$, $(2, n)$, $(n, 2)$, $(n-1, 3)$, $(3, n-1)$, c'est-à-dire 1 + une matrice antidiagonale, dont on trouve le déterminant. Il manque le signe de $\det S$… + - + - Changement de variable $s = k+r$. + - Vérifier que $(\zeta^{s^2})$ est un vecteur propre, en utilisant que $2$ est inversible modulo $n$. +#+END_proof +# ID:nil #+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. @@ -685,10 +695,12 @@ Soient $n\in{\N}$ avec $n\geq 2$, $\zeta=e^{2i\pi/n}$ et $S=\Big(\zeta^{(r-1)(s- #+end_exercice +# ID:7731 #+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 $F_u(\lambda)$ est un sous-espace vectoriel stable par $u$ et qu'il admet un supplémentaire stable par $u$. + - On écrit $\pi_u=(X-\lambda)^pQ$ avec $(X-\lambda)\wedge Q=1$. Montrer que $E=F_u(\lambda)\oplus\op{Ker}Q(u)$. + - E On suppose de plus que $u$ commute avec $v$. On note $p_{\lambda}$ le projecteur sur $F_u(\lambda)$ parallèlement 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. @@ -696,15 +708,20 @@ Soient $E$ un $\C$-espace vectoriel de dimension finie et $v,w\in\mc{L}(E)$. On #+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 $\forall k\in\N^*,\ \op{tr}(M^k)=0$. +#+begin_exercice [ENS MP 2024 # 44] :todo: +Soit $A,B$ dans $\M_n(\C)$. + - Montrer que $M\in\M_n(\C)$ 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_proof + - + - Quand on développe $(AB-BA)^k$, dans la trace, chaque produit a autant de $A$ que de $B$. En utilisant la propriété $A^2 B = ABA$ (et la trace), on a la même trace que n'importe quel produit alterné. + - !! +#+END_proof +# ID:nil # Classique #+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 $\op{Ker}P_i(A)$. @@ -712,7 +729,7 @@ Soit $A\in\M_n(\C)$ de polynôme caractéristique $\chi_A=\prod_{i=1}^r\underbra - 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 - +# ID:6402 #+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. @@ -727,73 +744,125 @@ Pour $A\in\mc{U}_n(\mathbb{K})$, on pose $\ln(A)=\sum_{p=1}^{+\i}\frac{(-1)^{p-1 #+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 $B$. +Soient $A$, $B$, $C\in\M_n(\R)$. On considére l'équation $(E)\colon X-AXB=C$ 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_proof + - Il n'est pas possible que $X = AXB$ sans que $X$ soit nulle : prendre une base de trigonalisation de $B$ (espace caractéristique par espace caractéristique), et montrer que $X$ s'annule dessus, par récurrence. + - On veut trouver un $X$ tel que $AXB = X$, sous l'hypothèse d'une valeur propre $\la$ pour $B$ et $\frac{1}{\la}$ pour $A$. !! +#+END_proof -#+begin_exercice [ENS MP 2024 # 48] +# ID:7732 +#+begin_exercice [ENS MP 2024 # 48] :todo: 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_proof +C'est du Jordan. +#+END_proof +# ID:nil # Cf année précédente #+begin_exercice [ENS MP 2024 # 49] Déterminer les $M$ de $\M_n(\R)$ telles que $M$ soit semblable à $2M$. #+end_exercice +# ID:7766 #+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_proof +Les valeurs propres de $M_k$ doivent tendre vers $1$, mais il n'y a qu'un nombre fini de valeurs propres possibles d'un polynôme à coefficients entiers, assez proches de $1$, donc les valeurs propres de $A$ sont de module $1$. Ce sont en fait des racines de l'unité, et elle admettent des racines $k$-ièmes pour tout $k$, dans cet ensemble fini. Ce n'est pas possible, (pour $n!$ essentiellement). Pour $k$ assez grand, la racine $k$-ième considérée est unipotente aussi. + +Donc les seules valeurs propres de $A$ sont $0$ et $1$, donc $0$ car inversible : $A$ unipotente. On sait que $A$ admet une racine $p$-ième $B$ unipotente dans la même base de trigonalisation, de plus $B$ est un polynôme en $A$. Si $C$ est une autre racine $p$-ième unipotente, on a $C^p = B^p$ et $C, B$ commutent : $(C-B)(\sum B^k C^{p-1-k}) = O_n$. Mais la somme est inversible, car co-trigonalisable, donc $B=C$. Mais clairement, cette racine $p$-ième ne peut pas être entière pour tout $p$, de par son expression : si $À = I_n + N$, c'est $I_n + \frac{N}{p} + N^2 +\dots$. +#+END_proof +# ID:7733 #+begin_exercice [ENS MP 2024 # 51] Montrer que toute matrice de $\mathrm{GL}_n(\C)$ admet une racine carrée. #+end_exercice +# ID:7767 #+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_proof + - C'est les espaces caractéristiques. + + Unicité : $D'D^{-1} = U'U^{-1}$, utilise que $D'$ commute avec $D'$ et $U'$, donc avec $M$, donc avec $D,U$, puisque ce sont des polynômes en $M$. + - On regarde les diagonales. Celles dont les valeurs propres sont dans $\bigcup \m U_n$ ont leurs images toutes co-diagonalisables. Comme l'application est polynomiale, il en va de même de toutes les matrice diagonales. + + On regarde les unipotentes $U$. Elles vérifient $U^n$ et $U^{-n}$ de tailles polynomiales, donc $\rho(U)$ n'a que des valeurs propres de module $1$. + + La somme des valeurs propres est une expression polynomiale en les coefficients de $U$, et elle est bornée, donc elle est constante, donc elle vaut toujours $n$, et les valeurs propres valent toutes $1$. +#+END_proof +# ID:7738 #+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 $j\in\llbracket 1,\beta_i-1}$. + - 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 $i\in\db{1,r}$ et $j\in\db{1,\beta_i-1}$. #+end_exercice +#+BEGIN_proof + - Si et seulement si $A,B$ commutent. + - Unicité : $D'D^{-1} = U'U^{-1}$, utilise que $D'$ commute avec $D'$ et $U'$, donc avec $M$, donc avec $D,U$, puisque ce sont des polynômes en $M$. + + - $P(D+N) = P(D) + P'(D)N + \dots + \frac{P^{(k)}(D)}{k!} N^k$ +#+END_proof +# ID:nil #+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 +# ID:nil #+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. + - Soit $(f_k)_{0\leq k\leq n}$ 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 $(f_k)_{0\leq k\leq n}$ 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 +# ID:7739 #+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 $\forall i\in\llbracket 1,n},\;f(e_i)\gt 0$. +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 $\db{1,n}$. Montrer que $(e_1,\ldots,e_n)$ est libre si et seulement s'il existe une forme lineaire $f$ sur $E$ telle que $\forall i\in\db{1,n},\;f(e_i)\gt 0$. #+end_exercice +#+BEGIN_proof +Si elle est libre, la base duale marche (prendre la somme des $e_i^*$). + +Si elle est liée, $\sum \la_i e_i = 0$, alors les $\la_i$ doivent tous avoir le même signe (sinon, $\sum_1 \la_i e_i = - \sum_1 \la_i e_j$, et écrire $\langle v,v\rangle\leq 0$). +#+END_proof #+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 $x,x'\in\R^n$, $\langle x,x'\rangle^m=\langle f(x),f(x')\rangle_E$. +Soient $n,m\geq 1$ des entiers. On note $\langle\;,\;\rangle$ le produit scalaire canonique sur $\R^n$. Montrer qu'il existe un espace préhilbertien $(E,\langle\;,\;\rangle_E)$ et une application $f\colon\R^n\ra E$ (non linéaire) tels que, pour tous $x,x'\in\R^n$, $\langle x,x'\rangle^m=\langle f(x),f(x')\rangle_E$. #+end_exercice +#+BEGIN_proof +Prendre $E = \vect_{x\in\R^n} e_x$, et $\langle e_x, e_y\rangle = \langle x,y\rangle^m$, c'est une forme bilinéaire. Il faut vérifier le caractère def pos, on est ramené à un nombre fini de $x,y$, et + $$\lN \sum \la_x e_x\rN^2 = \sum \la_x^2 \lN x\rN^{2m} + \sum \la_x \la_y \langle x,y\rangle^{m}$$ + +Autrement dit, si $A$ est pos (non def pos si les $x_i$ sont liés), la matrice dont les coefficients sont ses carrés est def pos. ?? +#+END_proof +# ID:7789 #+begin_exercice [ENS MP 2024 # 58] -Trouver un espace prehilbertien $(E,\langle\;,\;\rangle)$ et $f\colon\R\ra E$ tels que, pour tous $x,y\in\R$, $\exp\left(-\frac{(y-x)^2}{2}\right)=\langle f(x),f(y)\rangle$. +Trouver un espace préhilbertien $(E,\langle\;,\;\rangle)$ 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_proof + $f(x) = t\mapsto e^{-(x-t)^2}$, avec $\langle f,g\rangle = \int_{\R} f(t) g(t)\dt$. +#+END_proof #+begin_exercice [ENS MP 2024 # 59] @@ -802,7 +871,7 @@ Soient $m,n\in\N^*$ tels que $n\lt m$. On munit $\R^m$ de sa structure euclidien - 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 - +# ID:7740 #+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 $I_{n,k}=\inf_{(a_0,\ldots,a_n)\in\R^{n+1}}\int_0^1\left(t^k- \sum_{i=0}^na_it^{\lambda_i}\right)^2dt$. @@ -810,52 +879,92 @@ Soient $(\lambda_n)_{n\geq 0}\in\R^{\N}$ une suite strictement croissante vérif 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_proof + - La distance est un déterminant de Gram. + - On trouve que $\sum \frac{1}{\la_i}$ diverge. +#+END_proof +# ID:7741 #+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 $u\circ u^*=C\,\mathrm{id}$. + - Soit $u\in\mc{L}(V)$ qui préserve l'orthogonalité : 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 $u\circ u^*=C\,\op{id}$. #+end_exercice +#+BEGIN_proof + - + - + - +#+END_proof +# ID:7742 #+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 $\Lambda=\Big{\{}\sum_{i=1}^n\lambda_ie_i,\,(\lambda - {1 \leq i\leq n}\in\Z^n\Big{\}}$. +Soient $E$ un espace euclidien de dimension $n$ et $(e_1,\ldots,e_n)$ une base de $E$. On pose $\Lambda=\Big{\{}\sum_{i=1}^n\lambda_ie_i,\,(\lambda_i)_{1 \leq i\leq n}\in\Z^n\Big{\}}$. - 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 +# ID:7743 #+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_proof +Ils sont diagonalisables, à valeurs propres réelles, donc d'ordre $1$, ou $2$. +#+END_proof +# ID:7768 #+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. + - Déterminer une condition nécessaire et suffisante sur $a\in\R$ pour que la matrice $\begin{pmatrix}1&a\\ a&1\end{pmatrix}$ soit positive, puis 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_proof + - $(1-X)^2 - a^2 = X^2 - 2X + 1 - a^2$. La somme des valeurs propres vaut $2$, on regarde le produit, qui vaut $1-a^2$, donc si et seulement si $a\leq 1$. + - On pose $g_n(a,b,c) = 1 + 2 (abc)^n - a^{2n} + b^{2n} + c^{2n}$, on en cherche le minimum, dans le compact vérifiant les