From 9764e7844774ebab2720ed815b8d7004d651c538 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?S=C3=A9bastien=20Miquel?= Date: Sat, 8 Jun 2024 18:04:50 +0200 Subject: [PATCH] Miscs --- Exercices 2022.org | 18 +- Exercices 2022.pdf | Bin 804337 -> 545033 bytes Exercices 2023.org | 800 ++++++++++++++++++++++++------------- Exercices 2023.pdf | Bin 1122995 -> 1143659 bytes Exercices XENS MP 2023.pdf | Bin 737131 -> 584814 bytes 5 files changed, 524 insertions(+), 294 deletions(-) diff --git a/Exercices 2022.org b/Exercices 2022.org index 76591a3..d611780 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: <16-04-24 18:59> +# Time-stamp: <03-06-24 17:16> * Meta :noexport: @@ -23,6 +23,7 @@ #+EXCLUDE_TYPES: indication +#+exclude_types: proof * ENS @@ -2364,21 +2365,6 @@ Soit $(X_n)_{n\in\N}$ une suite de variables aléatoires indépendantes de même -# ID:6499 -#+BEGIN_exercice -Soit $p\in \interval]{0, \frac{1}{2}}[$, et $(X_n)_{n\in\N^*}$ une suite de variables aléatoires indépendantes de même loi $\mc R(p)$. On pose $S_n = X_1 + \dots + X_n$. - 1. Montrer qu'il existe $t_0\gt 0$ tel que $pe^{t_0} + (1-p)e^{-t_0}\lt 1$. - 2. Montrer qu'il existe $\a,\b\in \interval]{0, 1}[$ tels que - $$\forall n\in\N,\forall k\in\Z,\, P(S_n\geq k)\leq \a^k \b^n.$$ - 3. Montrer que $S_n$ tend vers $-\i$ presque sûrement. -#+END_exercice -#+BEGIN_proof - 1. La dérivée en $0$ est $2 p - 1\lt 1$. - 2. $P(S_n\geq k) = P(e^{tS_n}\geq e^{tk})\leq \frac{E(e^{t S_n})}{e^{tk}}$, d'où le résultat, avec $t = t_0$. On a $\b\lt 1$, d'après la première question. - 3. 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+141,7 @@ python3 -c "import torch; print(f'device name [0]:', torch.cuda.get_device_name( ** Options #+OPTIONS: latex:verbatim - +#+exclude_types: proof *** All # #+OPTIONS: toc:t @@ -158,6 +158,19 @@ python3 -c "import torch; print(f'device name [0]:', torch.cuda.get_device_name( #+exclude_tags: autre #+export_file_name: Exercices XENS MP 2023 +*** Centrale + +# #+select_tags: cent +# #+export_file_name: Exercices Centrale 2023 + +*** Mines + +# #+select_tags: mines +# #+export_file_name: Exercices Mines 2023 + + + + * ENS MP-MPI :xens: ** Algèbre @@ -343,11 +356,7 @@ Soit $z\in\C$ annulé par un polynôme unitaire à coefficients entiers. Soit $Q #+BEGIN_proof On note $P$ le polynôme unitaire qui annule $z$ (polynôme minimal, via lemme de Gauss). -Pour $z^2$, je vois mal quoi faire, si ce n'est $P = \prod (X - x_i^2)$. - -Par dimension, on sait que $Q(z)$ admet un polynôme annulateur dans $\Q[X]$. - -!! +En passant par la matrice compagnon, on sait que $z$ est valeur propre d'une matrice à coefficients entiers, donc $Q(z)$ aussi. #+END_proof @@ -533,6 +542,10 @@ On pose $A=\left(\begin{array}{cc}p & a+i b \\ a-i b & m\end{array}\right)$. Mon #+END_exercice #+BEGIN_proof On a une matrice hermitienne, de déterminant $1$. Donc diagonalisable ? !! + +Les équations donnent : on cherche quatre complexes $u,v,w,z$ tels que $u\ol{v} + w\ol{z} = a+ib$ et $p = |u|^2 + |w|^2$ et $m = |v|^2 + |z|^2$. On peut remplacer la dernière équation par $uz - vw = 1$, ou $|uz - vw| = 1$. + +Dans $\Q[i]$, on peut résoudre $u\ol{v} + w \ol{z} = a + ib$ et $uz - vw = 1$, quels que soient $u, w\in \Q[i]$, puis en multipliant par la racine de $\frac{|u|^2 + |w|^2}{p}$, on vérifie l'équation $p = \dots$. Mais on a un élément de $\sqrt{D}\Q[i]$. On peut le multiplier par un élément de $\m U$, donc il faudrait un élément de $\sqrt{D} \Q[i]\cap \m U$, c'est-à-dire montrer que $D$ est une somme de quatre carrés (de rationnels). #+END_proof @@ -598,11 +611,9 @@ Soient $A\in\M_n(\C)$ et $m$ la multiplicité de $0$ dans $\chi_A$. On suppose q + pour tout $k\geq 1$, il existe $M\in\M_n(\C)$ telle que $M^k = A$. #+END_exercice #+BEGIN_proof - $(iii)\Rightarrow (ii)$ - - $(iii)\Rightarrow (i)$ est simple, via les noyaux itérés. - - $(i)\Rightarrow (iii)$ : Décomposition des noyaux, on est ramené au cas $A$ inversible. !! + + $(iii)\Rightarrow (ii)$ + + $(iii)\Rightarrow (i)$ est simple, via les noyaux itérés. + + $(i)\Rightarrow (iii)$ : Décomposition des noyaux, on est ramené au cas où $A$ inversible, puis réduction, au cas où $A$ est $\la I_n + N$, puis méthode de DL. #+END_proof @@ -683,17 +694,22 @@ On note $H$ l'ensemble des matrices de $\M_2(\R)$ de trace nulle. #+END_proof +# Relier à 2416 #+BEGIN_exercice [ENS 2023 # 57] -Soient $E$ un espace vectoriel réel de dimension finie, $h_1$ et $h_2$ deux éléments de $\mc{L}(E)$. On suppose qu'il existe une norme sur $E$ pour laquelle $h_1$ et $h_2$ sont des isométries et que $\left[h_1, h_2\right]=h_1 h_2 h_1^{-1} h_2^{-1}$ commute avec $h_1$ et $h_2$. Montrer que l'espace des vecteurs de $E$ fixes par $h_1$ et $h_2$ admet un supplémentaire dans $E$ stable par $h_1$ et $h_2$. +Soient $E$ un espace vectoriel réel de dimension finie, $h_1$ et $h_2$ deux éléments de $\mc{L}(E)$. On suppose qu'il existe une norme sur $E$ pour laquelle $h_1$ et $h_2$ sont des isométries et que $\left[h_1, h_2\right]=h_1 h_2 h_1^{-1} h_2^{-1}$ commute avec $h_1$ et $h_2$. Montrer que l'espace $F$ des vecteurs de $E$ fixes par $h_1$ et $h_2$ admet un supplémentaire dans $E$ stable par $h_1$ et $h_2$. #+END_exercice #+BEGIN_proof -On peut supposer que l'ensemble $F$ des points fixes est de dimension $1$. Donc est le noyau d'une forme linéaire $\phi$. +Pour la norme euclidienne. + + Si $h_1 = h_2$ : On prend $F^{\bot}$. + + Si $h_1$ et $h_2$ commutent. On a $F = F_1 \cap F_2$, $F^{\bot} = F_1^{\bot} + F_2^{\bot}$, mais $F_1^{\bot}$ est stable par $h_1$, et par $h_2$, car $h_2$ commute avec $h_1$, donc stabilise $F_1$. + + L'ensemble $F_{com}$ des points fixes de $[h_1,h_2]$ contient $F_0$, et son orthogonal est stable par $h_1$ et $h_2$. Par ailleurs, dans $F_{com}$, $h_1$ et $h_2$ commutent. -Notons $C$ le commutateur. On a $Ch_2 = h_1h_2 h_1^{-1}$. +Pour une autre norme. + + Si $h_1 = h_2$, on prend $(\sum_{k=0}^n h^k) (I_n - h) = I_n - h^n$ est bornée, donc $\frac{1}{n}\sum_{k=0}^n h^k$ vaut $\op{Id}$ sur les points fixes, et tend vers $0$ sur $\Im (I_n - h)$. Alors $\Im (I_n - h)$ est stable, et supplémentaire. -Si $h_1$ et $h_2$ commutent. - -Si $h_1 = h_2$. !! + Rmq : Cet espace est l'ensemble des éléments tels que $(\sum^n h^k)(x)$ soit bornée. + + Si $h_1$ et $h_2$ commutent, $F = F_1 \cap F_2$, on considère $G = \Im (I_n - h_1) + \Im (I_n - h_2)$. $G$ est stable par commutativité. $G$ est en somme directe avec $F$ : si $x \in G\cap F$, $x = x_1 + x_2$, alors en appliquant $h_1$, $\sum_{k,\l} h_1^k h_2^{\l} (x) \sim n^2$, mais $\sum h_1^k h_2^{\l} (x_1) = O(n)$, idem pour l'autre. + + Même chose que le précédent. #+END_proof #+BEGIN_exercice [ENS 2023 # 58] @@ -702,7 +718,7 @@ Soit $A\in\M_n(\C)$ et $\la_1,\dots,\la_n$ ses valeurs propres. 2. Montrer que $|\det A|\leq n^{n/2}\sup |a_{ij}|$. #+END_exercice #+BEGIN_proof - 1. !! + 1. C'est l'inégalité de Schur, dont la seule preuve que je trouve repose sur la décomposition de Schur : $A$ est unitairement semblable à une matrice triangulaire supérieure : on choisit un vecteur propre, puis on recommence dans l'orthogonal… 2. IAG probablement. #+END_proof @@ -717,7 +733,7 @@ Easy, on a $\langle x, p(v_i)\rangle = \langle p(x), v_i\rangle = \langle u_i, x On munit ${\R}[X]$ du produit scalaire $(P,Q)\mapsto\left\langle P,Q\right\rangle=\int_0^{+\i}P(t)Q(t)e^{-t}\, dt$. On pose $F=\text{Vect}\,(X,X^2,\ldots,X^n)$ et on note $Q$ la projection orthogonale de $1$ sur $F$. On écrit $Q=-\sum_{k=1}^na_kX^k$ et $P=1+\sum_{k=1}^na_k(X+1)\ldots(X+k)$. - - Déterminer $\left\langle Q-1,X^k\right\rangle$ pour $k\in\db{1,n]\!]$ et montrer que $P(k)=0$ pour $k\in[\![1,n}$. + - Déterminer $\left\langle Q-1,X^k\right\rangle$ pour $k\in\db{1,n}$ et montrer que $P(k)=0$ pour $k\in\db{1,n}$. - Calculer $\inf_{(a_1,\ldots,a_n)\in{\R}^n}\int_0^{+\i}(1+a_1x+ \cdots+a_nx^n)^2e^{-x}\,dx$. #+end_exercice #+BEGIN_proof @@ -726,16 +742,21 @@ On écrit $Q=-\sum_{k=1}^na_kX^k$ et $P=1+\sum_{k=1}^na_k(X+1)\ldots(X+k)$. #+END_proof -#+BEGIN_exercice [ENS 2023 # 61] -Soient $(E,\langle\rangle)$ un espace euclidien, $m \in \N^* \et u, u_1, \ldots, u_m$ des vecteurs de $E$. Montrer que $u \in \R^+ u_1+\cdots+\R^+ u_m$ si et seulement si pour tout $x \in E$, $\left\{x \in E ; \forall i \in \db{1, m },\left\langle u_i, x\right\rangle \leq 0\right\} \subset\{x \in E ;\langle u, x\rangle \leq 0\}$. +#+BEGIN_exercice [ENS 2023 # 61] Lemme de Farkas +Soient $(E,\langle\cdot\rangle)$ un espace euclidien, $m \in \N^* \et u, u_1, \ldots, u_m$ des vecteurs de $E$. Montrer que $u \in \R^+ u_1+\cdots+\R^+ u_m$ si et seulement si pour tout $x \in E$, $\left\{x \in E \mid \forall i \in \db{1, m },\left\langle u_i, x\right\rangle \leq 0\right\} \subset\{x \in E \mid \langle u, x\rangle \leq 0\}$. #+END_exercice #+BEGIN_proof - $\Rightarrow$ : Easy. + + $\Rightarrow$ : Easy. + + $\Leftarrow$ : Si les vecteurs $u_i$ sont libres, on peut prendre un élément $x$ orthogonal à tous sauf $1$, qui implique que la coordonnée de $u$ en $u_i$ est bien positive. - $\Leftarrow$ : Si les vecteurs $u_i$ sont libres, on peut prendre un élément $x$ orthogonal à tous sauf $1$. + Si $u$ n'est pas dans le cone convexe, alors il existe un hyperplan qui le sépare du cone, et on peut alors construire un vecteur $x$ qui a un produit scalaire $\leq 0$ avec tous les $u_i$, mais un produit scalaire $\lt 0$ avec $u$. +#+END_proof - Sinon, si $u_m$ est combinaison linéaire des précédents, avec un coefficient $\lt 0$. -!! +#+BEGIN_theorem +Si $C$ est un cone convexe fermé, $C^{\circ} = \{y\mid \forall x\in C,\, \langle y, x\rangle\leq 0\}$. Montrer que $\big(C^{\circ}\big)^{\circ} = C$. +#+END_theorem +#+BEGIN_proof +Utilise la projection sur un fermé. #+END_proof @@ -812,7 +833,11 @@ Soit $n\in{\N}^*$. On pose $J=\begin{pmatrix}0_n&-I_n\\ I_n&0_n\end{pmatrix}$. Soit $A \in \mc{S}_n(\R)$. On note $\lambda_1 \leq \cdots \leq \lambda_n$ les valeurs propres de $A$ non nécessairement distinctes. Montrer que $\forall k \in \db{1,n}, \sum_{i=1}^k \lambda_i \leq \sum_{i=1}^k a_{i, i} \leq \sum_{i=1}^k \lambda_{n+1-i}$. #+END_exercice #+BEGIN_proof -!! +On montre que toutes les valeurs propres de la matrice extraite sont $\geq$ que celles de $A$. + +D'une part la plus petite est $\geq \la_1$. D'autre part, la seconde est $\min_{\dim F = 2} \max_{x\in F} \langle Ax, x\rangle$, qui est supérieur au même min sur tous les espaces. + +On obtient l'autre inégalité en passant à l'opposé. #+END_proof #+BEGIN_exercice [ENS 2023 # 71] @@ -835,7 +860,7 @@ Soit $A\in{\cal S}_n({\R})$. On définit $p(A)$ comme la dimension maximale d'un #+begin_exercice [ENS 2023 # 73] On note $\lambda_1(M)\leq\cdots\leq\lambda_n(M)$ le spectre ordonne d'une matrice $S$ de ${\cal S}_n({\R})$. - Soient $A$ et $B$ dans ${\cal S}_n({\R})$ telles que $A+B\in{\cal S}_n^+({\R})$. Si $1\leq i,j\leq n$ et $i+j\geq n+1$, que dire du signe de $\lambda_i(A)+\lambda_j(B)$ -Soient $a\leq b$ deux réels, et $(O - {i\in I}$ une famille d'ouverts de $\R$ telle que $[a,b]\subset\bigcup_{i\in I}O_i$. On note $X$ l'ensemble des $x\in[a,b]$ tels qu'il existe une partie finie $J\subset I$ vérifiant $[a,x]\subset\bigcup_{j\in J}O_j$. Montrer que $X=[a,b]$. +Soient $a\leq b$ deux réels, et $(O_i)_{i\in I}$ une famille d'ouverts de $\R$ telle que $[a,b]\subset\bigcup_{i\in I}O_i$. On note $X$ l'ensemble des $x\in[a,b]$ tels qu'il existe une partie finie $J\subset I$ vérifiant $[a,x]\subset\bigcup_{j\in J}O_j$. Montrer que $X=[a,b]$. #+end_exercice #+BEGIN_exercice [ENS 2023 # 74] @@ -918,11 +943,11 @@ Soit $A\in\M_n(\mathbb{K})$, ou $\mathbb{K}=\R$ ou $\mathbb{K}=\C$. Montrer que #+end_exercice #+begin_exercice [ENS 2023 # 85] - - On note $D$ le disque unite du plan euclidien $\R^2$. Démontrer qu'il existe une suite $(C - {i\in\N}$ de parties de $D$ telle que : + - On note $D$ le disque unite du plan euclidien $\R^2$. Démontrer qu'il existe une suite $(C_i)_{i\in\N}$ de parties de $D$ telle que : + pour tout $i\in\N$, l'ensemble $C_i$ soit un carré de $\R^2$ dont les cotes sont parallèles aux axes ; + les $C_i$ soient d'intérieurs deux a deux disjoints ; + $\sum_{i\in\N}\mathrm{Aire}(C_i)=\pi$. - - On note $C=[-1,1]^2$. Démontrer qu'il existe une suite $(D - {i\in\N}$ de parties de $C$ telle que : + - On note $C=[-1,1]^2$. Démontrer qu'il existe une suite $(D_i)_{i\in\N}$ de parties de $C$ telle que : + pour tout $i\in\N$, l'ensemble $D_i$ soit un disque ferme de $\R^2$ ; + les $D_i$ soient d'intérieurs deux a deux disjoints ; + $\sum_{i\in\N}\mathrm{Aire}(D_i)=4$. @@ -1000,7 +1025,7 @@ Soient $\left(a_n\right)$ et $\left(b_n\right)$ deux suites réelles de limite 1 #+begin_exercice [ENS 2023 # 93] -Soient $\alpha\gt 0$ et $(a - {n\in{\N}}$ une suite strictement décroissante a valeurs dans $]0,1[$. Soit $(u - {n\in{\N}}$ une suite définie par $u_0\gt 0$ et $\forall n\in{\N}$, $u_{n+1}=u_n(u_n^{\alpha}+a_n)$. Montrer qu'il existe un unique $u_0\gt 0$ tel que la suite $(u - {n\in{\N}}$ converge vers un réel strictement positif. +Soient $\alpha\gt 0$ et $(a_n)_{n\in{\N}}$ une suite strictement décroissante a valeurs dans $]0,1[$. Soit $(u_n)_{n\in{\N}}$ une suite définie par $u_0\gt 0$ et $\forall n\in{\N}$, $u_{n+1}=u_n(u_n^{\alpha}+a_n)$. Montrer qu'il existe un unique $u_0\gt 0$ tel que la suite $(u_n)_{n\in{\N}}$ converge vers un réel strictement positif. #+end_exercice #+begin_exercice [ENS 2023 # 94] @@ -1105,18 +1130,44 @@ Déterminer les applications $f$ de $\R$ dans $\R$ telles que, pour tout entier $f^2$ et $f^3$ polynomiales, donc $f$ est une fraction rationnelle, $f\in\Q(x)$ et $f^2\in \Q[X]$ impliquent $f\in\Q[X]$. #+END_proof +# ID:7085 #+begin_exercice [ENS 2023 # 106] Soit $p\gt 1$ un réel. Montrer qu'il existe une constante $k_p\gt 0$ telle que, pour tout $(x,y)\in\R^2$ tel que $|x|^p+|y|^p=2$, on ait $(x-y)^2\leq k_p\,(4-(x+y)^2)$. #+end_exercice +#+BEGIN_proof +On veut montrer que $\frac{(x-y)^2}{4 - (x+y)^2}$ est majorée. -#+begin_exercice [ENS 2023 # 107] -Soit $f\colon\R\ra\R$. On note $f^*(s)=\sup_{x\in\R}\,(sx-f(x))$ et $f^*(x)=\sup_{s\in\R}\,(sx-f^*(s))$. +Rmq : l'inégalité de Hölder donne $(x+y)^2 \leq 4$ avec égalité si et seulement si $(x,y) = (1,1)$. Ce que l'on peut retrouver par de l'optimisation sous contrainte. -Montrer que $f^*(x)=\sup_{a\text{ affine }\leq f}a(x)$. +Si une suite tend vers $+\i$, nécessairement $x_n + y_n \ra 2$, donc nécessairement $(x_n,y_n)\ra (1,1)$. + +On considère $f(x,y) = 4 - (x+y)^2$, on peut écrire un DL à l'ordre $2$ en $(1,1)$, que l'on peut trouver algébriquement : $f(x,y) = -(x-1)^2 - (y-1)^2 - 2(x-1) - 2(y-1) - 2(xy - 1)$, où $xy-1 = (x-1)(y-1) +x-1 + y-1$, donc $f(x,y) = - (x-1)^2 - (y-1)^2 -4(x-1) - 4(y-1) - 2 (x-1)(y-1)$. + +Ensuite, on considère $g(x,y) = x^p + y^p$, et un DL à l'ordre $2$, qui donne $g(x,y) - g(1,1) = p (x-1) + p (y-1) + p (p-1)(x-1)^2 + p (p-1)(y-1)^2 + o_0(\lN \dots\rN^2)$. + +En l'injectant dans $f$, on obtient $f(x,y) = (x-1)^2 \big(-1 + 4(p-1)\big) + (y-1)^2 \big(-1 + 4(p-1) \big) - 2 (x-1)(y-1)$, et par ailleurs $(x-1) = -(y-1)$, donc on trouve $f(x,y) = (x-1)^2 \big(-1 + 4(p-1) + 1\big) + (x-1)^2 \big(-1 + 4(p-1) + 1\big)(y-1)^2$. +#+END_proof + + +# ID:7086 +#+begin_exercice [ENS 2023 # 107] Dualité de Legendre +Soit $f\colon\R\ra\R$. On note $f^*(s)=\sup_{x\in\R}\,(sx-f(x))$ et $f^{**}(x)=\sup_{s\in\R}\,(sx-f^*(s))$. + +Montrer que $f^{**}(x)=\sup\limits_{\substack{a\text{ affine }\\ a \leq f}}a(x)$. #+end_exercice +#+BEGIN_proof +Supposons que $f\gt 0$, et montrons que $f^{**}\geq 0$, c'est-à-dire que pour tout $x$, il existe $s$ tel que $sx - f^*(s)\geq 0$, mais $f^*(0) = \sup_x (-f(x))$, d'où le résultat. + +Cela démontre que $f^{**}(x)\geq \sup\limits_{\substack{a\text{ affine }\\ a \leq f}}a(x)$. + +Par ailleurs $f^{**}$ est convexe. + +Par ailleurs, $f^{**}\leq f$ : On a $f^{**}(x) = \sup_s sx - f^*(s)$, et pour tout $s$, $sx - f^*(s)\leq f(s) \ssi f^*(s)\geq sx - f(s)$, ce qui est immédiat. +#+END_proof + #+begin_exercice [ENS 2023 # 108] -Soient $I$ un ensemble fini et $(P - {i\in I}$ une famille de polynômes réels stable par derivation. On définit une fonction signe par $\op{sign}(x)=\dfrac{x}{|x|}$ si $x\neq 0$ et $\op{sign}(0)=0$. +Soient $I$ un ensemble fini et $(P_i)_{i\in I}$ une famille de polynômes réels stable par derivation. On définit une fonction signe par $\op{sign}(x)=\dfrac{x}{|x|}$ si $x\neq 0$ et $\op{sign}(0)=0$. Pour $\eps\in\{-1,1,0\}^I$, soient $A_{\eps}=\{t\in\R\ ;\ \forall i\in I,\ \op{sign}(P_i(t))= \eps(i)\}$ et @@ -1181,7 +1232,7 @@ Pour $n\in\N$, on note $A_n=\int_0^{\frac{\pi}{2}}\cos^{2n}(x)\,dx$ et $B_n=\int - En déduire que $\sum_{k=1}^{+\i}\frac{1}{n^2}=\frac{\pi^2}{6}$ puis que $\sum_{k=1}^n\frac{1}{k^2}=\frac{\pi^2}{6}+O\left(\frac{1}{n}\right)$. #+end_exercice -#+BEGIN_exercice [ENS 2023 # 113] +#+BEGIN_exercice [ENS 2023 # 113] :sup: Soit $f\colon \R^+ \ra \R$ une fonction continue et presque périodique c'est-à-dire telle que, pour tout $\epsilon\gt 0$, il existe $T\gt 0$ tel que : $\forall x \in \R^+, \forall n \in \N,|f(x+n T)-f(x)| \leq \epsilon$. Soit $f\colon \R^+ \ra \R$ continue et presque périodique. 1. Montrer que $f$ est uniformément continue sur $\R^+$. @@ -1189,10 +1240,13 @@ Soit $f\colon \R^+ \ra \R$ continue et presque périodique. #+END_exercice #+BEGIN_proof 1. Easy. - 2. !! + 2. Si la quantité donnée n'admettait pas de limite, on trouverait deux suites extraites, à distance $\eps$ l'une de l'autre. + + Pour $\eps/2$, il existe $T$. Alors $g(nT)$ est proche à $\eps/2$ près de $g(T)$. Mieux que ça, la fonction est bornée, donc la différence entre $g(nT)$ et $g(nT + u)$ est petite. #+END_proof -#+begin_exercice [ENS 2023 # 114] +# Relier à je ne sais quoi +#+begin_exercice [ENS 2023 # 114] :sup: Soit $f$ une fonction continue par morceaux et croissante de $[0,1]$ dans $\R$. Montrer que $\int_0^1f(x)e^{i\lambda x}dx\underset{\lambda\to+\i}{=}O \left(\frac{1}{\lambda}\right)$. #+end_exercice @@ -1220,7 +1274,7 @@ Quand on l'intègre, on obtient $\sum_{\sigma,\sigma'} \eps(\sigma') \eps(\sigma #+end_exercice #+begin_exercice [ENS 2023 # 117] -Soient $d,N\in\N$ tels que $N\gt d$. Soient $(P - {n\in\N}$ une suite de polynômes a coefficients réels de degré au plus $d$ et $x_1,...,x_N$ des réels distincts. On suppose que pour tout $j\in\{1,...,N\}$, la suite $(P_n(x_j))_{n\in\N}$ est bornée. Montrer que l'on peut extraire de $(P - {n\in\N}$ une suite $(Q - {n\in\N}$ qui converge uniformément sur $[0,1]$ vers un polynôme de degré au plus $d$. +Soient $d,N\in\N$ tels que $N\gt d$. Soient $(P_n)_{n\in\N}$ une suite de polynômes a coefficients réels de degré au plus $d$ et $x_1,...,x_N$ des réels distincts. On suppose que pour tout $j\in\{1,...,N\}$, la suite $(P_n(x_j))_{n\in\N}$ est bornée. Montrer que l'on peut extraire de $(P_n)_{n\in\N}$ une suite $(Q_n)_{n\in\N}$ qui converge uniformément sur $[0,1]$ vers un polynôme de degré au plus $d$. #+end_exercice #+begin_exercice [ENS 2023 # 118] @@ -1267,7 +1321,7 @@ On admet que $\|\ \|_{h^d}$ est une norme sur l'espace vectoriel ${\cal T}$ des $\|f\star g\|_2\leq c(d)\|f\|_{h^d}\|g\|_2$. - Déterminer tous les réels $d$ vérifiant la condition de la question précédente. - Soit $f$ de classe ${\cal C}^{\i}$ et $2\pi$-periodique. On pose, pour $k\in{\Z}$, $c_k(f)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\theta)e^{-ik\theta}{\rm d}\theta$ et, pour tout $d\in{\R}$, $\|f\|_{h^d}^2=\sum_{k\in{\Z}}|c_k(f)|^2(1+|k|)^{2d}$. Déterminer les $d\in{\R}$ tels que $\|f\|_{h^d}\lt +\i$. - - Soient $f$, $g$ de classe ${\cal C}^{\i}$ et $2\pi$-périoddiques et $d\in{\R}$. Calculer $\|f\star g\|_{h^d}$. + - Soient $f$, $g$ de classe ${\cal C}^{\i}$ et $2\pi$-périodiques et $d\in{\R}$. Calculer $\|f\star g\|_{h^d}$. #+end_exercice #+begin_exercice [ENS 2023 # 124] @@ -1294,7 +1348,7 @@ Soient $f:z\mapsto\sum_{n=0}^{+\i}z^n$ et $g:z\mapsto\sum_{n=0}^{+\i}z^{2^n}$. #+end_exercice #+begin_exercice [ENS 2023 # 127] -Soit $\alpha=(\alpha - {i\geq 1}$ une suite de $\Z$ nulle a partir d'un certain rang. Pour $n\geq 1$, on pose $u_n=\prod_{i\in\N^*}((in)!)