conditions. + + $\nabla g = (2n a^{n-1} (bc)^n - 2n a^{n-1} a^n) = 2n a^{n-1}\big((bc)^n - a^n\big) = 2n a^{n-1} (bc - a) (\sum (bc)^k a^{n-1-k})$. + + Si on atteint ce minimum à l'intérieur, on a soit $a = 0$ (on vérifie que c'est bon), soit $a^n = (bc)^n$, et de même pour les autres produits, de manière cyclique, qui implique que les termes sont nuls. + + Si on atteint ce minimum sur un bord $a = \pm 1$, c'est clair aussi. + + Sinon, on atteint ce minimum là où $g_1 = 0$, et les gradients sont colinéaires. On obtient que $a^{n-1}\sum (bc)^{n-1-k} a^{k} = c^{n-1}\sum (ab)^{n-1-k} c^{k} = b^{n-1}\sum (ca)^{n-1-k} b^{k}$, donc $\sum \left(\frac{a}{bc}\right)^k = \sum \left(\frac{b}{ac}\right)^k = \dots$. En fonction de la parité de $n$, on trace la fonction $\frac{1-x^{n+1}}{1-x}$. Si $n$ est impair, elle est strictement croissante (on peut calculer sa dérivée), donc tous les $\frac{a}{bc}$ sont égaux. On conclut. + + Si $n$ est pair. chaque valeur a deux antécédents, donc deux des quotients sont égaux, ce qui donne deux des carrés égaux, ce qui conclut. +#+END_proof +# ID:7769 #+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 $\gt 0$. + - s 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 $A$ possède 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_proof + - Celui de valeur propre maximale maximise $\frac{\lN AX\rN}{\lN X\rN}$. Si il n'était pas à coefficients positifs, on pourrait l'augmenter. Pour le $\gt 0$, c'est une question d'ordre du $DL$. + - La trace est $\gt 0$, et on a deux valeurs propres réelles. On envoie le cône du quadrant supérieure dans lui-même, donc il y a un point fixe. + - D'après la question précédente, il existe un vecteur propre réel, à coefficients positifs. S'il existait une valeur propre rationnelle, les deux le seraient. Le produit vaut $2$, et la somme est entière donc la valeur propre vaut $\pm 1$, donc on a un vecteur entier $\vv{p}{q}$, qui retombe sur lui-même, mais $\begin{pmatrix}a_i & 1 \\ 1 & 0\end{pmatrix}\vv{p}{q} = \vv{a_1 p + q}{p}$, donc à chaque étape, le premier coefficient est trop grand. +#+END_proof -#+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 $\left(\begin{array}{cc}a&b\\ -b&a\end{array}\right)$. +# ID:7770 +#+begin_exercice Réduction des endomorphismes normaux [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 +# ID:7771 #+begin_exercice [ENS MP 2024 # 67] Montrer que $\text{SO}_3(\Q)$ est dense dans $\text{SO}_3(\R)$. #+end_exercice +# ID:nil #+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 @@ -868,135 +977,215 @@ Soient $A,B\in\mc{S}_n(\R)$. Montrer que $\colon\inf\limits_{\|G\|=1}\|AG-GB\|=\ #+end_exercice +# ID:7788 #+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 $(K_x)_{x\in X}$. +Soient $X$ un ensemble et $K\colon X\times X\ra\R$. 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 $K_x\colon y\mapsto K(x,y)$. Soit $E$ le sous-espace de $\R^X$ engendré 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 +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 $\forall x,y\in X$, $K(x,y)=\left\langle f(x),f(y)\right\rangle$. + - Montrer que cela définit bien un produit scalaire sur $E$. + - Montrer qu'il existe $f\colon X\ra E$ telle que $\forall x,y\in X$, $K(x,y)=\left\langle f(x),f(y)\right\rangle$. #+end_exercice +#+BEGIN_proof + - C'est clair. + - C'est clair. +#+END_proof #+begin_exercice [ENS MP 2024 # 71] Soient $p\geq 1$ et $A,B\in\mc{S}_p^{++}(\R)$. - - Montrer que $\op{tr}\left(I_p-A^{-1}B\right)\leq\ln\left(\frac{\det A}{ \det B}\right)$. + - 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 $\sum_{m=1}^n\left\langle u_m,B_m^{-1}u_m\right\rangle \leq\sum_{i=1}^p\ln\left(1+\frac{\lambda_i}{\lambda}\right)$. + 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 +# ID:7772 #+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. +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 +#+BEGIN_proof + - + - + - Décomposer en symétrique + antisymétrique, puis réduire : les symétriques sont diagonalisable, donc diagonale, et si leur valeurs propres sont $\leq 1$, on trouve. + + De même les antisymétriques sont diagonalisables. + - Si on est dans le centre, on commute avec toutes les matrices de $\M_n(\C)$. +#+END_proof ** Analyse +# ID:7757 #+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_proof + - Oui. + - Ce sont les parties convexes, contenant un voisinage de $0$, classique. +#+END_proof +# ID:nil #+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 $\lambda\in\R$, $\phi(\lambda x)=|\lambda|\phi(x)$. + + 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 $K$ un partie de $E$ convexe, compacte, d'intérieur 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 +# ID:nil #+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 $G=\left\{\sum_{k=1}^pa_k.u_k,\ (a_1,\ldots,a_p)\in\Z^p\right\}$. +Soit $G$ un sous-groupe de $(\R^n,+)$ dans lequel $0$ est un point isolé. 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 +# ID:7759 #+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 $\sup\{k\in\N\ ;\ \exists G\subset\R^n,\ |G|=k,\ f(G)= \mc{P}(G)\}$. +Soit $n\in\N^*$. Soit $E$ l'ensemble des pavés 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_proof +L'ensemble $\{\pm e_i\}^n$ permet d'atteindre $k = 2n$. Réciproquement, c'est la valeur maximale : si un ensemble a strictement plus de $2n$ points, on prend celui d'abscisse maximale, celui d'abscisse minimale, idem pour la seconde coordonnée, etc, tout pavé contenant tous ces points contient tous les autres. +#+END_proof +# ID:7760 #+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. + - Soient $E$ un espace vectoriel normé et $K$ un compact convexe non vide de $E$. Soit $(f_i)_{i\in\N}$ 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_proof + - Prendre un élément du compact, appliquer les fonctions. S'il n'y a qu'une seule fonction, il faut utiliser le caractère affine : prendre la moyenne des éléments, de sorte que $|f(m_n) - m_n|\ra 0$, et une valeur d'adhérence de $m_n$. + + Donc $f_1$ a des points fixes, et l'ensemble des points fixes de $f_1$ est stable par $f_2$, etc. + - Oui : Il s'agit de montrer que si chaque intersection fini de compacts $K_i$ est non vide, il en va de même de toute intersection. Assume not : une intersection $K_i$ de compact est vide. + + On peut recouvrir $K$ par un nombre fini de boules de rayon $1$. Une de ces boules intersecte toute intersection finie de points fixes de $f_i$. Puis on recouvre cette boule par des boules de rayon $\frac{1}{2}$, etc. + + On a un point d'accumulation. Il est dans toutes les intersections. +#+END_proof + +# ID:7782 #+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. +Soient $H$ le groupe (pour la composition) des homéomorphismes de $\R$ sur $\R$, $H^+$ le sous-groupe des homéomorphismes 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 $\forall f,g,h\in G$, $f\leq g\ \Longrightarrow\ h\circ f\leq h\circ g$. + - Montrer qu'on peut munir tout sous-groupe $G$ 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_proof + - Si $g^n = \op{Id}$, alors si $n$ impair, $g = \op{Id}$, et si $n$ pair, $g^2 = \op{Id}$. + + Si deux fonctions vérifient $f^2 = \op{Id}$ et $g^2 = \op{Id}$, et leur composée aussi, alors elles commutent, donc elles sont égales. + - Prendre une énumération de $\Q$, et mettre l'ordre lexicographique sur $(f(u_n))$. + - On peut créer une application $H\ra\Q$ injective qui préserve l'ordre. Alors, $H$ agit sur cette partie de $\Q$ (l'image), de manière croissante. On peut s'assurer que cette action est continue (si on a un point d'accumulation, on a deux suites, une croissante et une décroissante qui converge vers ce point si les limites étaient différentes, aucun élément du groupe ne serait envoyé entre ces deux limites : dans la construction, quand on place un élément entre $a$ et $b$, le mettre en $\frac{a+b}{2}$). + + Alors, on prolonge $H$ par continuité, puis de manière linéaire. + + Par ailleurs, c'est bien un morphisme, car la composée de deux fonctions de ce type l'est encore. +#+END_proof +# ID:7783 #+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 $\R^m$. +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 $\R^m$. + 1. sV2 On suppose que $\forall x\in F,\, \lN f(x)\rN\gt \lN x\rN$, montrer que $0$ n'appartient pas à l'enveloppe convexe de $f(F)$. + 2. Montrer que l'on peut prolonger $f$ en une application $1$-lipschitzienne de $F\cup\{x\}$ dans $\R^m$. #+end_exercice +#+BEGIN_proof + 1. + 1. Par récurrence sur $k$. Pour $k = 2$, c'est clair. Sinon on applique l'hypothèse de récurrence à deux sous parties de $F$ privées d'un élément XX + + Le point $x$ vérifie $\lN x- x_i\rN = \delta_i$ + + On augmente des boules autour des $f(x_i)$, de rayon $t\delta_i$, à partir de $t = 0$. On prend $x_c$ le premier point d'intersection, atteint pour $t_c$. On veut montrer que $t_c \leq 1$. + + Le point $x_c$ est sur le bord d'un certain nombre des disques, et par ailleurs, $\sum_{J} \la_j (f(x_j)-x_c) = \vec 0$, avec des $\la_j\geq 0$, autrement dit, $x_c$ est dans l'enveloppe convexe des $f(x_j)$. + + Si on suppose par l'absurde que $t_c\gt 1$. On peut supposer que $x_c = 0$, alors $\lN f(x_j)\rN\gt \lN x_j\rN$. Cela implique que les $\langle f(x_j), f(x_i)\rangle \gt \langle x_i,x_j\rangle$. + + Alors $\lN \sum \la_j f(x_j)\rN\gt \lN \sum \la_j x_j\rN^2$, ce qui empêche $\sum \la_j x_j = \vec 0$. +#+END_proof +# ID:7761 #+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 $D=\bigcap_{N\geq 1}D_N$. + $$D_N=\left\{x\in\R^d\;;\;\forall k\in\Z^d\setminus\{0\}, \|k\|\leq N\Rightarrow|\left\langle x,k\right\rangle|\geq\frac{ \gamma}{\|k\|^{\tau}}\right\} \quad \et \quad D=\bigcap_{N\geq 1}D_N.