^{\alpha_i}$. +Soit $\alpha=(\alpha_i)_{i\geq 1}$ une suite de $\Z$ nulle a partir d'un certain rang. Pour $n\geq 1$, on pose $u_n=\prod_{i\in\N^*}((in)!)^{\alpha_i}$. - Déterminer, selon la valeur de $\alpha$, le rayon de convergence $R$ de la série entière $\sum_{n\geq 1}u_nz^n$. Dans la suite, on note $f$ la somme de cette série entière. @@ -1335,7 +1389,8 @@ Soit $(a, b) \in \R \times \R^-$ tel que $\forall x \in[0,1], 1+a x+b x^2 \geq 0 2. Si $a \in \R^{-*}$, montrer que $n \int_0^1\left(1+a x+b x^2\right)^n\dx \underset{n \ra+\i}{\longrightarrow}-\frac{1}{a}$. #+END_exercice #+BEGIN_proof - + 1. Si $a\gt 0$, si $a = 0$. + 2. #+END_proof @@ -1657,7 +1712,7 @@ Soient $m \geq 1$ et $r \geq 1$ deux entiers. On munit l'ensemble des morphismes Le faire pour $m = p$, puis lemme Chinois. #+END_proof -# ID:6836 +# ID:7044 #+begin_exercice [ENS 2023 # 166] Deux joueurs $A$ et $B$ lancent une pièce truquée donnant pile avec une probabilité égale à $5/9$. Les règles de gain sont les suivantes : pile rapporte $5$ euros et face $4$ euros. Pour $n\in\N^*$, chacun des joueurs effectue $9n$ lancers indépendants ; on note $A_n$ (resp. $B_n$) la variable aléatoire donnant le gain du joueur $A$ (resp. $B$). - Trouver un équivalent, lorsque $n$ tend vers $+\i$, de $\mathbf{P}\left(A_n=B_n\right)$. @@ -1665,7 +1720,7 @@ Deux joueurs $A$ et $B$ lancent une pièce truquée donnant pile avec une probab - Vers quoi tend $\mathbf{P}\left(A_n\lt B_n\right)$ ? #+end_exercice #+BEGIN_proof - - On a $P(A_n = B_n) = P(A_n = 9n - B_n) = P(A_n + B_n = 9n)$, et la somme est une loi binomiale. + - IDK, Cela ne dépend que du nombre de Pile obtenus, pas du gain… Éventuellement, on tend vers une loi normale… - C'est clair. - Découle des questions précédentes. #+END_proof @@ -1682,11 +1737,13 @@ On a $P(S_{2n} = n+k)\leq P(S_{2n} = n-k)$, puis on montre que $P(S_{2n}\geq n+ Soit $X$ une variable aléatoire à valeurs dans $\N$ telle que $\mathbf{E}(X)=1$, $\mathbf{E}\left(X^2\right)=2$ et $\mathbf{E}\left(X^3\right)=5$. Quelle est la valeur minimale de $\mathbf{P}(X=0)$ ? #+END_exercice #+BEGIN_proof -On a $E(X) E(X^3)\geq E(X^2)^2$. En notant $e = P(X=1)$, on a $E(X 1_{X\gt 1}) E(X^3 1_{X\gt 1})\geq E(X^2 1_{X\gt 1})^2$, donc $(1-e)(5-e) (2-e)^2$, qui donne $e\leq \frac{1}{2}$. +On a $E(X) E(X^3)\geq E(X^2)^2$. En notant $e = P(X=1)$, on a $E(X 1_{X\gt 1}) E(X^3 1_{X\gt 1})\geq E(X^2 1_{X\gt 1})^2$, donc $(*)$ $(1-e)(5-e) \geq (2-e)^2$, qui donne $e\leq \frac{1}{2}$. Comme $E(X) = 1$, on doit avoir $P(X=0)\geq \frac{1}{4}$, mais le cas d'égalité ne donne pas les bonnes valeurs : mais $E(X) = 1$, $E(X^2) = \frac{3}{2}$ et $E(X^3) = \frac{5}{2}$. -Si on suppose que $e = \frac{1}{2}$, on peut prendre $Y$ qui vaut $3$ avec probabilité $\frac{1}{6}$ et $0$ avec probabilité $\frac{1}{3}$. +Si on suppose que $e = \frac{1}{2}$, on doit avoir égalité dans Cauchy-Schwarz qui donne $(*)$, donc $X$ ne prend qu'une seule valeur $\gt 1$. On peut prendre $Y$ qui vaut $3$ avec probabilité $\frac{1}{6}$ et $0$ avec probabilité $\frac{1}{3}$, et on a les bonnes valeurs (et c'est la seule façon). + +On a aussi $E(X 1_{X\geq 1})^2\leq E(1_{X\geq 1}) E(X^2)$, donc $(1 - r)\geq \frac{1}{2}$, $r\leq \frac{1}{2}$ !! Manque : on ne peut pas faire mieux… #+END_proof @@ -1746,7 +1803,7 @@ Dans tout l'énonce, on fixe un entier $p\geq 1$. - Soient $X_1,\ldots,X_n$ des variables aléatoires i.i.d. suivant la loi uniforme sur $\{-1,1\}$. Soit $(a_1,\ldots,a_n)\in\R^n$. On pose $X=\sum_{i=1}^na_iX_i$. Montrer que $\mathbf{E}(X^{2p})\leq(2p)^p(\mathbf{E}(X^2))^p$. - Montrer que $\mathbf{E}(X^{2p})\leq p^p(\mathbf{E}(X^2))^p$. -- Soit $(a - {k\geq 1}$ une suite réelle telle que $\sum_{k=1}^{+\i}a_k^2=1$. Soient $x\in\R$ et $Y_x=\sum_{k=1}^na_k\cos(kx)\,X_i$. +- Soit $(a_k)_{k\geq 1}$ une suite réelle telle que $\sum_{k=1}^{+\i}a_k^2=1$. Soient $x\in\R$ et $Y_x=\sum_{k=1}^na_k\cos(kx)\,X_i$. Montrer que $\omega\mapsto\int_0^{2\pi}Y_x(\omega)^{2p}\,dx$ prend au moins une valeur inférieure ou égal a $2\pi p^p$. #+end_exercice @@ -1791,11 +1848,11 @@ Soit $(X_n)_{n\geq 1}$ une suite de variables aléatoires indépendantes de loi #+begin_exercice [ENS 2023 # 181] -Soient $(\Omega,\mc{A},\mathbf{P})$ un espace probabilise et $(m - {k\in\N}$ une suite de réels positifs de somme 1. On considère un arbre aléatoire sur cet espace tel que chaque noeud ait un nombre aléatoire $X$ de successive avec, pour tout $k\in\N$, $\mathbf{P}(X=k)=m_k$. Ces variables aléatoires correspondant au nombre de succcesseurs sont mutuellement indépendantes. On note $X_1$ la variable aléatoire comptant le nombre de succcesseurs de la racine. Caracteriser le fait que la longueur de l'arbre soit presque surement finie. +Soient $(\Omega,\mc{A},\mathbf{P})$ un espace probabilisé et $(m_k)_{k\in\N}$ une suite de réels positifs de somme 1. On considère un arbre aléatoire sur cet espace tel que chaque noeud ait un nombre aléatoire $X$ de successive avec, pour tout $k\in\N$, $\mathbf{P}(X=k)=m_k$. Ces variables aléatoires correspondant au nombre de succcesseurs sont mutuellement indépendantes. On note $X_1$ la variable aléatoire comptant le nombre de succcesseurs de la racine. Caracteriser le fait que la longueur de l'arbre soit presque surement finie. #+end_exercice #+begin_exercice [ENS 2023 # 182] -On construit iterativement et aléatoirement un arbre aléatoire sur l'ensemble de sommets $\left[\![1,n]\!\right]$ (graphe oriente) selon le procede suivant : a l'etape $k$, on choisit aléatoirementun point dans $\db{1,k}$ (avec probabilité uniforme) et on rajoute une arete orientée de ce point vers $k+1$. Ces choix s'effectuent de maniere indépendante les uns des autres. +On construit iterativement et aléatoirement un arbre aléatoire sur l'ensemble de sommets $\db{1,n}$ (graphe oriente) selon le procede suivant : a l'etape $k$, on choisit aléatoirementun point dans $\db{1,k}$ (avec probabilité uniforme) et on rajoute une arete orientée de ce point vers $k+1$. Ces choix s'effectuent de maniere indépendante les uns des autres. - On note $X_n$ la variable aléatoire donnant le nombre d'aretes partant du point $1$. Déterminer l'espérance et la variance de $X_n$. - On suppose $n\geq 2$. On note $S_n$ la variable aléatoire donnant le nombre de descendants (directs ou non) du sommet $2$. Déterminer la loi de $S_n$. - Calculer l'espérance du nombre de feuilles de l'arbre. @@ -1823,7 +1880,7 @@ Une variable aléatoire réelle $X$ est infiniment divisible si $X$ admet un mom #+end_exercice #+begin_exercice [ENS 2023 # 185] -On se donne une suite $(X - {i\geq 1}$ de variables aléatoires indépendantes. On suppose que pour tout $i\geq 1$, il existe $a_i\in\left]0,2\right]$ et $p_i\in[0,1]$ tels que $X_i$ soit a valeurs dans $\{0,a_i,-a_i\}$ et $\mathbf{P}(X_i=a_i)=\mathbf{P}(X_i=-a_i)=\frac{p_i}{2}$. +On se donne une suite $(X_i)_{i\geq 1}$ de variables aléatoires indépendantes. On suppose que pour tout $i\geq 1$, il existe $a_i\in\left]0,2\right]$ et $p_i\in[0,1]$ tels que $X_i$ soit a valeurs dans $\{0,a_i,-a_i\}$ et $\mathbf{P}(X_i=a_i)=\mathbf{P}(X_i=-a_i)=\frac{p_i}{2}$. - Quelle relation doivent vérifier $a_i$ et $p_i$ pour que $\mathbf{V}(X_i)=1$? Dans toute la suite, on suppose cette relation vérifiée et on pose $S_n=\sum_{i=1}^nX_i$. - Calculer la variance de $n^{-1/2}S_n$. - Montrer que $\mathbf{E}(\cos(n^{-1/2}tS_n))=\prod_{i=1}^n\mathbf{E}(\cos(n^{-1/2}tX_i)$. @@ -1876,7 +1933,7 @@ C'est simple : On peut passer d'un somme à un autre en au plus $\frac{n d}{2}$ *** Algèbre #+begin_exercice [ENS PSI 2023 # 191] -Soit $E$ un espace vectoriel de dimension $n$. Soit $u\in\mc{L}(E)$, annule par un polynôme $Q$ tel que $Q(0)=0$ et $Q'(0)\neq 0$. Montrer que $\op{Ker}u$ et $\op{Im}u$ sont supplementaires. +Soit $E$ un espace vectoriel de dimension $n$. Soit $u\in\mc{L}(E)$, annule par un polynôme $Q$ tel que $Q(0)=0$ et $Q'(0)\neq 0$. Montrer que $\op{Ker}u$ et $\op{Im}u$ sont supplémentaires. #+end_exercice #+begin_exercice [ENS PSI 2023 # 192] @@ -1889,7 +1946,7 @@ Soient $E$ un $\C$ espace vectoriel de dimension finie $n$, $f\in\mc{L}(E)$. On - Calculer $\phi_f^n(g)$ pour $g\in\mc{L}(E)$. - Montrer que $f^{n+1}\circ g-g\circ f^{n+1}=\sum_{k=0}^nf^k(f\circ g-g\circ f)f^{n-k}$. - On suppose $f$ non inversible. Montrer que $f$ est nilpotente si et seulement si $\phi_f$ l'est. - - Montrer que, si $f$ possede une unique valeur propre, alors $\phi_f$ est nilpotente. Étudier la réciproque. + - Montrer que, si $f$ possède une unique valeur propre, alors $\phi_f$ est nilpotente. Étudier la réciproque. #+end_exercice #+begin_exercice [ENS PSI 2023 # 194] @@ -1965,7 +2022,7 @@ $\mathbb{D}_{2n}=\{\sigma\in\mc{O}(E)\ ;\ \forall i\in\db{1,n }\,\sigma(v_i)\in\ - Si $\sigma\in\mathbb{D}_{2n}$ vérifie $\sigma(v_1)=v_i$, montrer que $\sigma(v_2)=v_{i-1}$ ou $\sigma(v_2)=v_{i+1}$. - En déduire que le cardinal de $\mathbb{D}_{2n}$ est $2n$. - Montrer que $\mathbb{D}_{2n}=\{\mathrm{id},r,sr,r^2,sr^2,r^3,sr^3,\ldots\}$ ou $s$ est une reflexion et $r$ une rotation d'angle $\mathrm{Arccos}\ (\langle v_1,v_2\rangle)$. - - On note $D=\bigcup_{n\geq 3}D_{2n}$. Montrer que pour tout $\sigma\in\mc{O}(E)$, il existe une suite $(\sigma - {k\geq 0}\in D^{\N}$ telle que $\sigma=\lim_{k\ra+\i}\sigma_k$. + - On note $D=\bigcup_{n\geq 