$$ -Montrer que $D$ est ferme et d'interieur vide. Qu'en est-il de $D_N$? +Montrer que $D$ est fermé et d'intérieur vide. Qu'en est-il de $D_N$? #+end_exercice +#+BEGIN_proof + $D$ est une intersection de fermés. Intérieur vide : il suffit de montrer que dans toute boule ouvert, on peut trouver un vecteur qui a un orthogonal à coordonnées entières. + + Pour $D_N$ : c'est une intersection de deux demi-hyperplans, opposés séparés d'une constante. Chacun retire au plus une direction : pour tout $x$, pour $\la$ assez grand, $\la x\in D_N$. +#+END_proof +# ID:7784 #+begin_exercice [ENS MP 2024 # 81] -Soient $E$ et $F$ deux espaces vectoriels normes. Soit $f:E\ra F$ telle que : $\forall r\in]0,1],$ $\forall x\in E,\ B\left(f(x),\frac{r}{2}\right)\subset f(B(x,r))\subset B(f(x),2r)$. +Soient $E$ et $F$ deux espaces vectoriels normés. Soit $f\colon E\ra F$ telle que : $\forall r\in \interval]{0, 1}]$, $\forall x\in E,\ B\left(f(x),\frac{r}{2}\right)\subset f(B(x,r))\subset B(f(x),2r)$. - Montrer que $f$ est continue et surjective. - - Que peut-on dire de l'image par $f$ d'un ouvert? D'un ferme? + - Que peut-on dire de l'image par $f$ d'un ouvert ? D'un fermé ? - 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_proof + - La continuité découle de la deuxième inclusion, la surjectivité de la première (il faut travailler, parce que $r\leq 1$). + - $f$ est clairement ouverte. Aussi, elle est $2$-lipshitzienne. Image d'un fermé : si $f(x_n)\ra a$, il est très possible que $x_n$ tende vers l'infini, bon, pour $\R\ra\R$ ça a l'air dur, mais pour $\R^2 \ra \R$, il suffit de prendre une fonction dont le graphe est un plan. + - Si on découple $\gamma$ en des points à des distances $\lt \frac{1}{2}$, on peut trouver une suite de points à des distances $\leq 1$ tel que $f(y_i) = \gamma_i$. On recommence, etc. On trouve $c$. +#+END_proof +# ID:nil : See 4272 #+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. + - Soit $X\subset\R^n$ un fermé 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 $f$ possède 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 $f\colon X\ra X$. 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 $f$ possède 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 $m\geq n$. + - 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 $\lim\limits_{x\ra+\i}\rho(x)=+\i$ 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_proof + - Équivalence des normes. + - + - Majoration : on a $\lN f(x) - f(y)\rN\leq \mu \lN x-y\rN + b$, et on peut retirer le $b$, car on est sur $\Z^n$. + + Minoration : Sinon, il existerait des $x_n,y_n$, avec $\lN x_n-y_n\rN$ arbitrairement grand, et $\lN f(x_n) - f(y_n)\rN$ borné, ce qui contredit la deuxième condition. + - !! + - +#+END_proof #+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 : +On munit $\R^2$ de sa structure euclidienne canonique. Soit $f\colon\R^2\ra\R^2$ un homéomorphisme. Pour $x\in\R^2$ et $r\gt 0$, on pose : $L_f(x,r)=\sup\big{\{}\|f(x)-f(y)\|\;;\;y\in\R^2,\ \|x-y\|\leq r \big{\}}$, @@ -1023,7 +1212,7 @@ Soit $v\in\R^n$. Montrer que l'application $f_v:M\in\mc{A}\mapsto a_{v,M}$ est c #+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 $\Pi_X(x)=\{y\in X\;;\;\forall z\in X,\;\|y-x\|\leq\|z-x\|\}$. +Soit $(E,\|\ \|)$ un espace vectoriel normé. 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 $\Pi_X(x)=\emptyset$? @@ -1031,6 +1220,7 @@ Soit $(E,\|\ \|)$ un espace vectoriel norme. Pour $X\subset E$ et $x\in E$, on n #+end_exercice +# ID:7748 #+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 $x_k\xrightarrow[k\ra+\i]{}x_*$. @@ -1039,28 +1229,51 @@ Soit $n\geq 1$ un entier, $L\in\left]0,1\right[$, $F\colon\R^n\ra\R^n$ une appli 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_proof + - trivial + - Trivial. + - La suite $|(x_k - x_*)_i|$ est décroissante, donc converge, d'où le résultat. +#+END_proof +# ID:7749 #+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 $\ell^{\i}$. + - Soit $(a_n)$ une suite réelle sommable. Montrer que l'application $f\colon x\mapsto\sum_{n=0}^{+\i}a_nx_n$ 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 $x\in\ell^{\i}$. + - 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 -. + 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_proof + - $|f(x)|\leq \sum |a_n| \lN x \rN_{\i}$ + - + - C'est la preuve de BW. + - C'est clairement linéaire, continue. On justifie que ce n'est pas de la forme précédente. Si c'était le cas, on aurait $\sum a_i = 1$ (prendre une suite constante), puis en prenant deux constantes, pour les termes paires et impaires, on obtient que l'une des deux sommes est $1$ l'autre est $0$. On recommence en coupant en $4$, en $8$, etc. + + On obtient que un seul des termes vaut $1$, et les autres $0$. + - On définit les $a_i$ comme l'image de $(0,\dots, 0, 1, 0 \dots)$, on vérifie que la famille est sommable, et une fois que l'on coïncide sur les suites nulles APCR, on coïncide, par $\eps$. +#+END_proof +# ID:7750 #+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 $P=0$. +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_proof +Écrire $P(X,Y) = \sum X^n P_n(Y)$. Prendre la valeur en $(n^n, n)$. +#+END_proof +# ID:7751 #+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_proof +Projeter $0$ sur $C$, et prendre une droite orthogonale. +#+END_proof #+begin_exercice [ENS MP 2024 # 91] @@ -1069,12 +1282,12 @@ 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 $z\in\Delta$, $2\left\langle u,z-x\right\rangle\leq\left\|z-x\right\|_2^2-\left\|z -x'\right\|_2^2+\left\|u\right\|_2^2$. + 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 $y_{k+1}=\pi(y_k-\gamma_kA^Tx_k)$. + 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 -$$\max_{x\in\Delta}\sum_{k=1}^N\left\langle x,Ay_k\right\rangle-\min_{y\in \Delta}\sum_{k=1}^N\left\langle x_k,Ay\right\rangle\leq o(N)\text{.}$$ + $$\max_{x\in\Delta}\sum_{k=1}^N\left\langle x,Ay_k\right\rangle-\min_{y\in \Delta}\sum_{k=1}^N\left\langle x_k,Ay\right\rangle\leq o(N)\text{.}$$ - 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 @@ -1082,7 +1295,7 @@ $$\max_{x\in\Delta}\sum_{k=1}^N\left\langle x,Ay_k\right\rangle-\min_{y\in \Delt #+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$. + $\forall(x,y)\in E^2,\,\left\|\lN T(x) - T(y)\rN- \lN x-y\rN\right| \leq C$. L'objectif est de montrer qu'il existe $h\in\R^+$ et un unique $u\in\mc{O}(E)$ tels que @@ -1092,10 +1305,17 @@ L'objectif est de montrer qu'il existe $h\in\R^+$ et un unique $u\in\mc{O}(E)$ t - 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_proof + - Simple. + - Simple. + - + - +#+END_proof +# Relier à l'X… #+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)$. +Soient $(E,\left\langle\,\ \right\rangle)$ un espace préhilbertien, $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$. @@ -1111,6 +1331,10 @@ Soient $n\geq 2$ et $I_n(\R)=\{A\in\M_n(\R)\,;\,\exists\lambda\in \op{Sp}(A),\,\ - 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_proof + - + - La condition donne que $A$ n'a qu'une seule valeur propre non nulle et $\R^n = E_{\la}\oplus E_0$. +#+END_proof #+begin_exercice [ENS MP 2024 # 95] @@ -1131,24 +1355,37 @@ Soit $n\in\N^*$. Soit $G$ un sous-groupe compact de $\op{GL}_n(\R)$. Pour tous $ #+end_exercice +# ID:7752 #+begin_exercice [ENS MP 2024 # 97] Déterminer les valeurs d'adherence des suites $(\cos n)$ et $(\cos^nn)$. #+end_exercice +# ID:7753 #+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_proof +Elle converge vers sa borne inférieure. +#+END_proof -#+begin_exercice [ENS MP 2024 # 99] +#+begin_exercice [ENS MP 2024 # 99] :todo: 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_proof +On écrit $z_n = r e^{i\theta_n}$, on a $\theta_{n+1} = \theta_n - r \sin \theta_n$. + +Si $r\leq 1$, cela converge vers $0$. Si +#+END_proof #+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_proof +C'est maximum quand $t = \frac{n}{\ln 2}$. Essayer de comparer à l'intégrale. +#+END_proof #+begin_exercice [ENS MP 2024 # 101] @@ -1156,12 +1393,13 @@ On fixe un entiers $n\geq 2$ et $(t_i)_{i\in\Z/n\Z}$ une famille d'éléments de #+end_exercice +# ID:nil #+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] +#+begin_exercice [ENS MP 2024 # 103] :todo: 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 @@ -1179,11 +1417,13 @@ Puis, sommation des équivalents des restes. #+END_proof +# ID:7762 #+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. +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)$. #+end_exercice +#+BEGIN_proof +On a $\sum_{n=0} \ln \left( 1 + \frac{1}{a^{2^n}}\right) = \ln \prod \left(1 + \frac{1}{a^{2^n}}\right) = \ln \sum_{k\geq 0} \frac{1}{a^k} = \ln \frac{a}{a-1}= \ln 1 + \frac{1}{a-1}$. +#+END_proof #+begin_exercice [ENS MP 2024 # 106] @@ -1203,6 +1443,7 @@ Soient $p\in]1,+\i[$ et $q\in\R$ tels que $\frac{1}{p}+\frac{1}{q}=1$. #+end_exercice +# ID:nil #+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}$. @@ -1210,11 +1451,19 @@ On admet l'irrationalite de $\pi$. Pour $n\in\N$, on pose $u_n=\frac{(-1)^n}{n^{ #+end_exercice +# Relier à 85 #+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$. + - 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_proof + - + - On suppose que $\sum a_n$ converge. Pour toute suite $c_n$, on a $\sqrt[n]{a_1\dots a_n}\leq \frac{1}{\sqrt[n]{c_1\dots c_n} n} \sum_{k=1}^n a_k c_k$, ce qu'on somme en une majoration de la forme $\sum_{k=1}^n a_k c_k \sum_{\l = k}^{+\i} \frac{1}{\sqrt[\l]{c_1\dots c_\l} \l}$. + + Si on prend $c_k = k$ + - +#+END_proof #+begin_exercice [ENS MP 2024 # 110] @@ -1305,7 +1554,7 @@ Montrer que $\forall x\in I,\ |f(x)|\leq|f(0)|e^{C|x|}+\dfrac{A}{C}\left(e^{C|x| #+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? +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 généralisations peut-on étudier? #+end_exercice @@ -1321,6 +1570,7 @@ Montrer que $I$ est unique. On la note $\int_{HK}f$. #+end_exercice +# ID:nil #+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 @@ -1386,22 +1636,26 @@ Montrer que $\{X_u(1)\;;\;u\in E\}=\R^n$ si et seulement si la matrice $(A|AB|\l #+end_exercice +# ID:nil #+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$. +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^{+*}$. + - Montrer 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}$$ + $$\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? + exclut $(n,m)=(0,0)$. Ces limites sont-elles égales? #+end_exercice +# ID:7786 #+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)$. +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_proof +C'est homogène, on peut supposer que $x+y+z = 1$, et montrer que $1 + 9 xyz \geq 4 (xy + yz+zx)$. On cherche le minimum de la différence, de gradient $(9yz - 4y - 4z, 9\dots,\dots)$. Sur un bord où $x = 0$ c'est clair, et sinon, il faut que les trois coordonnées soient égales : $9yz - 4y - 4z = 9xz - 4x - 4z$ donne $9z(y-x) = 4 (y-x)$, donc soit les coordonnées sont égales, soit $z = \dots$. +#+END_proof #+begin_exercice [ENS MP 2024 # 132] @@ -1416,35 +1670,47 @@ Démontrer que $\sup_{\R^+\times\R}(u-v)=\sup_{\{0\}\times\R}(u-v)$. #+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}$. +Soient $a\gt 0$, $n\geq 1$ et $x_1,\ldots,x_n\gt 0$. Calculer $\inf\limits_{\substack{y_1,\ldots,y_n\gt 0\\ y_1+\cdots+y_n\leq }}\sum\limits_{i=1}^n\frac{x_i}{y_i^a}$. #+end_exercice +#+BEGIN_proof + +#+END_proof #+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$. +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_i)_{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)$. +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$. + - 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_proof + - Prendre les $e_i$ et $v = -\sum e_i$ + - + - +#+END_proof +# ID:7785 #+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? + - 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)=\lN A - M_0\rN$. + - Rappeler le résultat sur les extrema sous contrainte. Que peut-on en déduire sur la matrice $M_0$ définie ci-dessus? #+end_exercice +#+BEGIN_remarque + - + - $0$ + - $S$ est fermé. + - On travaille sous $\det M = 0$, est la différentielle du déterminant est $H\mapsto \op{Tr}(\op{Com} M_0^T H)$, donc $\op{Com} M_0^T$ est colinéaire à $A-M_0$. +#+END_remarque -** Geometrie +** Géométrie #+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}$. @@ -1460,16 +1726,22 @@ On munit $\M_n(\R)$ de sa structure euclidienne canonique. ** Probabilités +# ID:7764 #+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)$. +Soit $E$ un espace vectoriel normé 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_proof +La fonction $\la\mapvo \lN u + \la v\rN + \lN u - \la v\rN$ est croissante sur $\R_+$. +#+END_proof +# ID:nil #+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$. +On considére une pièce 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 +# ID:nil #+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}$. @@ -1488,20 +1760,32 @@ On note $E=\db{1,n}$ et $\Delta$ la différence symétrique. Soit $p\in[0,1]$ et #+end_exercice +# ID:7787 #+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_proof + - Prendre le premier $1$. + - On tire pile ou face, et on choisit l'indice du milieu de la première apparition de $110$, ou de $011$ suivant le résultat. + + Considérons $P(X_{N-2} = 1)$ : Sachant qu'on tire pile, et sachant que $N = k\geq 2$, les termes d'avant $110$ peuvent être n'importe quoi sans cette séquence. Par exemple, si il reste trois places, sur les $2^3 = 8$ possibilités, une est exclue, donc $4$ façons de finir par $1$ et $3$ façons de finir par $0$. + + Par contre, sachant qu'on tire face, on regarde les termes avant $011$, on a $3$ façons de finir par $1$ et quatre par $0$. + + De plus, le nombre de séquences sans facteur $110$ est le même que le nombre de séquences sans facteur $011$. +#+END_proof +# ID:nil #+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 - +# ID:nil #+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$. @@ -1512,52 +1796,51 @@ On note ${\rm AA}=\{xy,\ (x,y)\in A^2\}$. #+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 $X$ une variable aléatoire réelle positive $L^2$. Montrer 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. Montrer 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_proof + - Cauchy-Schwarz : $E(X^2) P(X\geq\la E(X)) \geq E(X 1_{X\geq \la E(X)})^2$, et $E(X 1_{X\leq \la E(X)})\leq \la E(X)$. + - Si $\sum E(u_n)$ converge, pour $\eps = \frac{1}{N}$, $P(X\gt )$ +#+END_proof #+begin_exercice [ENS MP 2024 # 146] -Soient $\lambda\gt 0$ et $N_{\lambda}$ une variable de Poisson de paramêtre $\lambda$. +Soient $\lambda\gt 0$ et $N_{\lambda}$ une variable de Poisson de paramètre $\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 + - Montrer 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))$. - $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))$. + Montrer 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. + - Montrer que ${\cal H}(X)\geq 0$ avec egalite si et seulement si $X$ est constante. + - Soit $(p_i)_{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$. + - Montrer que $\sum_{i=1}^np_i\ln(p_i)\geq\sum_{i=1}^np_i\ln(q_i)$. Expliciter le cas d'égalité. + - Montrer que ${\cal H}(X)\leq\ln(n)$ avec egalite si et seulement si $X$ suit une loi uniforme. -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$. -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$. -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)$. + L'entropie de $(X,Y)$ est ${\cal H}(X,Y)=-\sum_{i,j=1}^np_{i,j}\ln(p_{i,j})$. + - Montrer 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}$. +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$. Montrer 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)$. + - Montrer 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}))$. @@ -1565,13 +1848,13 @@ Démontrer que $\frac{1}{n}\ln{\bf P}(S_n\geq na)\xrightarrow[n\ra+\i]{}\inf_{s #+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}$. +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)$. Montrer 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$, + - Montrer 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 @@ -1585,6 +1868,7 @@ Pour $x\gt 0$, on pose $\Gamma(x)=\int_0^{+\i}t^{x-1}e^{-t}\dt$. On pourra utili #+end_exercice +# ID:nil # Problème d'énoncé #+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 : @@ -1594,8 +1878,12 @@ Soient $n\geq 3$ un entier. Si $\sigma\in\mc{S}_n$, une suite alternante pour $\ 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_proof +Problème d'énoncé ? +#+END_proof +# ID:nil # Cf année précédente #+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 : @@ -1605,19 +1893,36 @@ Soit $(X_k)_{k\geq 1}$ une suite de variables aléatoires discretes réelles i.i #+end_exercice +# ID:7774 #+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}}}$. + - 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'évènement «$\left(\dfrac{S_n}{\ln(n)}\right)_{n\geq 1}$ converge» est presque sûr. #+end_exercice +#+BEGIN_proof + - $P\left(\left|S_{\l} - H_{\l}\right| \gt \frac{H_{\l}}{k}\right)\leq \frac{k^2 V(S_{\l})}{H_{\l}^2}$. $V(S_{\l}) = \sum_{k=1}^n \frac{1}{k(\l-k)}\leq \frac{H_{\l}}{\l}$ (en coupant au milieu), donc $P(\dots)\leq \frac{k^2}{H_{\l}\l}$ + On prend $\l = k^4$. + - D'après ce qui précède, presque sûrement, $\frac{S_{k^4}}{\ln (k^4)}\ra 1$, ce qui implique $\frac{S_n}{\ln n}\ra 1$. +#+END_proof +# ID:7773 #+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})$. + - 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 majoré 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\limits_{k = 0}^{\lfloor \frac{n-1}{2}\rfloor}\m 1_{S_{2k+1}=0}$ et en déduire que $\mathbf{E}(N_n)\underset{n\ra+\i}{=}O(\sqrt{n})$. #+end_exercice +#+BEGIN_proof + - + - L'ensemble des zéros est fini. + + Si la suite ne change pas de signe, il n'y a pas de zéro. + + Si on primitive, avec une dérivée qui commence négative, et qu'on met un terme $\gt 0$ au début, alors on peut avoir un zéro de plus. + + Si on reprimitive, avec un nouveau terme positif au début, on ne peut pas gagner un zéro, car on est positif au voisinage de $0$, si on gagnait un zéro, on serait nulle alors que la dérivée est positive. + - Multiplier par $\sum x^n$ et appliquer la question précédente. La probabilité qu'on s'annule est en $\frac{1}{\sqrt{k}}$. +#+END_proof #+begin_exercice [ENS MP 2024 # 157] @@ -1627,11 +1932,15 @@ Soit $(X_n)_{n\in\N}$ une suite de variables aléatoires à valeurs dans $\N$ te #+end_exercice +# ID:7765 #+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 +Soient $a\in]0,1[$ et $m\in\N^*$. à l'aide d'une interprétation 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 +#+BEGIN_proof +Si on multiplie par $a^m$, on a des évènements indépendants de probabilité $a u_k$, c'est la probabilité qu'il s'en produit au moins $m$. Cela vaut $1$. +#+END_proof * ENS PSI 2024 :autre: @@ -1754,7 +2063,7 @@ Soit $d\in{\N}^*$. On se place dans ${\cal M}_d({\R})$ muni du produit scalaire - 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. + - Montrer que ${\mathbb{P}}_d({\R})$ est fermé. 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})$. @@ -1768,7 +2077,7 @@ On note $R=\big(p_{i,j}^2\big)_{1\leq i,j\leq d}$. #+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 dit qu'une suite $(u_n)$ à valeurs dans un espace vectoriel normé $(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. @@ -1854,9 +2163,9 @@ Soit $f\in\mc C^0(\R^+,\R)$ carre intégrable. Pour $x\in\R^{+*}$, on pose $g(x) #+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^+$. + - Donner le rayon de convergence de la série entière $\sum n!