3}D_{2n}$. Montrer que pour tout $\sigma\in\mc{O}(E)$, il existe une suite $(\sigma_k)_{k\geq 0}\in D^{\N}$ telle que $\sigma=\lim_{k\ra+\i}\sigma_k$. #+end_exercice #+begin_exercice [ENS PSI 2023 # 202] @@ -2154,7 +2211,7 @@ Soit $n\geq 2$. On note $n=p_1^{s_1}\dots p_r^{s_r}$ sa decomposition en facteur _a) i)_ Montrer que si $d$ et $d'$ sont deux entiers premiers entre eux alors $A_d\cap A_{d'}=A_{dd'}$, et en déduire que $A_d$ et $A_{d'}$ sont indépendants. - On note $B=\{k\in\Omega,\ k\wedge n=1\}$. Exprimer $B$ en fonction des $A_{p_i}$ et en déduire une expression de $\mathbf{P}(B)$ puis de $|B|$. Cette valeur sera notée $\phi(n)$. - Soient $n$ et $m$ deux entiers premiers entre eux. Montrer que $\phi(mn)=\phi(m)\phi(n)$. - On note $\mc{U}=\bigcup_{n\in\N}\mathbb{U}_n$ ou $\mathbb{U}_n$ designe l'ensemble des racines $n$-ièmes de l'unite. Pour $z$ dans $\mc{U}$, on note $n_z=\inf\{n\in\N\,\ z\in\mathbb{U}_n\}$. - - Pour $z\in\C$ tel que $|z|=1$, montrer qu'il existe une suite $(z - {k\in\N}$ a valeurs dans $\mc{U}$ telle que $\lim_{k\to+\i}z_k=z$. + - Pour $z\in\C$ tel que $|z|=1$, montrer qu'il existe une suite $(z_k)_{k\in\N}$ a valeurs dans $\mc{U}$ telle que $\lim_{k\to+\i}z_k=z$. - Pour tout entier naturel $n$ on note $P_m=\{z\in\mc{U},\ n_z=m\}$. Montrer que $P_m$ est fini et de cardinal $\phi(m)$, et que si $n$ et $m$ sont distincts $P_m\cap P_n=\emptyset$. - Montrer que $\mc{U}=\bigcup_{m\in\N}P_m$. #+end_exercice @@ -2312,7 +2369,7 @@ Soit $\sum a_n$ une série convergente de réels positifs. Montrere que la séri #+end_exercice #+begin_exercice [ENS PC 2023 # 242] -Soit $(a - {n\in\N}$ une suite réelle telle que $\sum\exp(a_n)$ converge. +Soit $(a_n)_{n\in\N}$ une suite réelle telle que $\sum\exp(a_n)$ converge. Déterminer $\lim_{k\to+\i}\sum_{n=0}^{+\i}\exp(ka_n)$. #+end_exercice @@ -2331,7 +2388,26 @@ Soient $g:[0,1]\to\R$ continue et $F:[0,1]\to\R$ de classe $C^1$ telles que $:F( Soient $f\in\mc C^1\left(\R,\R\right)$ et $(a,b,c,d)\in\R^4$ tels que les fonctions $af'+bf$ et $cf'+df$ soient bornées. A quelle condition sur $(a,b,c,d)$ la fonction $f$ est-elle bornée? #+end_exercice -!! Page manquante. +#+BEGIN_exercice [ENS PC 2023 # 246] +Dans l'espace $E = \mc C^0([-1,1])$, pour $\b\in\R$, on considère $g_{\b}\colon x\mapsto \max(x-\b, 0)$. On pose $E = \vect_{\b} g_{\b} + \R \tilde{1}$. Soit $f$ une fonction lipschitzienne. Montrer que $\forall \eps\gt 0,\, \exists g\in E,\, \lN f-g\rN_{\i}\lt \eps$. +#+END_exercice + +#+BEGIN_exercice [ENS PC 2023 # 247] +Soit $f\in\mc C^{\i}(\R,\R)$. Soient $p\in\N$ et $x\in\R$ tels que $f^{(p)}()\neq 0$. Montrer que pour tout $\l\in\db{0, p}$ et tout $\eps\gt 0$, il existe $a\lt b$ tels que $\max(|x-a|, |x-b|)\leq \eps$ et pour tout $y\in \interval]{a, b}[$, on ait $f^{(\l)}(y)\neq 0$. +#+END_exercice + +#+BEGIN_exercice [ENS PC 2023 # 249] +Soit $A\in\M_2(\R)$, $x_0\in\R^2$ et $r\gt 0$. Calculer $\int_{\mc C}\langle Ax, xrangle \dx$, où $\mc C$ est le cercle de centre $x_0$ et de rayon $r$. +#+END_exercice + +#+BEGIN_exercice [ENS PC 2023 # 252] +Soit $f_n\colon [0,1]\ra\R_+$ des fonctions continues, et $f\colon x\mapsto \inf f_n(x)$. La fonction $f$ est-elle nécessairement continue ? +#+END_exercice + +#+BEGIN_exercice [ENS PC 2023 # 254] +Soit $\a\gt 0$. Étudier la convergence simple, normale et uniforme sur $\R$ de $\sum \frac{\sin (nx)}{n^{\a}}$. +#+END_exercice + #+begin_exercice [ENS PC 2023 # 256] Soit $g\in{\cal C}^0([0,1],{\R}_+^*)$. On définit $\Phi\colon x\in{\R}\mapsto\ln\left(\int_0^1e^{xt}g(t)\;{\rm d}t\right)$. @@ -2457,6 +2533,7 @@ On a $p(n)\leq p(n-1) + p(n-2) + \dots + p(1) + 1$, en considérant le (un) plus #+END_proof # Relier à Lucas : 3721 +# ID:7027 #+begin_exercice [X MP 2023 # 276] :sup: Soient $e_r\gt \cdots\gt e_2\gt e_1\geq 0$ des entiers, $n=\sum_{k=1}^r2^{e_k}$ et $X=\{s\in\N;\ 2^s\,|\,n!\}$. - Montrer que $\max X=n-r$. @@ -2550,7 +2627,7 @@ Soient $G$ un groupe et $T$ l'ensemble des éléments de $G$ d'ordre fini. Montrer que le sous-groupe qu'elles engendrent est isomorphe a $\mc{S}_3$. - Retrouver le résultat de la question précédente en considérant le quotient $A$ de $(\R^*)^3$ par la relation de colinearite, la bijection $f:A\to(\R^*)^2$ qui associe a la classe de $(x_1,x_2,x_3)$ le couple $(x_1/x_2,x_2/x_3)$, et enfin les permutations de $A$ induites par $(x_1,x_2,x_3)\mapsto(x_2,x_1,x_3)$ et $(x_1,x_2,x_3)\mapsto(x_1,x_3,x_2)$. - - Soit $n\geq 3$. Déterminer le groupe engendre par les bijections $(s - {1\leq i\leq n}$ de $(\R^*)^n$ définies par $s_i(t_1,...,t_n)=(t_1,...,t_{i-2},t_{i-1}\times t_i,t_i^{-1},t_i \times t_{i+1},t_{i+2},...,t_n)$ si $1\lt i\lt n$, $s_1(t_1,...,t_n)=(t_1^{-1},t_1\times t_2,t_3,...,t_n)$ et $s_n(t_1,...,t_n)=(t_1,...,t_{n-2},t_{n-1}\times t_n,t_n^{-1})$. + - Soit $n\geq 3$. Déterminer le groupe engendre par les bijections $(s_i)_{1\leq i\leq n}$ de $(\R^*)^n$ définies par $s_i(t_1,...,t_n)=(t_1,...,t_{i-2},t_{i-1}\times t_i,t_i^{-1},t_i \times t_{i+1},t_{i+2},...,t_n)$ si $1\lt i\lt n$, $s_1(t_1,...,t_n)=(t_1^{-1},t_1\times t_2,t_3,...,t_n)$ et $s_n(t_1,...,t_n)=(t_1,...,t_{n-2},t_{n-1}\times t_n,t_n^{-1})$. Ind. Considèrer $f:(\R^*)^{n+1}\to(\R^*)^n$ définie par $f(t_1,...,t_{n+1})=\left(\dfrac{t_2}{t_1},...,\dfrac{t_{n+1}}{t_n}\right)$ et chercher des bijections simples $s'_i$ de $(\R^*)^{n+1}$ telles que $s_i\circ f=f\circ s'_i$. #+end_exercice @@ -2675,6 +2752,7 @@ Interpolation de Hermite. #+END_proof +#+call: get_exo(7026) #+begin_exercice [X MP 2023 # 295] Soit $n\in\N$. Le polynôme $X^{n+1}-nX^n+1$ est-il irréductible dans $\Z[X]$? #+end_exercice @@ -2771,6 +2849,10 @@ Donc on est point fixe si et seulement si $(ad - bc)(bc - ad) = adbc \ssi (ad-bc #+begin_exercice [X MP 2023 # 302] Soit $R\in\M_n(\Z)$ non nulle et $M=I_n+3R$. Montrer que, pour tout $k\in\N^*$, $M^k\neq I_n$. #+end_exercice +#+BEGIN_proof +$R = \frac{A -I_n}{3}$, mais les valeurs propres de $R$ sont de module $1$, donc $N$ est diagonalisable, avec des valeurs propres $\lt 1$, donc $N^k \ra 0$, donc $N$ nilpotent, et comme $N$ diagonalisable, $N = O_n$. +#+END_proof + #+BEGIN_exercice [X MP 2023 # 303] :sup: Soient $E$ un $\R$-espace vectoriel de dimension finie, $p, u \in \mc{L}(E)$. On suppose que $p$ est un projecteur et que $p u+u p=u$. Montrer que $\op{tr}(u)=0$. @@ -2848,6 +2930,18 @@ Soit $M=(m_{i,j})_{1\leq i,j\leq n}\in\M_n(\R)$ telle que : $\forall i,j$, $m_{i #+begin_exercice [X MP 2023 # 310] Soient $E$ un $\mathbb{K}$-espace vectoriel de dimension finie, $f\in\mc{L}(E)$ cyclique, $F$ un sous-espace de $E$ stable par $f$. Montrer que l'induit par $f$ sur $F$ est cyclique. #+end_exercice +#+BEGIN_proof +Soit $F$ un sous-espace vectoriel stable. À $F$ on associe l'ensemble des polynômes $P$ tel que $P(u)(x)\in F$, c'est un idéal, qui contient $\chi_u$, donc correspond à un diviseur $D$ de $\chi_u$. + +Il y en a donc un nombre fini, et il en va de même de l'induit. + +Réciproquement, si $u$ a un nombre fini d'espaces stables, il est cyclique (si le corps est infini), puisque la réunion de ses espaces stables ne donne pas tout l'ensemble. + +Si le corps est fini, on a obtenu $F = \Ker \frac{\chi_u}{D}$ ; $F$ contient $D(u)(x)$, dont le polynôme minimal est $\frac{\chi_u}{D}$. Donc $\pi_F = \frac{\chi_u}{D}$. Le problème est d'empêcher que deux espaces stables correspondant à deux polynômes minimaux différents aient le même polynôme caractéristique. + +Si on admet que tout facteur irréductible de $\chi$ est un facteur de $\pi$, la propriété de monotonie permet d'exclure ce qui précède : si $\pi$ est un facteur irréductible, les $\Ker \pi^k(u)$ sont strictement croissants, de polynômes caractéristiques $\pi^{\l}$, donc nécessairement $\l = k$, et le lemme des noyaux permet de conclure. +#+END_proof + #+begin_exercice [X MP 2023 # 311] Soient $E$ un $\C$-espace vectoriel de dimension finie, $a,b\in\mc{L}(E)$. On suppose qu'il existe $f\in\mc{L}(\C,E)$ et $v\in\mc{L}(E,\C)$ telles que $ab-ba=fv$. @@ -2872,7 +2966,8 @@ Soient $E$ un $\C$-espace vectoriel de dimension finie et $f\in\mc{L}(E)$ tel qu - On revient au cas général. Montrer l'existence de $k\in\N^*$ et $p\in\N$ tels que $f^{p+k}=f^p$. #+end_exercice #+BEGIN_proof - - Les valeurs propres sont des $\m U_k$, et si elle n'était pas diagonalisable… + - Les valeurs propres sont des $\m U_k$, et si elle n'était pas diagonalisable, dans un espace caractéristique, on a $f = \la I_n + N$, prendre un élément dans le noyau de $N^2$ mais pas dans le noyau de $N$. + - Décompositions en espaces caractéristiques. #+END_proof @@ -2921,14 +3016,20 @@ Pour $n = 2$, On peut $P_a(x) + P_a'(y) = P_b(x) + P_b'(y) = \dots$ Existe-t-il $A\in\text{SO}_2(\Q)$ telle qu'il n'existe pas $B\in\text{SO}_2(\Q)$ vérifiant $B^2=A$? #+end_exercice #+BEGIN_proof -S'il en existe une, son opposé marche aussi. On a $\cos \theta = \sqrt{\frac{1 + \cos (2\theta)}{2}}$, si on pouvait appliquer ça à chaque fois, problème de taille du dénominateur. +Par l'absurde, si toute matrice a une racine carrée rationnelle. S'il en existe une, son opposé marche aussi. On a $\cos \theta = \sqrt{\frac{1 + \cos (2\theta)}{2}}$ : on passe de $\frac{p}{q}$ à $\sqrt{\frac{q+p}{2q}}$, donc le dénominateur diminue, donc fini par atteindre $1$ ou $2$, ce qui n'est pas possible. #+END_proof #+begin_exercice [X MP 2023 # 318] Soient $E$ un espace vectoriel euclidien, $f\in\mc{S}(E)$, $\Phi:\begin{array}{rcl}E&\ra&\R\\ v&\mapsto&\|f(v)\|^2-\langle f(v),v\rangle^2\end{array}$. Donner une condition nécessaire et suffisante pour que $\Phi$ admette un extremum. #+end_exercice #+BEGIN_proof -Si $\la$ est valeurs propres, $\Phi(t v_{\la}) = \la^2 t^2 - \la t^2 = (\la^2 - \la)t^2$. Il est donc nécessaire, ou bien que toutes les valeurs propres sont $\in [0,1]$, ou bien toutes dans le