\,z^{n^2}$ + - Existe-t-il une fonction $g\colon\R^+\ra\R^+$ telle que, pour toute fonction $f$ développable en série entière, $f(x)\underset{x\ra+\i}{=}o(g(x))$? + - Une fonction est dite analytique si elle est développable en série entière 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 @@ -2091,7 +2400,7 @@ Soit $A=(a_{i,j})_{(i,j)\in[1,n]^2}$ vérifiant $\forall(i,j)\in[1,n]^2,\ a_{i,j #+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. + - Donner un exemple de matrice $A$ qui vérifie $A^2=A^T$ et qui possède toutes les valeurs propres possibles trouvées à la question précédente. #+end_exercice @@ -2301,7 +2610,7 @@ Adapter la preuve précédente pour prouver $e^2\notin\Q$. #+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. +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$ possède une limite en $+\i$ que l'on déterminera. #+end_exercice @@ -2478,21 +2787,27 @@ Dans toute la suite, on suppose que $\theta=\dfrac{2\pi}{3}$. ** 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$. +#+begin_exercice [X MP 2024 # 265] :todo: +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) \quad \et \quad a^-(X)=\prod_{i=1}^{n-1}(x_{i+1}-x_i-1).$$ L'objectif est d'etablir que : $\sum\substack_{B\subset A\\ B\neq\emptyset}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)$. + - On suppose le résultat acquis. Trouver une expression de : $\alpha(A)=\sum\limits_{\substack{B\subset A\\ a_n\in B}}a^-(B)$. + - Établir le résultat cherché. + - On suppose $A=\db{1,n}$. Calculer : $\sum\limits_{\substack{B\subset A\\ B\neq\emptyset\\ B\cap(B+1)=\emptyset}}a^-(B)$. #+end_exercice +#+BEGIN_proof + - + - + - +#+END_proof #+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$. + - Soit $n$ un entier supérieur à l et premier avec 10. Montrer que $n$ possède un multiple dont l'écriture en base 10 n'a que des 9. + - On remarque que $\frac{1}{7}=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{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$ @@ -2500,6 +2815,7 @@ Expliquer. #+end_exercice +# ID:nil #+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$. @@ -2513,24 +2829,27 @@ On note enfin, pour tout $n\in\N^*$, $H_n=\sum_{k=1}^n\frac{1}{k}$. #+end_exercice +# ID:7734 #+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_proof +C'est simple : marche pour n'importe quel $m$, d'après le lemme Chinois. +#+END_proof +# ID:nil # Cf année précédente #+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)$. +Soit $n\in\N^*$. On note $\mc{F}$ l'ensemble des entiers naturels qui ne sont pas divisibles par le carré 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}$. + 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}$. @@ -2540,7 +2859,12 @@ Dans la suite, on fixe un entier $n\geq 1$ on suppose acquire, pour tout ensembl - 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_proof +#+END_proof + + +# ID:nil #+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}$. @@ -2584,7 +2908,7 @@ On pose $z=x_0+\sqrt{d}\,y_0$. Montrer que, pour tout $n\in\N^*$, il existe un u - 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. + - Conclure que $(*)$ possède des solutions non triviales. #+end_exercice @@ -2616,8 +2940,8 @@ Pour $t\in\N$, on pose $g_p(t)=\sum_{x\in\mathbb{F}_p}\chi(tx)e^{\frac{2i\pi x}{ #+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$. + - Montrer que $G$ possède un élément d'ordre 2. + - Montrer que $G$ possède 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. @@ -2892,30 +3216,44 @@ Montrer que $(e_1,...,e_n)$ est une base de vecteurs propres de $u$. ** Analyse +# ID:7735 #+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? +Soit $E$ un espace vectoriel normé. Que dire d'une partie $A$ de $E$ à la fois ouverte et fermée ? #+end_exercice +# ID:7744 #+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. +Trouver une partie $A$ de $\R$ telle que $A$, $A^{\circ}$, $\overline{A}$, $\stackrel{{\circ}}{\ol{A}}$ et $\ol{\stackrel{\circ}{A}}$ soient toutes distinctes. #+end_exercice +#+BEGIN_proof +Partir de $\{\frac{1}{k},\,k\in\N^*\}$. Ajouter un petit intervalle fermé autour de chaque point. + +Puis ajouter tous les rationnels de $[-1, 0]$. +#+END_proof +# ID:7745 #+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. + - Montrer que la boule unite fermée pour $N$ est fermée, bornée, d'intérieur non vide, convexe et symétrique par rapport à $0$. + - Soit $C$ une partie non vide de $E$, fermée, bornée, d'intérieur 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 +# ID:6707 #+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 $f$ est continue alors le graphe de $f$ noté $\Gamma_f$ est fermé dans $\R^2$. La réciproque est-elle vraie? - Montrer que si $\Gamma_f$ est compact alors $f$ est continue. #+end_exercice +#+BEGIN_proof + - + - +#+END_proof +# ID:nil # Année précédente #+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). @@ -2924,18 +3262,30 @@ Soient $E$ l'ensemble des polynômes à coefficients dans $\{-1,0,1\}$ et $A$ l' #+end_exercice +# ID:7746 #+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))$. + - Soit $h_1\colon E\ra\R$ définie par $h_1(f)=\sum\limits_{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\colon E\ra\R$ définie par $h_2(f)=\sup_{t\in[0,1]}g(f(t))$. -Déterminer les points de continuité de $h_2$. + Déterminer les points de continuité de $h_2$. #+end_exercice +#+BEGIN_proof + - Sans difficulté. + - On a $g_2(f) = g(\sup f)$. La fonction $f\mapsto \sup f$ est continue, donc si $\sup f$ est sur un point de continuité de $g$, on est continue. + + Sinon, clairement non continue. +#+END_proof -#+begin_exercice [X MP 2024 # 319] +# Déjà vu… +#+begin_exercice [X MP 2024 # 319] :todo: Existe-t-il une fonction continue $f\colon\C\ra\C$ telle que $f\circ f=\exp$? #+end_exercice +#+BEGIN_proof + + $f$ est surjective sur $\C^*$, non surjective. + + $f(f(x + 2ik\pi)) = f(f(x))$ +#+END_proof #+begin_exercice [X MP 2024 # 320] @@ -2945,9 +3295,16 @@ Existe-t-il une fonction continue $f\colon\C\ra\C$ telle que $f\circ f=\exp$? #+end_exercice +# ID:7747 #+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 +#+BEGIN_proof +D'après le CG. Il y a une suite de termes, de plus en plus grand. et une suite de termes, de plus en plus petits. On encadre un petit par deux grands. $G_1 \leq p \leq G_2$. + +Si la suite est positive, alors $G_1 u_{G_1} + G_2 u_{G_2}$ contient une proportion non triviale de $p u_p$, et sont majorés. C'est impossible. +#+END_proof + # ID:7668 #+BEGIN_exercice [X MP 2024 # 322] @@ -3043,6 +3400,7 @@ Soit $f\colon\N^*\ra\R$ telle que $f(mn)=f(m)+f(n)$ pour tous $m,n\geq 1$. #+end_exercice +# ID:nil #+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$. @@ -3091,7 +3449,7 @@ Si $f$ est une fonction de $[0,1]$ dans $\R$, on note $V(f)$ la borne supérieur #+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. + - 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$ possède 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$. @@ -3111,7 +3469,7 @@ Soient $f\in\mc C^{\i}(\R,\R)$ et $(a,b)\in\R^2$ avec $a\lt b$. Montrer l'équiv #+begin_exercice [X MP 2024 # 340] -Soient $F$ un ferme de $\R$, $O=\R\setminus F$. +Soient $F$ un fermé 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 @@ -3145,14 +3503,20 @@ Pour $r\in\R$ et $n\in\N$, soit $D_n(r)=\int_{-1}^1(1-x^2)^n\cos(rx)dx$. #+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{\}}$. +#+begin_exercice [X MP 2024 # 344] :todo: +Soient $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_proof +C'est $\int_{-\i}^{+\i} |f|$. Soit $\eps$, l'intégrale est proche de $\int_{-\i}^{+\i} |f| 1_{|f|\geq \eps}$ (c'est de la triche, cette indicatrice n'est pas intégrable sans Lebesgue), mais on peut approcher $f$ de manière uniforme par des fonctions polynomiales… +#+END_proof #+begin_exercice [X MP 2024 # 345] -Nature de $\int_0^{+\i}\frac{e^x}{e^{-x}+e^{2x}|\sin x|}dx$? +Nature de $\int_0^{+\i}\frac{e^x}{e^{-x}+e^{2x}|\sin x|}\dx$? #+end_exercice +#+BEGIN_proof +On est ramené à une série, de $\int_0^{\pi} \frac{e^{x+n\pi}}{e^{-x-n\pi} + e^{2x + 2n\pi} \sin x}$, dont on cherche un équivalent. En fait, plutôt couper en $\frac{\pi}{2}$. C'est $\int_0^{\frac{\pi}{2}} \frac{t e^x}{e^{-x}/t + e^{2x} t^2 \sin x}\dx$, dont on veut un équivalent quand $t\ra +\i$. +#+END_proof # ID:7696 @@ -3185,7 +3549,7 @@ Soit $q\in\R^*$. Soit $a\in\mc C^0(\R,\R^*)$. Soit $m,M$ deux réels vérifiant #+begin_exercice [X MP 2024 # 350] -Soit $\sum a_nz^n$ une série entiere dont le rayon de convergence appartient à $]0,+\i[$. +Soit $\sum a_nz^n$ une série entière dont le rayon de convergence appartient à $]0,+\i[$. #+end_exercice @@ -3231,7 +3595,7 @@ Soit $n\geq 2$. On pose $g_n=\sum_{k=0}^n\frac{1}{2^{4k}}\binom{2k}{k}^2$. Soit $\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$. + - Soit $\sum a_kz^k$ une série entière 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 @@ -3316,7 +3680,7 @@ Pour $f\in\mc C^{\i}(\R,\R)$, on pose $H(f):x\mapsto x^2f(x)-f''(x)$, $A_-(f):x\ #+begin_exercice [X MP 2024 # 367] - - Soit $f$ une fonction croissante de $[a,b]$ dans $[a,b]$. Montrter que $f$ possede un point fixe. + - Soit $f$ une fonction croissante de $[a,b]$ dans $[a,b]$. Montrter que $f$ possède un point fixe. - On s'intéresse 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. @@ -3372,7 +3736,7 @@ Soient $n\in\N^*$ et $r\in\db{0,n}$, $\mc{P}$ l'ensemble des projecteurs orthogo #+end_exercice -** Geometrie +** Géométrie #+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$. @@ -3412,6 +3776,7 @@ Soit $r\in\N^*$. Combien y a-t-il de facon d'apparier les entiers de $1$ à $2r$ #+end_exercice +# ID:7736 #+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. @@ -3419,16 +3784,17 @@ Soit $n\in\N^*$. #+end_exercice + +# ID:7737 #+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$. +Soit $N\in\N^*$. Dénombrer les fonctions $f\colon\db{0,2N}\ra \db{0,2N}$ telles que $f(0)=f(2N)=0$ et $\forall k\in \db{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\}$. +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$. @@ -3570,7 +3936,7 @@ Déterminer les $f\in{\cal C}^0({\R},{\R})$ telles que: $\forall(x,y)\in{\R}^2$, #+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}$. + - Soit $\phi\in{\cal S}({\R})$. Déterminer une condition nécessaire et suffisante sur $\phi$ pour que $\phi$ possède une primitive appartenant à ${\cal S}$. #+end_exercice @@ -3622,7 +3988,7 @@ Soient $E_{n,d}=\{(i_1,...,i_d)\in{\N}^d,\;i_1+\cdots+i_d=n\}$ et #+end_exercice -** Geometrie +** Géométrie #+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. @@ -3632,8 +3998,8 @@ Soient $n\in\N^*$ et $x_1,\ldots,x_{2n}$ des points distincts de $\R^2$. Montrer ** 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? +Soit $X$ une variable aléatoire de loi $\mc{G}(1/2)$. Pour tout $k\in\N^*$, on note $A_k$ l'évènement \lt \lt $X$ est multiple de $k$\gt \gt . + - Soient $(p,q)\in(2\N^*)^2$. Les évènements $A_p$ et $A_q$ sont-ils indépendants? - Même question pour $p$ et $q$ quelconques dans $\N^*$. #+end_exercice @@ -3915,14 +4281,14 @@ Déterminer une condition nécessaire et suffisante sur $f:[\,0,1\,]\ra\R$ afin #+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 $f$ est développable en série entière 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. + - Chercher les $f\in\mc{S}$ développables en série entière. - L'espace $\mc{S}$ est-il de dimension finie? #+end_exercice @@ -3941,12 +4307,12 @@ La fonction $f:x\mapsto\sum_{k=0}^{+\i}(-1)^kx^{k!}$ admet-elle une limite lorsq #+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$. +Soit $(a_{k,n})_{(k,n)\in\N^2}$ une famille de nombres complexes telle que, pour tout $n\in\N$, la série entière $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$. +Soient $U$ un voisinage de $0$ dans $\C$, $k\in\N$ et $f$ une fonction de $U$ dans $\C$ développable en série entière 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 @@ -3967,7 +4333,7 @@ On admet le theoreme d'approximation de Weierstrass. Soit $f\colon\R\ra\R$ une f #+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. +Ind. Chercher les solutions développables en série entière. #+end_exercice @@ -4005,14 +4371,14 @@ Soient $N\in\N^*$ et $f\in\mc C^0(\R^N,\R)$. Montrer qu'il existe une suite $(f_ ** 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. +On lance une pièce 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. +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]$. Interpréter d'un point de vue probabiliste. #+end_exercice @@ -4270,7 +4636,7 @@ Soient $n\in\N$ et $x_1\lt x_2\lt ...\lt x_n$ réels. On note $V=(x_i^{j-1})_{1\ - 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]$. +Ind. On pourra interpréter $V$ comme matrice de passage dans $\R_{n-1}[X]$. #+end_exercice @@ -4453,7 +4819,7 @@ Réciproquement, ces conditions suffisent, car $e^{tA} = \left(e^{\frac{tA}{n}}\ #+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. + - Montrer que $J$ est diagonalisable dans $\M_n(\C)$, et preciser ses éléments propres. - Déterminer les éléments propres de la matrice $A$. #+end_exercice @@ -4473,7 +4839,7 @@ Soient $a,b\in\R$ avec $a^2\neq b^2$. Diagonaliser si possible la matrice $A\in\ #+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$. + - Montrer 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 @@ -4481,22 +4847,22 @@ Soit $A=$ $\begin{pmatrix}0&1&0&0\\ 1&k&1&1\\ 0&1&0&0\\ 0&1&0&0\end{pmatrix}\in\ #+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. + - - Montrer 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)$. +Soit $A\in\M_n(\R)$ une matrice diagonalisable dans $\M_n(\C)$. Montrer 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. + - Montrer qu'il existe un unique couple $(\alpha,\beta)\in\R^2$ tel que $\omega^2=\alpha\omega+\beta$. + - Montrer 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$. Montrer 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 @@ -4681,7 +5047,7 @@ Soient $n\in\N^*$, $A$, $B\in\M_n(\R)$, $P\in\R[X]$ et $M=\begin{pmatrix}A&B\\ 0 #+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$. +Soient $E$ un espace préhilbertien 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 @@ -5016,7 +5382,7 @@ Soit $Q\in\R[X]$ non nul. Pour $P\in\R[X]$, on pose $\|P\|_Q=\sup\limits_{x\in[- #+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$. +Soient $(E,\|\ \|)$ un espace vectoriel normé, $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 @@ -5039,7 +5405,7 @@ Soient $C$ une partie convexe d'un espace norme $E$, $X$ une partie de $E$ telle #+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}$. + - 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 fermé de $\M_n(\R)$ et que l'adherence de $\mc{D}$ est $\mc{T}$. #+end_exercice @@ -5069,7 +5435,7 @@ Que dire de sa limite? #+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 pour tout $n\in\N^*$, $(E_n)$ possède 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 @@ -5303,7 +5669,7 @@ Soit $f(t)=\dfrac{1}{\sqrt{1+t^2}}$. Montrer que la derivée $n$-ieme de $f$ s' #+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$. +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}$ possède une limite quand $x\ra 0$. #+end_exercice @@ -5543,7 +5909,7 @@ On pose $f:x\mapsto\sum_{n=1}^{+\i}\frac{e^{-nx}}{n+x}$. #+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. + - Montrer que $f$ possède une limite finie en $1$ et la déterminer. #+end_exercice @@ -5613,7 +5979,7 @@ Rayon de convergence et somme de $\sum_{n\geq 1}\cos\left(\frac{2\pi n}{3}\right #+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$. +Montrer que la fonction $f:x\mapsto\ln(1+e^{-x})$ est développable en série entière au voisinage de $0$. #+end_exercice @@ -5631,15 +5997,15 @@ Soit, pour $n\in\N$, $a_n=\int_0^{\pi/2}\cos(t)^n\sin(nt)\dt$. Soit $f:x\mapsto\ #+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$. + - Soit $z\in\C$ tel que $|z|\neq 1$. Montrer que la fonction $t\mapsto\frac{1}{e^t-z}$ est développable en série entière 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 entière 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? + - On pose, pour $n\in\N^*$, $b_n=\frac{1}{n}\left(\sum_{k=1}^na_k\right)$. La suite $(b_n)$ possède-t-elle une limite? - Déterminer le rayon de convergence de $\sum b_nx^n$. #+end_exercice @@ -5652,7 +6018,7 @@ Pour $n\in\N$, soit $a_n=\int_0^{\pi/4}\tan^n(x)\dx$. - Montrer que $(a_n)_{n\ge #+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$. +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 entière $\sum_{n\geq 0}\dfrac{u_n}{n!}x^n$. #+end_exercice @@ -5695,7 +6061,7 @@ Soit $(a_n)$ définie par $a_0=a_1=1$ et $\forall n\geq 1,\,a_{n+1}=a_n+\dfrac{2 #+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. +Soient $f(z)=\sum_{n=0}^{+\i}a_nz^n$ la somme d'une série entière 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$. @@ -5710,7 +6076,7 @@ $$|F(z)|\geq|F(0)+a_pz^p|-\sum_{n=p+1}^{+\i}|a_n||z|^n.$$ #+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 + - Montrer que la série entière $\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$. @@ -5719,7 +6085,7 @@ Soit $(b_n)$ la suite définie par $b_0=1$ et $\forall n\in\N^*$, $b_n=\sum_{k=1 #+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$. + - Montrer que $f$ est développable en série entière 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)$. @@ -5730,7 +6096,7 @@ Soit $f:z\in\C\setminus\{1\}\mapsto\exp\left(\dfrac{z}{1-z}\right)$. 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$. +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 entière $\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)}$. @@ -5738,7 +6104,7 @@ Donner une expression simple de $h:x\mapsto S(x)\,e^{S(x)}$. #+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$. +Soit $(a_n)_{n\geq 2}$ une suite réelle telle que la série entière 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$. @@ -5755,7 +6121,7 @@ Soit $(a_n)$ une suite de réels positifs avec $a_0\gt 0$ et $a_1=1$. Soient $S: #+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$. +Soit $\sum a_nz^n$ une série entière 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[$. @@ -5763,7 +6129,7 @@ Soit $\sum a_nz^n$ une série entiere de rayon de convergence $R\gt 0$ et de som #+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. +Soient $P\in\R[X]$ de degre $2$ et $f:x\mapsto e^{P(x)}$. Montrer que $f$ est développable en série entière en $0$ et que deux coefficients consécutifs de ce développement ne sont jamais simultanement nuls. #+end_exercice @@ -5782,7 +6148,7 @@ Déterminer un équivalent de $p(n)=\big{|}\big{\{}(x,y,z)\in\N^3,\;x+2y+3z=n\bi #+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^-$. +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$ possède 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}$. @@ -5890,7 +6256,7 @@ Soient $\alpha\gt 0$ et $f:x\in\R^{+*}\mapsto\int_0^1\frac{dt}{x^{\alpha}+t^3}$. #+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 déterminant le développement en série entière de $f(x)$; - en montrant que $f$ est de classe $\mc C^1$ et vérifie une équation différentielle lineaire d'ordre $1$. #+end_exercice @@ -5924,7 +6290,7 @@ Trouver toutes les fonctions $f\colon\R^{+*}\ra\R$ d $\mathrm{\acute{e}ivables\ 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. + - Montrer que $(E)$ possède une unique solution $f$ sur $\R$ telle que $f(0)=1$ et qui soit la somme d'une série entière. - Donner une expression de $f$ à l'aide de fonctions usuelles. #+end_exercice @@ -5933,14 +6299,14 @@ On considére l'équation différentielle $(E):2xy''+y'-y=0$. #+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. +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 entière. #+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. +Déterminer une solution de $(E):y''+xy'+y=1$ développable en série entière au voisinage de 0. #+end_exercice @@ -5956,7 +6322,7 @@ 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. + - Déterminer les solutions de $(E)$ développables en série entière. #+end_exercice @@ -6056,7 +6422,7 @@ 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. - Montrer 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. + - En déduire que $f$ possède au plus un point fixe. #+end_exercice @@ -6084,7 +6450,7 @@ _Ind._ Considérer $\psi:t\mapsto f(tx)-\langle\nabla f(0),tx\rangle-\frac{1}{2} #+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}$. + - Soit $x\in E\setminus\{0\}$. Interpréter $df(x)$ en faisant intervenir la reflexion d'axe $\{x\}^{\perp}$. - En déduire que $df(x)$ conserve les angles. #+end_exercice @@ -6244,15 +6610,15 @@ On dispose de deux urnes $A$ et $B$, et de $2N$ boules numérotées de 1 à $2N$ #+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 . +On lance simultanement deux pièces equilibrées $n$ fois. Soit $E_n$ l'évènement \lt \lt les deux pièces 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. + - Déduire combien de fois en moyenne les pièces sont tombées sur Pile lorsque l'évènement $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)$. +Soient $A$ et $B$ deux évènements. 