complémentaire. +Si $\la$ est valeur propre, $\Phi(t v_{\la}) = \la^2 (t^2 - t^4)$, donc pas de minimum. + +Si toutes les valeurs propres sont $\gt 0$, $\lN f(x)\rN\ra +\i$. + +Çela marche pareil si l'une des valeurs propres est nulle. + +Si une est négative l'autre positive, vérifier avec le cas $n=2$ qu'on peut prendre des valeurs arbitrairement grandes : prendre $v$ tel que $\langle v, f(v)\rangle = 0$. #+END_proof @@ -2938,6 +3039,12 @@ On considère dans $\M_{2n}(\R)$ les matrices $J=\begin{pmatrix}0&-I_n\\ I_n&0\e - On note $\mc C$ l'ensemble des $K\in\M_{2n}(\R)$ telles que $K^2=-I$ et $K^TJ\in\mc{S}_n^{++}(\R)$. Soit $K\in\mc C$. Montrer que $K+J$ est inversible et que $(K+J)^{-1}(K-J)$ est symétrique. - Soit $K\in\mc C$. On pose $S=(K+J)^{-1}(K-J)$. Montrer que $SJ+JS=0$. #+end_exercice +#+BEGIN_proof + - + - $K+J = K + K^T J K = (I_n + K^T J)K$. En écrivant ce qu'on veut, et en multipliant par les inverses, on obtient qu'on veut $KJ + J K^T = O_n$, et on utilise $K^{-1} = -K$. + - Remplacer le $S$ de droite par sa transposée, multiplier par les inverses, et développer. +#+END_proof + #+begin_exercice [X MP 2023 # 320] Montrer que $\forall(A,B)\in\mc{S}_n^+(\R)^2$, $\det(A+B)\geq\max(\det(A),\det(B))$. @@ -2946,12 +3053,17 @@ Montrer que $\forall(A,B)\in\mc{S}_n^+(\R)^2$, $\det(A+B)\geq\max(\det(A),\det(B Simple ? Écrire $A = PD P^T$ et $B = P P^T$, si $\det B\gt 0$. #+END_proof -# À relier -#+begin_exercice [X MP 2023 # 321] +# À relier à un de l'année précédente +#+begin_exercice [X MP 2023 # 321] Golden-Thompson inequality Soient $A,B\in\mc{S}_n(\R)$. - Montrer que $\op{tr}\left(e^Ae^B\right)\gt 0$. - Montrer que $\op{tr}\left(e^{A+B}\right)\leq\op{tr}\left(e^Ae^ {B}\right)$. #+end_exercice +#+BEGIN_proof + - On se ramène à $A$ diagonale, et $e^B$ est définie positive, donc de coefficients diagonaux $\gt 0$. + - Cf https://terrytao.wordpress.com/2010/07/15/the-golden-thompson-inequality/, et un énoncé de l'année précédente. Super dur, et utilise au finale que $e^{A+B} = \lim (e^{A/p} e^{B/p})^p$. +#+END_proof + #+BEGIN_exercice [X 2023 # 322] Soit $t_1, \ldots, t_n$ des réels. @@ -2971,13 +3083,13 @@ Soit $t_1, \ldots, t_n$ des réels. - Montrer que $\lN A\rN=\sup_{(X,Y)\in B_f(0,1)^2}|\left\langle AX,Y\right\rangle|$. - On prend $A=\Big(\dfrac{1}{i+j+1}\Big)_{0\leq i,j\leq n}$ dans $\M_{n+1}(\R)$. Pour $X=(x_0\cdots x_n)^T$ et $Y=(y_0\cdots y_n)^T$ dans $\R^{n+1}$, donner une interpretation de $\left\langle AX,Y\right\rangle$ a l'aide d'une intégrale faisant intervenir $P:t\in[0,2\pi]\mapsto\sum_{k=0}^nx_ke^{ikt}$ et $Q:t\in[0,2\pi]\mapsto\sum_{k=0}^ny_ke^{ikt}$. - En déduire que $\|\!|\!|A|\!|\leq 2\pi$. - - Montrer que l'on a meme $\|\!|A|\!|\!|\leq\pi$. + - Montrer que l'on a même $\|\!|A|\!|\!|\leq\pi$. #+end_exercice #+BEGIN_proof - - - - $\langle AX, Y\rangle = $ -#+END_proof + - + #+END_proof @@ -3015,14 +3127,24 @@ Soit $d\in\N^*$. On munit $\R^d$ de la structure euclidienne canonique. On défi Montrer que la suite de terme général $\prod_{k=0}^nu_k$ converge. -Soit $(M_n)_{n\geq 0}$ une suite de matrices de $\M_d(\R)$. On suppose que la série de terme général $\|M_n-I_d\|$ converge. On pose, pour $n\in\N$, $B_n=M_0\times M_1\times\cdots\times M_n$. + - E Soit $(M_n)_{n\geq 0}$ une suite de matrices de $\M_d(\R)$. On suppose que la série de terme général $\|M_n-I_d\|$ converge. On pose, pour $n\in\N$, $B_n=M_0\times M_1\times\cdots\times M_n$. - Montrer que la suite $(B_n)_{n\geq 0}$ converge. - Soit $\sigma$ une permutation de $\N$. Que peut-on dire de la suite de terme général $M_{\sigma(0)}\times\cdots\times M_{\sigma(n)}$? - - Soit $E=\left\{\prod_{k=0}^{+\i}M_{\sigma(k)},\ \sigma\in\mc{S}( \N)\right\}$. Existe-t-il une suite de matrices pour laquelle $E$ n'est pas ferme? - - Soit $k\in\N^*$. Existe-il $(M_n)_{n\geq 0}\in(\M_d(\R))^{\N}$ telle que $E$ possede exactement $k$ composantes connexes? + - Soit $E=\left\{\prod_{k=0}^{+\i}M_{\sigma(k)},\ \sigma\in\mc{S}( \N)\right\}$. Existe-t-il une suite de matrices pour laquelle $E$ n'est pas fermé ? + - Soit $k\in\N^*$. Existe-il $(M_n)_{n\geq 0}\in(\M_d(\R))^{\N}$ telle que $E$ possède exactement $k$ composantes connexes ? #+end_exercice #+BEGIN_proof -!! + - + - On a $B_{n+1} - B_n = \big(\prod M_i\big) (I_n - M_n)$, dont la série converge. + - Pour la même raison, elle converge. + - Non, si on a une suite de $\sigma_\l$, on peut en extraire une permutation. + - Si les $M_n$ commutent, $E$ est un point. + + On pourrait aussi les prendre presque toutes égales à $I_n$. Si on prend des diagonales commutantes et deux matrices de permutations inverses, et une matrice diagonale avec un zéro, soit le zéro est à son emplacement d'origine, soit le zéro est en $\sigma(1)$, soit le zéro est en $\sigma^{-1}(1)$. Donc on peut obtenir $2$, ou $3$ composantes connexes (pas sûr…) + + On peut fondamentalement remplacer le $0$ par un élément très petit. + + Si on prend comme $\sigma$ un produit de transpositions (de sorte que $P_{\sigma} = P_{\sigma}^{-1}$), et qu'on met quatre fois cette matrice de transposition. #+END_proof @@ -3042,26 +3164,31 @@ On définit la longueur d'un intervalle borne $I$ de bornes $a$ et $b$ par $\ell # À relier à Brouwer #+begin_exercice [X MP 2023 # 329] -Dans $\R^2$, on note $D$ le disque unite ferme pour la norme infinie, $C$ la sphere unite pour la norme infinie. On cherche a montrer qu'il n'existe pas de fonction continue $r:D\to C$ telle que la restriction de $r$ a $C$ soit l'identite. - - On considère une fonction $f\colon\R^2\to\R$, antisymmetrique (i.e. $f(x,y)=-f(y,x)$), et $A=(a_{i,j})_{i,j\leq n}$ une matrice réelle telle que : $\forall i,j\in\db{1,n-1}$, +Dans $\R^2$, on note $D$ le disque unite fermé pour la norme infinie, $C$ la sphère unité pour la norme infinie. On cherche à montrer qu'il n'existe pas de fonction continue $r\colon D\to C$ telle que la restriction de $r$ a $C$ soit l'identité. + 1. On considère une fonction $f\colon\R^2\to\R$, antisymétrique (i.e. $f(x,y)=-f(y,x)$), et $A=(a_{i,j})_{i,j\leq n}$ une matrice réelle telle que : $\forall i,j\in\db{1,n-1}$, + $f(a_{i,j},a_{i+1,j})+f(a_{i+1,j},a_{i+1,j+1})+f(a_{i+1,j+1},a_{i,j+1})+f(a_{i,j+1},a_{i,j})=0$. -$f(a_{i,j},a_{i+1,j})+f(a_{i+1,j},a_{i+1,j+1})+f(a_{i+1,j+1},a_{i,j+1})+f(a_{i,j+1},a_{i,j})=0$. + Montrer que : + $\sum_{i=0}^{n-1}f(a_{i,1},a_{i+1,1})+\sum_{j=0}^{n-1}f(a_{n,j},a_{n,j+1})+\sum_{i=0}^{n-1}f(a_{i+1,n},a_{i,n})+\sum_{j=0}^{n-1}f(a_{1,j+1},a_{1,j})=0$ + 2. Soit $M\in\M_{n+2}(\R)$ une matrice de la forme $\begin{pmatrix}1&1&\cdots&\cdots&1\\ 1&&&&3\\ \vdots&&M'&&\vdots\\ 1&&&&3\\ 1&2&\cdots&\cdots&2\end{pmatrix}$ ou $M'\in\M_n(\R)$ -Montrer que : + est à coefficients dans $\{1,2,3\}$. Montrer qu'au moins un des petits carrés de $M$ comporte trois valeurs différentes. + 3. Montrer qu'on dispose d'un $\eta\gt 0$ tel que, pour tous $x$, $y\in D$ vérifiant $\|x-y\|_{\i}\leq\eta$, on a $\|r(x)-r(y)\|\leq\frac{1}{10}$. + 4. Soit alors $n\in\N$ tel que $\frac{2}{n-1}\leq\eta$. Pour tous $i$, $j\in\db{1,n}$, on pose $v_{i,j}=\left(1-2\frac{i-1}{n-1},1-2\frac{j-1}{n-1}\right)$. -$\sum_{i=0}^{n-1}f(a_{i,1},a_{i+1,1})+\sum_{j=0}^{n-1}f(a_{n,j},a_{n,j+1})+\sum_{i=0}^{n-1}f(a_{i+1,n},a_{i,n})+\sum_{j=0}^{n-1}f(a_{1,j+1},a_{1,j})=0$ - - Soit $M\in\M_{n+2}(\R)$ une matrice de la forme $\begin{pmatrix}1&1&\cdots&\cdots&1\\ 1&&&&3\\ \vdots&&M'&&\vdots\\ 1&&&&3\\ 1&2&\cdots&\cdots&2\end{pmatrix}$ ou $M'\in\M_n(\R)$ - -est a coefficients dans $\{1,2,3\}$. Montrer qu'au moins un des petits carrés de $M$ comporte trois valeurs differentes. - - Montrer qu'on dispose d'un $\eta\gt 0$ tel que, pour tous $x$, $y\in D$ vérifiant $\|x-y\|_{\i}\leq\eta$, on a $\|r(x)-r(y)\|\leq\frac{1}{10}$. - - Soit alors $n\in\N$ tel que $\frac{2}{n-1}\leq\eta$. Pour tous $i$, $j\in\db{1,n}$, on pose - -$v_{i,j}=\left(1-2\frac{i-1}{n-1},1-2\frac{j-1}{n-1}\right)$. - -Montrer que, pour tous $i$, $j\in\db{1,n-1}$, $v_{i,j},v_{i+1,j},v_{i+1,j+1},v_{i,j+1}$ sont contenus dans une boule de rayon $1/10$. - - En utilisant une fonction bien choisie de $C$ dans $\{1,2,3\}$, aboutir a une contradiction et conclure. - - Utiliser ce résultat pour montrer que toute fonction continue de $D$ dans $D$ admet un point fixe. + Montrer que, pour tous $i$, $j\in\db{1,n-1}$, $r(v_{i,j})$, $r(v_{i+1,j})$, $r(v_{i+1,j+1})$, $r(v_{i,j+1})$ sont contenus dans une boule de rayon $1/10$. + 5. En utilisant une fonction bien choisie de $C$ dans $\{1,2,3\}$, aboutir à une contradiction et conclure. + 6. Utiliser ce résultat pour montrer que toute fonction continue de $D$ dans $D$ admet un point fixe. #+end_exercice +#+BEGIN_proof + 1. + 2. Utiliser 1. avec $f(x,y) = \sign(x-y)$. + 3. UC. + 4. Vide. + 5. Découper $C$ régulièrement. + 6. Par l'absurde, si $f$ n'a pas de point fixe, on considère $g$ qui à tout $x$ associe l'intersection de la demi-droite $[x, f(x))$ avec le cercle (ou l'inverse). +#+END_proof + #+BEGIN_exercice [X 2023 # 330] @@ -3080,7 +3207,7 @@ On dit qu'une famille $\left(D_t\right)_{t \in \R^+}$ de disques fermés de $\R^ #+END_proof #+begin_exercice [X MP 2023 # 331] -Dans tout l'enonce, $\mathbb{K}$ designe $\R$ ou $\C$. On se donne une $\mathbb{K}$-algèbre $A$ de dimension finie, et on identifie $\mathbb{K}$ a une sous-algèbre de $A$ via $\lambda\mapsto\lambda.1_A$. On suppose donnée sur $A$ une norme multiplicative $\|\ \|$, autrement dit une norme vérifiant $\forall(a,b)\in A^2,\ \|ab\|=\|a\|\,\|b\|$. On suppose $\mathbb{K}=\C$. +Dans tout l'énonce, $\mathbb{K}$ désigne $\R$ ou $\C$. On se donne une $\mathbb{K}$-algèbre $A$ de dimension finie, et on identifie $\mathbb{K}$ a une sous-algèbre de $A$ via $\lambda\mapsto\lambda.1_A$. On suppose donnée sur $A$ une norme multiplicative $\|\ \|$, autrement dit une norme vérifiant $\forall(a,b)\in A^2,\ \|ab\|=\|a\|\,\|b\|$. On suppose $\mathbb{K}=\C$. - Soit $x\in A$. Montrer qu'il existe un $z_0\in\C$ tel que $\forall z\in\C,\ \|z_0-x\|\leq\|z-x\|$. - On suppose $\|a\|=2$ pour $a=z_0-x$. Montrer que $\|a-e^{\frac{2ikx}{n}}\|\geq 2$ pour tout $(n,k)\in\N^*\times\N$. - En déduire que $\|a-1\|=2$. @@ -3089,7 +3216,7 @@ Dans tout l'enonce, $\mathbb{K}$ designe $\R$ ou $\C$. On se donne une $\mathbb{ Dans la suite, on suppose que $\mathbb{K}=\R$. - Est-ce que $A$ est nécessairement égale a $\R$? - - On admet qu'il existe une $\R$-algèbre $\mathbb{H}$ ayant une base de la forme $(1,i,j,k)$ ou $i,j,k$ anticommutent deux a deux et $i^2=j^2=k^2=-1$. On considère la symétrie $x\mapsto\overline{x}$ par rapport a $\R$ parallélément a $\op{Vect}_{\R}(i,j,k)$, et on considère la norme $N:q\mapsto\sqrt{\overline{q}q}$. Montrer que $N$ est bien définie, est effectivement une norme, et qu'elle est multiplicative. + - On admet qu'il existe une $\R$-algèbre $\mathbb{H}$ ayant une base de la forme $(1,i,j,k)$ ou $i,j,k$ anticommutent deux a deux et $i^2=j^2=k^2=-1$. On considère la symétrie $x\mapsto\overline{x}$ par rapport a $\R$ parallèlement à $\op{Vect}_{\R}(i,j,k)$, et on considère la norme $N:q\mapsto\sqrt{\overline{q}q}$. Montrer que $N$ est bien définie, est effectivement une norme, et qu'elle est multiplicative. - Montrer que $A$ est isomorphe, en tant que $\R$-algèbre, a $\R$, $\C$ ou $\mathbb{H}$. #+end_exercice @@ -3101,9 +3228,9 @@ Dérivée discrète. #+END_proof #+begin_exercice [X MP 2023 # 333] -Pour $n\geq 2$, on note $\ell_n=\min\left\{k\in\db{1,n} \mid \prod_{i=1}^k\left(1-\frac{i}{n}\right)\leq\frac{1}{2}\right\}$. - - Montrer que $\ell_n=o(n)$. - - Donner un équivalent de $\ell_n$. +Pour $n\geq 2$, on note $\l_n=\min\left\{k\in\db{1,n} \mid \prod_{i=1}^k\left(1-\frac{i}{n}\right)\leq\frac{1}{2}\right\}$. + - Montrer que $\l_n=o(n)$. + - Donner un équivalent de $\l_n$. #+end_exercice #+BEGIN_proof - C'est montrer que $\forall c,\, \prod_{i=1}^{cn} \big(1 - \frac{i}{n}\big)\leq \frac{1}{2}$ APCR. Ou bien par comparaison $\sum/\int$, ou somme de Riemann un peu technique. @@ -3145,9 +3272,16 @@ On a $a_{n+1} - a_n \sim a_n^3$, donc $\sum a_n^3$ converge. Il faut trouver un Soit $\sum u_n$ une série convergente de réels positifs. Existe-t-il une suite $(v_n)_{n\geq 0}$ de réels positifs tendant vers $+\i$ telle que la série $\sum u_nv_n$ converge? #+end_exercice +# Relier au précédent #+begin_exercice [X MP 2023 # 339] Soit $(x_n)$ une suite réelle. On suppose que $(x_ny_n)$ est sommable pour toute suite réelle $(y_n)$ de carré sommable. Montrer que $(x_n)$ est de carré sommable. #+end_exercice +#+BEGIN_proof +Prendre $y_n = \frac{x_n}{\sum_{k=1}^n x_k^2} = \frac{x_n}{S_n}$ ? On a bien $\sum x_n y_n$ qui diverge, mais il semble pas nécessaire que $\sum y_n^2$ converge. + +Mais, en fait on va parfois prendre $y_n = 0$, et parfois l'autre (sur une plage $[N, 2N]$). On prend $y_n \neq 0$, sur une place $[N, 2N]$, où on sait que $S_N\gt 2^N$. +#+END_proof + #+begin_exercice [X MP 2023 # 340] Soit $\sigma$ une permutation de $\N^*$. Déterminer la nature de la série $\sum\frac{\sigma(n)}{n^2}$. @@ -3197,8 +3331,10 @@ puis IPP. $\forall x\neq y\in\R$, $f\left(\frac{x+y}{x-y}\right)=\frac{f(x)+f(y)}{f(x)-f(y)}$. #+end_exercice #+BEGIN_proof - - - - !! + - Classique ; En passant au $\ln$. + - On obtient $f$ impaire, on peut supposer $f$ croissante. On obtient $f(1) = 1$. + + Puis, en posant $y = xy$, on obtient $\frac{f(x) + f(xy)}{f(x) - f(xy)} = \frac{f(1) + f(y)}{1 - f(y)}$, qui donne $f(xy) = f(x) f(y)$. #+END_proof @@ -3247,7 +3383,7 @@ Calculer $\int_0^1\frac{\ln(1+t)}{1+t^2}dt$. #+begin_exercice [X MP 2023 # 352] Pour $n\in\N^*$, on note $L_n$ la derivée $n$-ième de $(X^2-1)^n$. - Soit $n\in\N^*$. Montrer que $:\forall P\in\R_{n-1}[X]$, $\int_{-1}^1PL_n=0$. - - Montrer que $L_n$ possede $n$ racines distinctes $x_1\lt x_2\lt \cdots\lt x_n$ dans $]-1,1[$. + - Montrer que $L_n$ possède $n$ racines distinctes $x_1\lt x_2\lt \cdots\lt x_n$ dans $]-1,1[$. - Montrer qu'il existe $\alpha_1,\ldots,\alpha_n\in\R$ tels que $:\forall P\in\R_{2n-1}[X]$, $\int_{-1}^1P=\sum_{i=1}^n\alpha_iP(x_i)$. #+end_exercice @@ -3260,9 +3396,14 @@ Pour $n\in\N$, on pose $I_n=\sum_{k=0}^n(-1)^k {n\choose k}^3$. #+end_exercice #+BEGIN_proof - Changement de variable, à extraire ? - - Il suffit ${n\choose k}^3 \leq \frac{1}{2}\big({n\choose k-1}^3 + {n\choose k+1}^3\big)$, $\frac{1}{k^3 (n-k)^3}\leq \frac{1}{2}\big(\frac{1}{(n-k)^3(n-k+1)^3} + \frac{1}{k^3 (k+1)^3}\big)$ - Par l'AM-GM, il suffit $\frac{1}{k^3 (n-k)^3}\leq \frac{1}{(n-k+1)^3 (k+1)^3}$, ce qui est faux. !! - - + - $n$ multiple de $4$ signifie qu'il y a un terme central, et qu'il a un signe positif. + + Il suffit de montrer que, pour $k-1 \geq n$, ${n\choose k}^3 \leq \frac{1}{2}\big({n\choose k-1}^3 + {n\choose k+1}^3\big)$. + + Si c'est vrai sans les puissances $3$, on aura $\frac{x^3 + y^3}{2}\geq \big(\frac{x+y}{2}\big)^3\geq \dots$. + + Et sans les puissances $3$, ça devrait être simple… + - Partir de $\sin (x+y) = \sin x \cos y + \sin y \cos x$, développer par le binôme. On obtient, $\sin^{2n + k}(x) \cos^k y \sin^{2n + (2n-k)}(y) \cos^{2n-k}(x) {2n\choose k}$, puis c'est le produit des intégrales sur $x$ et sur $y$, et c'est du Wallis. #+END_proof @@ -3318,13 +3459,15 @@ Soit $f\colon\R\to\R^+$ intégrable, de classe $\mc C^1$, telle que $\int_{-\i}^ #+begin_exercice [X MP 2023 # 360] On note $F$ l'ensemble des fonctions de $[0,1]$ dans $[0,1]$, $C$ l'ensemble des fonctions continues de $F$. On note aussi $I=\{f\in F\;;\ \forall a\in[0,1],\ \{x\in[0,1],\ f(x)\leq a\}$ est ferme$\}$ et $S=\{f\in F\;;\ \forall a\in[0,1],\ \{x\in[0,1],\ f(x)\geq a\}$ est ferme$\}$. -Pour $f\in F$ et $n\in\N$, soit $L_n(f):x\in[0,1]\mapsto\inf_{y\in[0,1]}\left(f(y)+n|x-y|\right)\in[0,1]$. +Pour $f\in F$ et $n\in\N$, soit $L_n(f)\colon x\in[0,1]\mapsto\inf_{y\in[0,1]}\left(f(y)+n|x-y|\right)\in[0,1]$. - Montrer que $C=I\cap S$. - Montrer que, si $f\in F$, $L_n(f)$ est une suite croissante d'applications continues. - Soit $f\in F$. Montrer que $f\in I$ si et seulement s'il existe une suite $(f_n)_{n\geq 0}$ de fonctions de $C$ telle que pour tout $x\in[0,1]$, $f(x)=\sup_{n\in\N}f_n(x)$. #+end_exercice #+BEGIN_proof -!! + - $\subset$ clair ; $\supset$ : dans l'intersection, on a $f^{-1}([a,b])$ fermé, donc par complémentaire, $f^{-1}\big(\interval]{a, b}[\big)$ ouvert. + - La croissance est claire, la continuité est classique. + - À faire, … #+END_proof @@ -3368,7 +3511,7 @@ Pour $x$ réel, on pose $f(x)=\sum_{n=1}^{+\i}\dfrac{x^n}{1-x^n}$ sous reserve d # Relier à un précédent #+begin_exercice [X MP 2023 # 364] - Soient $U$ un voisinage de $0$ dans $\C$, et $f:U\to\C$ somme d'une série entière. Soit $k\in\N^*$ tel que $f(z)=O(z^k)$ quand $z$ tend vers $0$. Montrer que, pour $r$ voisin de $0^+$, il existe au moins $2k$ nombres complexes $z$ de module $r$ tels que $f(z)$ soit un nombre réel. - - Soient $A$ et $B$ deux polynômes a coefficients réels dont toute combinaison linéaire a coefficients réels est scindée ou nulle. Soient $x\lt y$ deux racines de $A$. Montrre que $[x,y]$ contient au moins une racine de $B$. + - Soient $A$ et $B$ deux polynômes a coefficients réels dont toute combinaison linéaire a coefficients réels est scindée ou nulle. Soient $x\lt y$ deux racines de $A$. Montrer que $[x,y]$ contient au moins une racine de $B$. #+end_exercice @@ -3416,12 +3559,15 @@ Pour $x$ et $q$ dans $]0,1[$, on pose $(x,q)_n=\prod_{k=0}^{n-1}(1-q^kx)$. #+begin_exercice [X MP 2023 # 369] Soit $p$ un nombre premier. Pour tout $F\in\mathbb{F}_p[X]$, on pose $|F|=p^{\deg F}$. - - Soit $s\in\C$ tel que $\op{Re}s\gt 1$. Montrre que la famille $\big{(}|F|^{-s}\big{)}$, indexée par les polynômes $F\in\mathbb{F}_p[X]$ unitaires, est sommable et calculer sa somme, qu'on notera $z(s)$. - - On note $A$ l'ensemble des polynômes unitaires de $F\in\mathbb{F}_p[X]$ sans facteur carré, c'est-a-dire tels que : $\forall D\in\mathbb{F}_p[X]$, $D^2|F\Rightarrow\deg D=0$. Montrre que $\sum_{F\in A}|F|^{-s}=\frac{z(s)}{z(2s)}$. + - Soit $s\in\C$ tel que $\op{Re}s\gt 1$. Montrer que la famille $\big(|F|^{-s}\big)$, indexée par les polynômes $F\in\mathbb{F}_p[X]$ unitaires, est sommable et calculer sa somme, qu'on notera $z(s)$. + - On note $A$ l'ensemble des polynômes unitaires de $F\in\mathbb{F}_p[X]$ sans facteur carré, c'est-a-dire tels que : $\forall D\in\mathbb{F}_p[X]$, $D^2|F\Rightarrow\deg D=0$. Montrer que $\sum_{F\in A}|F|^{-s}=\frac{z(s)}{z(2s)}$. - En déduire, pour tout $d\in\N$, la proportion de polynômes sans facteur carré parmi les polynômes unitaires de degré $d$ de $\mathbb{F}_p[X]$. #+end_exercice #+BEGIN_proof -!! todo + - Sommabilité est simple, la somme est + $$z(s) = \sum_{d} \frac{p^d}{p^{ds}} = \sum_{d} \frac{1}{p^{d (s-1)}} = \frac{1}{1 - 1/p^{s-1}} = \frac{p^{s-1}}{p^{s-1} - 1}.$$ + - $z(2s)$ correspond à la somme sur tous les carrés. Par multiplicativité, on a le résultat. + - !! todo #+END_proof @@ -3438,7 +3584,7 @@ Soit $f$ continue sur $[0,1]$ et $g:x\mapsto\int_0^1\frac{f(t)}{1+xt}dt$ pour $x #+begin_exercice [X MP 2023 # 371] - Déterminer le domaine de définition de $f\colon x\mapsto\frac{2}{\sqrt{\pi}}\int_0^{\pi/2}(\cos t)^{2x}\,dt$. - - Montrre, pour tout réel $x\gt 0$, l'égalité $f(x)=\frac{2}{\sqrt{\pi}}\int_0^{+\i}\frac{u\exp\!\left(-u^2 \big{(}x+\frac{1}{2}\big{)}\right)}{\sqrt{1-e^{-u^2}}}du$. + - Montrer, pour tout réel $x\gt 0$, l'égalité $f(x)=\frac{2}{\sqrt{\pi}}\int_0^{+\i}\frac{u\exp\!\left(-u^2 \big{(}x+\frac{1}{2}\big{)}\right)}{\sqrt{1-e^{-u^2}}}du$. #+end_exercice #+begin_exercice [X MP 2023 # 372] @@ -3467,7 +3613,7 @@ Soient $\alpha,\beta\gt 0$. Pour $x\gt 0$, on pose $I(x)=\int_0^{+\i}t^{\beta-1} #+begin_exercice [X MP 2023 # 375] - Soient $K$ un segment et $f:K\to K$ une fonction continue croissante. Montrer que $f$ admet un point fixe. - - On considère l'équation différentielle non linéaire $(E):\ x'=\cos(x)+\cos(t)$. On admet que pour tout $a\in\R$ il existe une unique solution $\phi_a$ de $(E)$ sur $\R$ vérifiant $\phi(0)=a$, et que, pour tous $a,b$ réels distincts, les fonctions $\phi_a$ et $\phi_b$ ne coincident en aucun point. Montrer que $(E)$ possede une solution $2\pi$-périoddique. + - On considère l'équation différentielle non linéaire $(E):\ x'=\cos(x)+\cos(t)$. On admet que pour tout $a\in\R$ il existe une unique solution $\phi_a$ de $(E)$ sur $\R$ vérifiant $\phi(0)=a$, et que, pour tous $a,b$ réels distincts, les fonctions $\phi_a$ et $\phi_b$ ne coincident en aucun point. Montrer que $(E)$ possède une solution $2\pi$-périoddique. #+end_exercice # ID:6896 @@ -3504,7 +3650,7 @@ Soient $v\colon \R\to\R$ une fonction continue à support compact et $\omega\in\ Soient $q_1,q_2$ deux fonctions continues de $\R^+$ dans $\R$ telles que $q_1\leq q_2$. On considère l'équation différentielle $(E_i):y^{''}+q_i(t)\,y=0$ pour $i\in\{1,2\}$. - Soient $y_1,y_2$ des solutions respectives de $(E_1)$ et $(E_2)$ sur $I$. Soient $\alpha\lt \beta$ deux zeros de $y_1$. Montrer que $y_2$ s'annule dans $[\alpha,\beta]$. - Soient $q:\R^+\to\R$ continue, $m,M$ deux réels strictement positifs tels que $m\leq q\leq M$. Soient $\alpha\lt \beta$ deux zeros consecutifs d'une solution non nulle $x$ de $y''+q(t)\,y=0$. - - Montrer que les zeros de $x$ fortner une suite strictement croissante $(t - {n\in\N}$. + - Montrer que les zeros de $x$ fortner une suite strictement croissante $(t_n)_{n\in\N}$. - Montrer que $\frac{\pi}{\sqrt{M}}\leq t_{n+1}-t_n\leq\frac{\pi}{\sqrt{m}}$ pour tout $n\in\N$. #+end_exercice @@ -3530,6 +3676,7 @@ Soient $v\colon \R\to\R$ une fonction continue à support compact et $\omega\in\ #+END_proof +# ID:7034 #+begin_exercice [X MP 2023 # 380] :sup: On munit $\R^2$ de sa structure euclidienne canonique. On considère le carré de coins $\{0,1\}\times\{0,1\}$. On choisit trois points $A$, $B$ et $C$ sur ce carré. - Montrer qu'il existe une disposition des points $A$, $B$ et $C$ maximisant l'aire du triangle $ABC$. @@ -3568,16 +3715,19 @@ Déterminer le nombre moyen de 2-cycles, de 3-cycles, de $p$-cycles, d'une permu # Relier à l'autre #+begin_exercice [X MP 2023 # 384] - Montrer que $\forall x\in{\R}^{+*},\ \frac{e^{-x}}{(1-e^{-x})^2}\lt \frac{1}{x^2}$. - - Soit $n\in{\N}^*$. On appelle partition de $n$ toute liste decroissante $(\lambda - {1\leq k\leq n}$ d'entiers naturels non nuls de somme $n$. On note $P(n)$ le nombre de telles listes. + - Soit $n\in{\N}^*$. On appelle partition de $n$ toute liste décroissante $(\lambda_k)_{1\leq k\leq n}$ d'entiers naturels non nuls de somme $n$. On note $P(n)$ le nombre de telles listes. Montrer que $P(n)\leq 2^{n-1}$. - On fixe $n\geq 1$ et on considère une variable aléatoire $X$ suivant la loi uniforme sur l'ensemble des partitions de $n$. On fixe $k\in{\N}^*$ et $j\in{\N}$. On pose $N_k=|\{i\in\db{1,n}:X_i=k\}|$. - Exprimer ${\bf P}(N_k\geq j)$ comme un quotient $\frac{P(a)}{P(b)}$ pour des entiers $a$ et $b$ a preciser. + Exprimer ${\bf P}(N_k\geq j)$ comme un quotient $\frac{P(a)}{P(b)}$ pour des entiers $a$ et $b$ à préciser. - Calculer $\sum_{i=1}^niN_i$. #+end_exercice #+BEGIN_proof -!! todo + - Revient à $(\ln u)^2 u \lt (1-u)^2$, pour $u\in \interval]{0, 1}[$. En posant $u = 1-h$ et en développant en série, par CSSA, c'est bon. + - Par récurrence, bof. + - L'évènement $(N_k \geq j)$ correspond à «il y a au moins $j$ occurrences de $k$», cela correspond à $\frac{P(n - kj)}{P(n)}$. + - C'est $n$. #+END_proof @@ -3611,7 +3761,7 @@ Soient $b,n\in{\N}^*$. On considère $(B_i)_{1\leq i\leq n}$ des variables aléa #+end_exercice #+begin_exercice [X MP 2023 # 389] -Si $n\in\N^*$, $\sigma\in\mc{S}_{2n}$ et $k\in\{1,\ldots,2n\}$, on note $s(\sigma,k)$ le segment de $\C$ qui joint les points $e^{\frac{ik\pi}{n}}$ et $e^{\frac{i\sigma(k)\pi}{n}}$. On note $b(\sigma)$ le nombre de segments qui ne croisent aucun autre segment (ou on dit que deux segments se croisent s'ils ont un point d'intersection qui n'est pas une extremite). +Si $n\in\N^*$, $\sigma\in\mc{S}_{2n}$ et $k\in\{1,\ldots,2n\}$, on note $s(\sigma,k)$ le segment de $\C$ qui joint les points $e^{\frac{ik\pi}{n}}$ et $e^{\frac{i\sigma(k)\pi}{n}}$. On note $b(\sigma)$ le nombre de segments qui ne croisent aucun autre segment (ou on dit que deux segments se croisent s'ils ont un point d'intersection qui n'est pas une extrémité). Pour $n\in\N^*$, soit $\sigma_n$ une variable aléatoire suivant la loi uniforme sur $\mc{S}_{2n}$. Déterminer $\mathbf{E}(b(\sigma_n))$ et en donner un équivalent. #+end_exercice @@ -3654,8 +3804,22 @@ Soit $n\geq 1$. - On se donne quatre variables aléatoires indépendantes $X_n,Y_n,A_n,B_n$ suivant chacune la loi uniforme sur $\db{1,n}^2$. On note $p_n$ la probabilité pour que $X_n\neq Y_n$, $A_n\neq B_n$ et les droites $(X_nY_n)$ et $(A_nB_n)$ soient parallèles. Montrer que $p_n=O\big(\frac{\ln n}{n^2}\big)$ quand $n\ra+\i$. #+end_exercice #+BEGIN_proof -!! - - C'est la probabilité que $\frac{a_n - b_n}{c_n - d_n} = \frac{p}{q}$, c'est-à-dire $p(c_n - d_n) = q (a_n - b_n)$. Les différences suivent des lois + - C'est la probabilité que $\frac{a_n - b_n}{c_n - d_n} = \frac{p}{q}$, c'est-à-dire $p(c_n - d_n) = q (a_n - b_n)$. + + Il faut que $a_n - b_n$ soit un multiple non nul de $p$, + + $\sum_{k\neq 0} P(a_n - b_n = kp)P(c_n - d_n = kq) = \sum_{k\neq 0} \sum_{i,j=1}^n \frac{1_{i = kp + j}}{n^2} \sum_{u, v} \frac{1_{u = kq + v}}{n^2}$ + On peut supposer $k\gt 0$, quitte à multiplier par $2$. Pour $k=1$ le produit vaut $\frac{(n-1 - p)(n-1-q)}{n^4}$. + + On a donc $\sum_{k\geq 1} \frac{(n-1 - kp) (n-1-kq)}{n^4}$, qu'on peut identifier à $\sum_{k\geq 1} \frac{(n - kp) (n-kq)}{n^4}$. + + C'est $\frac{1}{n^2} \sum_{k\geq 1} \big(1 - \frac{kp}{n}\big)\big(1 - \frac{kq}{n}\big)$, qui est une somme de Riemann, pour la fonction $f\colon x\mapsto (1 - px) (1-qx)$, nulle dès que $\max(qx, px)\geq 1$. + + On trouve $\frac{\max(p,q) + |p-q|}{6 \max(p,q)^2}$, peut-être. Mais le $\Theta(\frac{1}{\max(p,q)})$ est clair. + + En particulier, c'est en $O(\frac{1}{n})$ + - On somme sur les $r$. + On a $\sum_{r} \frac{1}{(n h(r))^2}$ $= \sum_{k = h(r)} \frac{k}{n^2 k^2}$, d'où le résultat. #+END_proof @@ -3702,7 +3866,7 @@ Relier à un précédent. On munit l'ensemble $\mc{S}_n$ des permutations de $[1,n]$ de la probabilité uniforme. Soit $X_n$ la variable aléatoire donnant le nombre de points fixes d'une permutation aléatoire $\sigma\in\mc{S}_n$. - Calculer $\mathbf{P}(X_n=0)$. - Déterminer la loi de $X_n$. - - Étudier la convergence en loi de la suite $(X - {n\in\N^*}$. + - Étudier la convergence en loi de la suite $(X_n)_{n\in\N^*}$. - Calculer les espérance et variance de la variable aléatoire $X_n$. #+end_exercice @@ -3722,18 +3886,18 @@ Soit $M=\begin{pmatrix}a&-b&-c&-d\\ b&a&d&-c\\ c&-d&a&b\\ d&c&-b&a\end{pmatrix}$ #+end_exercice #+BEGIN_exercice [X MP 2023 # 400] - Soit $n\geq 3$ un entier. Si $k\in\Z$, on note $\overline{k}$ la reduction de $k$ modulo $n$. Soient $X_1,\ldots,X_n$ des variables aléatoires indépendantes a valeurs dans $\Z/n\Z$ telles que, pour tout $k\in\db{1,n}$, $X_k$ suit la loi uniforme sur $\{\overline{1},\overline{2},\overline{3}\}$. Soit $F$ l'application aléatoire de $\Z/n\Z$ dans lui-meme telle que, pour tout $k\in\db{1,n}$, $F(\overline{k})=\overline{k}+X_k$. Calculer la probabilité que $F$ soit bijective. + Soit $n\geq 3$ un entier. Si $k\in\Z$, on note $\overline{k}$ la reduction de $k$ modulo $n$. Soient $X_1,\ldots,X_n$ des variables aléatoires indépendantes a valeurs dans $\Z/n\Z$ telles que, pour tout $k\in\db{1,n}$, $X_k$ suit la loi uniforme sur $\{\overline{1},\overline{2},\overline{3}\}$. Soit $F$ l'application aléatoire de $\Z/n\Z$ dans lui-même telle que, pour tout $k\in\db{1,n}$, $F(\overline{k})=\overline{k}+X_k$. Calculer la probabilité que $F$ soit bijective. #+END_exercice #+begin_exercice [X MP 2023 # 401] -On cherche a collectionner $N$ jouets. À chaque achat, chaque jouet a une probabilité uniforme d'être obtenu. Pour $i\in\db{1,N}$, on note $T_i$ le temps d'attente pour obtenir $i$ jouets différents. +On cherche à collectionner $N$ jouets. À chaque achat, chaque jouet a une probabilité uniforme d'être obtenu. Pour $i\in\db{1,N}$, on note $T_i$ le temps d'attente pour obtenir $i$ jouets différents. - Calculer l'espérance de $T_N$. - Calculer la variance de $T_N$. - Montrer que $\forall\eps\gt 0$, $\mathbf{P}\left(\left|\frac{T_N}{N\ln N}-1\right|\geq\eps \right)\longrightarrow 0$ quand $N\ra+\i$. #+end_exercice #+begin_exercice [X MP 2023 # 402] -Soit $(X - {n\in\N^*}$ une suite i.i.d. de variables aléatoires réelles centrées. +Soit $(X_n)_{n\in\N^*}$ une suite i.i.d. de variables aléatoires réelles centrées. On suppose que $\mathbf{E}(X_1^4)\lt +\i$. - Montrer que $\mathbf{E}\left(\left(X_1+\cdots+X_n\right)^4\right)=O(n^2)$. @@ -3741,7 +3905,7 @@ On suppose que $\mathbf{E}(X_1^4)\lt +\i$. #+end_exercice #+begin_exercice [X MP 2023 # 403] -Soient $x\in\R^{+*}$, $(X - {k\geq 1}$ une suite i.i.d. de variables aléatoires suivant la loi $\mc{P}(x)$. Pour $n\in\N^*$, soient $S_n=\sum_{k=1}^nX_k,T_n=\frac{S_n-n}{\sqrt{n}}$. +Soient $x\in\R^{+*}$, $(X_k)_{k\geq 1}$ une suite i.i.d. de variables aléatoires suivant la loi $\mc{P}(x)$. Pour $n\in\N^*$, soient $S_n=\sum_{k=1}^nX_k,T_n=\frac{S_n-n}{\sqrt{n}}$. - Montrer que $\int_0^{+\i}\mathbf{P}(T_n\geq x)dx=\sqrt{n}\left( \frac{n}{e}\right)^n\frac{1}{n!