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 @@ -6312,9 +6678,9 @@ Soient $X$ une variable aléatoire suivant la loi de Poisson de paramêtre $\lam #+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_p)$ ou $A_p$ est l'évènement $\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. + - On note $B$ l'évènement $\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 @@ -6334,12 +6700,12 @@ Soit $(X_n)_{n\geq 1}$ une suite de variables i.i.d. de loi de Bernoulli ${\cal #+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. +Soit $(E_n)_{n\geq 0}$ une suite d'évènements 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}$. + - Soit $n\in\N^*$. Donner le développement en série entière 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[$. @@ -6362,7 +6728,7 @@ Soient $k\in\N$ avec $k\geq 2$, $a_0,\ldots,a_{k-1}\in\,]0,1[^k$ tels que $a_0+\ #+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. +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$ possède une esperance et la calculer. #+end_exercice @@ -6383,7 +6749,7 @@ Soient un espace probabilise $(\Omega,{\cal A},{\bf P})$ et une variable aléato #+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. + - Rappeler le développement en série entière 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 @@ -6392,7 +6758,7 @@ Soient un espace probabilise $(\Omega,{\cal A},{\bf P})$ et une variable aléato #+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? + - Pour $n\in\N^*$, on note $A_n$ l'évènement $(n|X)$. Les évènements $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 @@ -6504,13 +6870,13 @@ Soit $X$ une variable aléatoire réelle discrete. #+begin_exercice [Mines MP 2024 # 853] Soient $k\leq n\in\N$. Un parking dispose de $n$ places consécutives numérotées 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. +Pour $j\in\db{1,n-k+1}$, $B_j$ designe l'évènement $\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$. +Montrer que la somme $f$ de la série entière $\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 @@ -6521,7 +6887,7 @@ Montrer que la somme $f$ de la série entiere $\sum S_nt^n$ est au moins défini #+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$. + - Interpréter geometriquement $A$. - Donner l'image du plan $P$ d'équation $x-y-z=0$ par $A$. #+end_exercice @@ -6749,7 +7115,7 @@ En considérant une division euclidienne, montrer que : $\forall k\in\N,\ A^k\in 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. + - Soient $A,B\in\M_n(\C)$ telles que $AC=CB$. Montrer que $A$ et $B$ possèdent $r$ valeurs propres communes en tenant compte des multiplicités. - Que peut-on dire dans - quand $r=n$? #+end_exercice @@ -6793,7 +7159,7 @@ Pour $P,Q\in E$, on note $\Phi(P,Q)=\sum_{k=0}^4\big(P(k)+(-1)^kP(-k)\big)\big(Q #+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? +Soit $E$ un espace euclidien de dimension $3$. On considére une isométrie indirecte $f$. Montrer 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 @@ -6801,25 +7167,25 @@ Soit $E$ un espace euclidien de dimension $3$. On considére une isométrie indi 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. + - Montrer 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}$. +Soit $(E,\langle\,\ \rangle)$ un espace préhilbertien réel et $F$ un sous-espace vectoriel de $E$. + - Montrer 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}$. + - Pour $E$ préhilbertien, 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)$. + - Montrer que $\colon\ G$ est inversible si et seulement si $(x_1,\ldots,x_p)$ est libre. + - Montrer que $\op{rg}(G)=\op{rg}(x_1,\ldots,x_p)$. #+end_exercice @@ -6828,12 +7194,12 @@ Soit $(E,\langle\,\ \rangle)$ un espace euclidien et $F$ une partie fermée, non 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. + - Montrer 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$. + - Montrer que $\Gamma(x)$ est reduit à un seul élément, que l'on notera $p(x)$. + - Montrer que $p(x)$ est caractérise par $\colon\forall y\in F,\ \langle x-p(x),y-p(x)\rangle\ \leq 0$. #+end_exercice @@ -6850,7 +7216,7 @@ On munit $\R^3$ de sa structure euclidienne canonique. Soit $u$ l'endomorphisme #+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]$. + - Montrer 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 @@ -6914,7 +7280,7 @@ Soit $A\in\mc{S}_n(\R)$. On dit que $A\in\mc{S}_n^{++}(\R)$ lorsque, pour toute #+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. + - Montrer que $N_{\phi}$ est une norme si et seulement si $\phi^{-1}(\{0\})$ est d'intérieur vide. - Montrer que $N_{\phi}$ et $\|\ \|_{\i}$ sont équivalentes si et seulement si $\phi^{-1}(\{0\})$ est vide. #+end_exercice @@ -6929,7 +7295,7 @@ Soient $E$ un $\R$ espace vectoriel, $N_1$ et $N_2$ deux normes sur $E$. #+begin_exercice [Mines PSI 2024 # 909] -Soit $(E,\|\ \|)$ un espace vectoriel norme. Une suite $(u_n)\in E^{\N}$ est de Cauchy si +Soit $(E,\|\ \|)$ un espace vectoriel normé. 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. @@ -6937,7 +7303,7 @@ Soit $(E,\|\ \|)$ un espace vectoriel norme. Une suite $(u_n)\in E^{\N}$ est de 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. + - Montrer que, si $(u_n)$ est de Cauchy et possède 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 @@ -7125,12 +7491,12 @@ Pour $a\in\R$, on considére la suite de fonctions définie par $f_0=1$ et, pour 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$. + - En utilisant le theoreme de Fubini, montrer que $S$ est développable en série entière 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}$. + - Déterminer le rayon de convergence $R$ de la série entière $\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 @@ -7149,7 +7515,7 @@ Soit $f\ :x\mapsto\sum_{n\geq 0}\binom{2n}{n}x^n$. 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. + - Montr'er que $F$ est développable en série entière et déterminer ce développement. #+end_exercice @@ -7162,13 +7528,13 @@ 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$. + - Soit $:x\mapsto\sum_{n=0}^{+\i}m_nx^n$. Montrer que le rayon de convergence de cette série entière 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$. + - Montrer que $g$ est développable en série entière au voisinage de $0$. - Donner un encadrement du rayon de convergence. #+end_exercice @@ -7195,13 +7561,13 @@ On pose $f(x,s)=\sum_{n=1}^{+\i}\frac{x^n}{n^s}$. 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$. + - Donner le rayon de convergence $R$ de la série entière $\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. +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 entière. - 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[$. @@ -7262,7 +7628,7 @@ Soit $f\,\colon\,\alpha\mapsto\int_0^{+\i}\frac{dt}{t^{\alpha}(t+1)}$. #+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. + - 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)$ possède une limite et la déterminer. - Trouver un équivalent de $u_n$ en $+\i$. #+end_exercice @@ -7270,7 +7636,7 @@ Soit $f:x\mapsto\int_0^{+\i}\arctan(xt)\,e^{-t}\dt$. #+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$. + - Montr e que $f$ est développable en série entière au voisinage de $0$. - Calculer ce développement et en déduire une expression $f(x)$. #+end_exercice @@ -7279,8 +7645,8 @@ Soit $f:x\mapsto\int_0^{+\i}\frac{1}{x+e^t}dt$. 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? + - Déterminer les solutions de $(E)$ développables en série entière et preciser le rayon. + - La fonction $g$ est-elle développable en série entière? #+end_exercice @@ -7332,7 +7698,7 @@ Soit $f\colon\R\ra\R$ une fonction de classe $\mc C^{\i}$ telle que $f(0)=0$. So #+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. + - Montrer que $(E)$ n'admet pas de solution développable en série entière. - Résoudre l'équation différentielle sur $]0,+\i[$. - Montrer qu'il existe une unique solution tendant vers $0$ en $0^+$. #+end_exercice @@ -7357,7 +7723,7 @@ On s'intéresse aux solutions $f:x\mapsto\sum_{n\geq 0}a_nx^n$ de l'équation di #+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 les solutions de $(E)$ non nulles développables en série entière. Preciser le rayon de convergence. - Déterminer l'ensemble des solutions de $(E)$ sur un intervalle raisonnable. - Les raccorder entre elles. #+end_exercice @@ -7397,7 +7763,7 @@ Soit $F=\left\{f_{a,b,c},\ (a,b,c)\in\R^3\right\}$. #+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 $J$ est développable en série entière 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 @@ -7406,7 +7772,7 @@ Soit $J:x\mapsto\int_0^{\pi}\cos(x\sin(\theta))\,d\theta$. #+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[$. + - Monter que $f$ est de classe $\mc C^1$ sur $\Omega$. - Comparer $f(1/x,y)$ et $f(x,y)$. Donner une interprétation 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 @@ -7454,7 +7820,7 @@ On considére initialement une urne contenant une boule blanche et une boule rou #+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. + - 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'évènement : 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$. @@ -7480,7 +7846,7 @@ Déterminer la loi de $v_p(X)$ puis son esperance. #+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. +On effectue des lancers avec une pièce dont la probabilité de donner pile est $p\in]0,1[$. On lance la pièce 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 numérotées de $0$ à $n$ dans l'une. Donner la loi de $Y$ ou $Y$ est le numéro de la boule tirée dans l'urne. Calculer ensuite l'esprance de $Y$ ainsi que sa variance. @@ -7506,7 +7872,7 @@ Soit $(X_k)$ une suite de variables aléatoires i.i.d. suivant la loi de Bernoul 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))$. + - Établir une relation de récurrence lineaire d'ordre deux vérifiée par $(\mathbf{P}(T=n))$. - Calculer l'esperance de $T$. #+end_exercice @@ -7556,7 +7922,7 @@ Ind. Exprimer $\mathbf{E}(A_{n+1})$ et appliquer la formule des probabilités to 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\}$. + - Généraliser à $B_k=\{a_1+a_2+\cdots+a_k\;;\;a_1,...,a_k\in A\}$. #+end_exercice @@ -8125,7 +8491,7 @@ Soit $E$ un espace euclidien. Soit $u\in{\cal S}^{++}(E)$. Montrer qu'il existe #+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})$. + - Soit $A\in{\cal M}_2({\C})$. Montrer que $\left\{Q(A)\ ;\ Q\in{\C}[X]\right\}$ est un fermé 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 @@ -8180,7 +8546,7 @@ Pour $n\in\N^*$, on pose $x_n=\sum_{k=1}^n\dfrac{1}{k},H_n=\sum_{k=1}^n\dfrac{1} - 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. + - Établir la convergence de $(u_n)$ et preciser sa limite. #+end_exercice @@ -8437,15 +8803,15 @@ Soit $F:x\mapsto\sum_{n=1}^{+\i}\mathrm{e}^{-n^2x^2}$. Déterminer les limites e 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)}$. + - Montrer que $f$ est continue sur $\R\setminus\Z$ et 1-periodique. + - Montrer que $f$ vérifie $(*)$. + - Montrer que, si $g$ est continue sur $\R$, 1-periodique et vérifie $(*)$ alors $g$ est nulle. + - Montrer 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? +Preciser le domaine de définition de $f:x\mapsto\sum_{n\geq 0}e^{-n}e^{in^2x}$. Montrer que l'application $f$ est de classe $\mc C^{\i}$ sur $\R$. Est-elle développable en série entière? #+end_exercice @@ -8456,7 +8822,7 @@ Preciser le domaine de définition de $f:x\mapsto\sum_{n\geq 0}e^{-n}e^{in^2x}$. #+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^{+*}$. + - Montrer 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^+$. @@ -8472,23 +8838,23 @@ Trouver $f$ de classe $\mc C^{\i}$ au voisinage de 0 telle que $\forall n\in\N$, #+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. + - Montrer que $f$ est de classe $\mc C^{\i}$. + - Montrer que $f$ est développable en série entière. #+end_exercice #+begin_exercice [Mines PC 2024 # 1125] -On s'intéresse à la série entiere suivante $:f(x)=\sum_{n=1}^{+\i}u_nx^n$ avec $u_n=\int_1^{+\i}e^{-t^n}\dt$. +On s'intéresse à la série entière 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 le domaine de convergence de la série entière. - 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})}$. + - Montrer 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]$. + - Montrer 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 @@ -8496,7 +8862,7 @@ Soit $N$ un entier qui n'est pas un carre parfait. On pose $a=\sqrt{N}$. 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. + - Montrer que $x\mapsto\op{cotan}(x)-\frac{1}{x}$ est développable en série entière. - Quel est le lien entre les deux dernieres questions? On pourra poser $z=2i\pi x$. #+end_exercice @@ -8510,7 +8876,7 @@ Soit $S:x\mapsto\sum_{n=0}^{+\i}\frac{x^n}{\binom{2n}{n}}$. #+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)$. +Montrer qu'il existe une fonction $\phi$ développable en série entière en 0 vérifiant au voisinage de 0 : $\phi'\left(x\right)=x+\phi^2\left(x\right)$. #+end_exercice @@ -8665,7 +9031,7 @@ Ind. Considérer $g:t\mapsto\int_a^b\left(\int_c^tf(x,y)dy\right) dx$. #+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. + - Trouver les solutions de $(E)$ développables en série entière et déterminer leur rayon de convergence. - Écrire ces fonctions à l'aide des fonctions usuelles. #+end_exercice @@ -8755,7 +9121,7 @@ Dans un magasin, on a $n$ caisses et $np$ clients. Chaque client choisit une cai #+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}$. +Soient $A$ et $B$ deux évènements. Montrer que $|\mathbf{P}(A)\mathbf{P}(B)-\mathbf{P}(A\cap B)|\leq\dfrac{1}{4}$. #+end_exercice @@ -8864,7 +9230,7 @@ Soient $X$ et $Y$ deux variables aléatoires discretes à valeurs strictement po #+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. + - Soit $k\in\N$. Exprimer l'évènement $(S\geq k)$ à l'aide des évènements $(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 @@ -8912,7 +9278,7 @@ On note $\mathbb{U}$ le groupe des nombres complexes de module $1$. Soit $q$ un #+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$. + - Montrer que seuls les polynômes de degre 1 possèdent 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 @@ -9050,10 +9416,10 @@ Soient $n\in{\N}^*$ et $p$ un entier premier impair. On note ${\rm GL}_n({\Z})$ #+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$. + - Montrer que $\chi_A(B)$ et $\chi_B(A)$ sont inversibles. + - Montrer 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. + - Montrer 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 @@ -9062,7 +9428,7 @@ Soient $n\in{\N}^*$ et $p$ un entier premier impair. On note ${\rm GL}_n({\Z})$ - 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)$. + - 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 entièrement 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 @@ -9070,15 +9436,15 @@ On appelle representation de degre $n$ un morphisme de groupes de $\mc{S}_3$ dan #+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$. + - Montrer 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)$. + - Montrer 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]$. + - Montrer 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 @@ -9298,7 +9664,7 @@ Notons $\mc C$ l'espace des fonctions continues de $[0,1]$ dans $\R$ muni de la #+begin_exercice [Centrale MP 2024 # 1227] - - Soient $E$ et $E'$ deux espaces vectoriels normes et $u\in\mc{L}(E,E')$. + - Soient $E$ et $E'$ deux espaces vectoriels normés et $u\in\mc{L}(E,E')$. Montrer que $u$ est continue sur $E$ si et seulement si elle est continue en $0$. @@ -9483,7 +9849,7 @@ Pour $n\in\N^*$, soient $\Omega(n)$ le nombre de facteurs premiers de $n$ compte #+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}$. + - Retrouver le développement en série entière 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}$. @@ -9498,14 +9864,14 @@ Montrer que $\left|\pi-S_n+(-1)^n\left(\dfrac{1}{n}-\dfrac{1}{n^2}\right)\right| - 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 : + - Soit $\sum a_nz^n$ une série entière 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 entière 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 entière 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$. +Soient $R\gt 0$ et $\mc{A}_R$ l'ensemble des fonctions $f\colon\R\ra\R$ développables en série entière 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. @@ -9515,8 +9881,8 @@ Soient $R\gt 0$ et $\mc{A}_R$ l'ensemble des fonctions $f\colon\R\ra\R$ dévelop #+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$. +Est-elle développable en série entière 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 entière au voisinage de $0$. #+end_exercice - Étudier la réciproque. @@ -9524,7 +9890,7 @@ Est-elle développable en série entiere au voisinage de $0$? #+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$. +Soit $f$ une fonction développable en série entière au voisinage de $0$, telle que $f(0)\neq 0$. Montrer que la fonction $\frac{1}{f}$ est développable en série entière 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[$. @@ -9578,11 +9944,11 @@ Soient $A\in{\cal S}^{++}_n({\R})$, $b\in{\R}^n$ et $f$ la fonction de ${\R}^n$ #+end_exercice -** Geometrie +** Géométrie #+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]$. + - 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 intérieurs 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 @@ -9629,7 +9995,7 @@ Ind. Pour $\eps\gt 0$ fixe, considérer une partie finie $I\subset E$ telle que #+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? +Une suite $(Z_n)_{n\geq 1}$ de variables aléatoires entières 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 diff --git a/Exercices 2024.pdf b/Exercices 2024.pdf index 09b99419f377fed3929dbb11d338b76625d5ec82..29517a1425a8f3d09278e2c6f94d4b2b225c68c5 100644 GIT binary patch delta 895565 zcmV)GK)%1d#zz0ZMvx`}H<#gl0W5!W+cp+{&#&NaA4<(e2oCOOHxIiRr{m6~^;Vuv zW|BV0w&LZkBefEprv3Fj;1#4nk+S05Cc#5Sv_qW>@Zo$Pa3q>0*9lF&c|=EE|9SrC z$#bg`(1d{$$@!av2}PvViJ^u7=gIk1@+v)fzuL}BdYN6WcC$H?G-Z^D*-wAx-;TOK z`#J$ZxKPa5pEEFoaY*Le5{<8ggY%iN>GI-cRa(EQ-Mn4q*=sJT^NznO9nu7nt+Vbf zeyChnVii-)19d)#VU_9A*Hpr64hl?q^4+4e`{Jyd2AZ&*31$HUwrHv_@qV@5b&~CQ zWP%X@Uf&MGaVqa!bBQB?|B`<>0I^b5Z!W`CT&MrIxpui<6v=n|C$Hv;(Ui>QlE#NIa{3WmpeuiZX3C?yv?_({d$L@>sIgR$%2QYpD1`fM^x*HOcxc; zCk_NNk>q$Jxlb1bj2=aD70VeTT9TVrT=y(QaN6Jo{Q!cKlrYv19D_)35z!-vhDdM` z2o4DRaDt0_`cD=OGVXurBMR72dwLW$BK@(FJXWMHJA&uMSy1q3!nmH?z>q+I%0};~ zd+8k&_0DGmX%$--5eNtb4pHw!G&E>XQSU?)aCINOqf2D`Shp~Y>`5Kh#|UX1PMFaL z9PUU$Wg~WY>|R7XkKtKIu@ixC2oR#!`5e$-qS%Qj7(~G&u~UDwvMiXM*g098Jv*Jy 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