}$. - On admet que, pour tout $x\in\R$, $\mathbf{P}(T_n\geq x)\underset{n\ra+\i}{\longrightarrow} \frac{1}{\sqrt{2\pi}}\int_x^{+\i}e^{-t^2/2}dt$. Retrouver la formule de Stirling. #+end_exercice @@ -3761,7 +3925,7 @@ Soit $E$ un espace vectoriel de dimension finie. Montrer qu'il existe une base d #+end_exercice #+begin_exercice [X PSI 2023 # 406] -Soient $E$ un $\C$-espace vectoriel de dimension finie $n$, $f$ un endomorphisme de $E$ diagonalisable. Montrer que $f$ possede $n$ valeurs propres distinctes si et seulement s'il existe $v\in E$ tel que $(v,f(v)\cdots,f^{n-1}(v))$ soit libre. +Soient $E$ un $\C$-espace vectoriel de dimension finie $n$, $f$ un endomorphisme de $E$ diagonalisable. Montrer que $f$ possède $n$ valeurs propres distinctes si et seulement s'il existe $v\in E$ tel que $(v,f(v)\cdots,f^{n-1}(v))$ soit libre. #+end_exercice #+begin_exercice [X PSI 2023 # 407] @@ -3774,7 +3938,7 @@ Trouver $\text{Vect}(\mc{O}_n(\R))$. #+begin_exercice [X PSI 2023 # 409] -Soit $A\in\M_n(\R)$ tel que $A^2\in\mc{S}_n(\R)$. - A-t-on nécessairement $A\in\mc{S}_n(\R)$? - Trouver une condition nécessaire supplementaire pour avoir $A\in\mc{S}_n(\R)$. +Soit $A\in\M_n(\R)$ tel que $A^2\in\mc{S}_n(\R)$. - A-t-on nécessairement $A\in\mc{S}_n(\R)$? - Trouver une condition nécessaire supplémentaire pour avoir $A\in\mc{S}_n(\R)$. #+end_exercice *** Analyse @@ -4011,7 +4175,7 @@ i) $\forall A\in\M_n(\R),\ f\big{(}AA^T\big{)}\geq 0$; ii) $\exists B\in\mc{S}_n^+(\R),\ \forall A\in\M_n(\R),\ f(A)= \mathrm{Tr}(AB)$; -iii) $\exists m\in\N,\ \exists(X - {i\in\db{1\,:\,m}}\in \M_{n,1}(\R)^m,\ \forall A\in\M_n(\R),\ f(A)=\sum_{i=1}^m \mathrm{Tr}\,\big{(}X_i^TAX_i\big{)}$. +iii) $\exists m\in\N,\ \exists(X_i)_{i\in\db{1\,:\,m}}\in \M_{n,1}(\R)^m,\ \forall A\in\M_n(\R),\ f(A)=\sum_{i=1}^m \mathrm{Tr}\,\big{(}X_i^TAX_i\big{)}$. #+end_exercice #+begin_exercice [X PC 2023 # 460] @@ -4021,7 +4185,7 @@ Sont $A\in\mc{S}_n^{++}\left(\R\right)$ et $B\in\mc{A}_n\left(\R\right)$. Montre *** Analyse #+begin_exercice [X PC 2023 # 461] -Si $I$ est un intervalle de $\R$, on note $|I|$ sa longueur. Montrer qu'il existe une famille $(I - {j\in A}$ d'intervalles de $\R$, non reduits a un point, deux a deux disjoints et tels que +Si $I$ est un intervalle de $\R$, on note $|I|$ sa longueur. Montrer qu'il existe une famille $(I_j)_{j\in A}$ d'intervalles de $\R$, non reduits a un point, deux a deux disjoints et tels que $\Q\subset\bigcup_{j\in A}I_j$ et $\sum_{j\in A}|I_j|=42$. #+end_exercice @@ -4039,7 +4203,7 @@ $\forall t\in\R$, $A(t)=P\mathrm{diag}(e^{i2\pi\lambda_1t},\ldots,e^{i2\pi\lambd #+end_exercice #+begin_exercice [X PC 2023 # 464] -Soit $A\in\M_n(\R)$. On définit une suite de matrices par $M_0=A$ et, pour tout $k\in\N$, $M_{k+1}=M_k-M_k^2$. On etudie la convergence eventuelle de $(M - {k\geq 0}$. +Soit $A\in\M_n(\R)$. On définit une suite de matrices par $M_0=A$ et, pour tout $k\in\N$, $M_{k+1}=M_k-M_k^2$. On étudie la convergence éventuelle de $(M_k)_{k\geq 0}$. - Étudier le cas ou $A$ admet une valeur propre réelle $\lambda\lt 0$ ou $\lambda\gt 1$. - Étudier le cas ou $A$ est nilpotente. - Étudier le cas ou $A=\lambda I+N$ avec $N\neq 0$, $N^2=0$ et $0\lt \lambda\lt 1$. @@ -4116,13 +4280,13 @@ Soit $(u_n)$ une suite bornée. Montr er qu'il y a equivalence entre : #+begin_exercice [X PC 2023 # 478] - Soit $\left(u_n\right)_{n\geq 0}$ une suite bornée telle que $\lim_{n\to+\i}\sum_{p=0}^{+\i}\frac{u_{n2^p}}{2^p}=1$. -Que peut-on en dedaire sur la suite $\left(u_n\right)_{n\geq 0}$? - - Soit $(v - {n\in\N}$ une suite réelle bornée. On suppose $\lim_{n\to+\i}\left(v_n-\frac{1}{2}v_{2n}\right)=\frac{1}{2}$. Que dire $\left(v_n\right)_{n\in\N}$? +Que peut-on en deduire sur la suite $\left(u_n\right)_{n\geq 0}$? + - Soit $(v_n)_{n\in\N}$ une suite réelle bornée. On suppose $\lim_{n\to+\i}\left(v_n-\frac{1}{2}v_{2n}\right)=\frac{1}{2}$. Que dire $\left(v_n\right)_{n\in\N}$? #+end_exercice #+begin_exercice [X PC 2023 # 479] -- Soient $a\in\N^*$ et $n\in\N$. Montr er qu'il existe des entiers $c_j$, avec $0\leq j\leq a-1$, tels que $\sum_{k=0}^{+\i}\frac{k^n}{k!^a}=\sum_{k=0}^{+\i}\frac{\sum_{j=0}^ {a-1}c_jk^j}{k!^a}$. - - Montr er que les $c_j$ sont uniquées (on traitera d'abord le cas $a=2$). + - Soient $a\in\N^*$ et $n\in\N$. Montrer qu'il existe des entiers $c_j$, avec $0\leq j\leq a-1$, tels que $\sum_{k=0}^{+\i}\frac{k^n}{k!^a}=\sum_{k=0}^{+\i}\frac{\sum_{j=0}^ {a-1}c_jk^j}{k!^a}$. + - Montrer que les $c_j$ sont uniques (on traitera d'abord le cas $a=2$). #+end_exercice #+begin_exercice [X PC 2023 # 480] @@ -4223,7 +4387,7 @@ Pour $ n\in\N^*$ et $ x\in\R^+$, on pose $ f_n(x)=\cos\biggl{(}\dfrac{x}{\sqrt{n #+end_exercice #+begin_exercice [X PC 2023 # 497] -Soit $(f - {n\in\N}$ une suite de fonctions appartenant a $\mc C^3(\R,\R)$ et $C$ une constante réelle positive. On suppose : (i) $\sup_{n\in\N}\|f_n^{(3)}\|_{\i}\leq C$, (ii) $\lim_{n\to+\i}\|f_n\|_{\i}=0$. +Soit $(f_n)_{n\in\N}$ une suite de fonctions appartenant a $\mc C^3(\R,\R)$ et $C$ une constante réelle positive. On suppose : (i) $\sup_{n\in\N}\|f_n^{(3)}\|_{\i}\leq C$, (ii) $\lim_{n\to+\i}\|f_n\|_{\i}=0$. - Montrer que $\lim\|f_n'\|_{\i}=\lim\|f_n^{''}\|_{\i}=0$. - Les résultats precedents restent-ils vrais si on ne fait plus l'hypothese (i)? #+end_exercice @@ -4338,13 +4502,12 @@ On tire une piece $n$ fois indépendamment avec probabilité de faire pile $1/n$ #+end_exercice #+begin_exercice [X PC 2023 # 516] -- Montrer que $\forall x\in\R$, $\mathrm{ch}(x)\leq e^{x^2/2}$. - - Soient $X_1,\ldots,X_n$ des variables i.i.d. suivant la loi uniforme sur $\{-1,1\}$. On pose, pour $n\in\N^*$, $S_n=X_1+\cdots+X_n$. Soit $\lambda\in\R^{+*}$. Montrer que $\mathbf{P}(S_n\geq\lambda)\leq e^{-\lambda^2/2n}$._Algèbre_ + - Montrer que $\forall x\in\R$, $\mathrm{ch}(x)\leq e^{x^2/2}$. + - Soient $X_1,\ldots,X_n$ des variables i.i.d. suivant la loi uniforme sur $\{-1,1\}$. On pose, pour $n\in\N^*$, $S_n=X_1+\cdots+X_n$. Soit $\lambda\in\R^{+*}$. Montrer que $\mathbf{P}(S_n\geq\lambda)\leq e^{-\lambda^2/2n}$. #+end_exercice - -* Mines +* Mines :mines: ** Algèbre @@ -4436,7 +4599,7 @@ en Soit $A$ un anneau commutatif. Si $I$ est un ideal de $A$, on note $R(I)=\{x\in A\ ;\ \exists n\in\N,\ x^n\in I\}$. - Montrer que $R(I)$ est un ideal de $A$ contenant $I$. - - Soient $I$ et $J$ deux ideaux de $A$. Montrer : + - Soient $I$ et $J$ deux idéaux de $A$. Montrer : $R(I\cap J)=R(I)\cap R(J)$ ; $R(I)+R(J)\subset R(I+J)$. - Pour cette question, $A=\Z$. Montrer que l'ensemble des entiers naturels non nuls tels que $R(n\Z)=n\Z$ est l'ensemble des entiers naturels non nuls dont la decomposition primaire ne comporte aucun facteur premier d'exposant au moins égal a 2. @@ -4452,10 +4615,10 @@ Soient $n\in\N^*$, $z_1,\ldots,z_n$ des nombres complexes non nuls de meme modul #+end_exercice #+begin_exercice [Mines 2023 # 534] -- Montrer que, pour tout $n\in\N$, il existe $P_n\in\R[X]$ tel que$\forall\theta\in\Big{]}0,\frac{\pi}{2}\Big{[},\ \frac{\sin((2n+1)\theta)}{\sin^{2n+1}( \theta)}=P_n(\text{cotan}^2\theta).$ +- Montrer que, pour tout $n\in\N$, il existe $P_n\in\R[X]$ tel que $\forall\theta\in\Big{]}0,\frac{\pi}{2}\Big{[},\ \frac{\sin((2n+1)\theta)}{\sin^{2n+1}( \theta)}=P_n(\text{cotan}^2\theta)$. - Déterminer les racines de $P_n$ et calculer leur somme. - - Montrer que, pour $\theta\in\Big{]}0,\frac{\pi}{2}\Big{[}$, $\text{cotan}^2\theta\lt \frac{1}{\theta^2}\lt \text{cotan}^2\theta+1.$ - - Déduire de ce qui precede la valeur de $\sum_{n=1}^{+\i}\frac{1}{n^2}.$ + - Montrer que, pour $\theta\in\Big{]}0,\frac{\pi}{2}\Big{[}$, $\text{cotan}^2\theta\lt \frac{1}{\theta^2}\lt \text{cotan}^2\theta+1$. + - Déduire de ce qui precede la valeur de $\sum_{n=1}^{+\i}\frac{1}{n^2}$. #+end_exercice #+begin_exercice [Mines 2023 # 535] @@ -4467,9 +4630,9 @@ Interpolation de Lagrange. #+begin_exercice [Mines 2023 # 536] Soit $P\in\C[X]$. - - À quelle condition $P$ realise-t-il une surjection de $\C$ sur $\C\,?$ - - À quelle condition $P$ realise-t-il une surjection de $\R$ sur $\R\,?$ - - À quelle condition $P$ realise-t-il une surjection de $\Q$ sur $\Q\,?$ + - À quelle condition $P$ réalise-t-il une surjection de $\C$ sur $\C\,?$ + - À quelle condition $P$ réalise-t-il une surjection de $\R$ sur $\R\,?$ + - À quelle condition $P$ réalise-t-il une surjection de $\Q$ sur $\Q\,?$ #+end_exercice #+begin_exercice [Mines 2023 # 537] @@ -4577,6 +4740,10 @@ Soit $E$ un $\mathbb{K}$-espace vectoriel de dimension finie. On dit que $h\in\m #+begin_exercice [Mines 2023 # 555] Soient $A\in\M_n(\R)$ et $M\in\M_n(\R)$. On suppose que toutes les matrices semblables a $A$ appartiennent au commutant de $M$. Déterminer $M$. Meme question dans $\M_n(\C)$. #+end_exercice +#+BEGIN_proof +Si $A$ admet un espace caractéristique non trivial, $M$ stabilise tous les espaces qui ont la même dimension, ce qui implique, par intersections judicieuses, que $M$ stabilise toutes les droites. +#+END_proof + # ID:7010 #+begin_exercice [Mines 2023 # 556] :sup: @@ -4601,11 +4768,12 @@ Soient $K_1$,..., $K_n$ des segments non triviaux disjoints. #+end_exercice #+begin_exercice [Mines 2023 # 559] -- Déterminer le rang de $\,\text{Com}(A)$ en fonction du rang de $A$. + - Déterminer le rang de $\,\text{Com}(A)$ en fonction du rang de $A$. - Calculer $\,\text{Com}\,\left(\text{Com}(A)\right)$ lorsque $A\in\mathrm{GL}_n(\R)$. - - Montrer que si $X$ est un vecteur propre de $A$ associe a une valeur propre non nulle, alors $X$ est un vecteur propre de $\,(\,\text{Com}(A))^T$. + - Montrer que si $X$ est un vecteur propre de $A$ associé à une valeur propre non nulle, alors $X$ est un vecteur propre de $(\text{Com}(A))^T$. #+end_exercice + #+begin_exercice [Mines 2023 # 560] Soit $n\in\N^*$. Soit $D$ l'ensemble des matrices $M\in\M_n(\mathbb{K})$ telles que $m_{i,j}=0$ si $i$ et $j$ sont de parites differentes. - Montrer que $D$ est une sous-algèbre de $\M_n(\mathbb{K})$. @@ -4660,7 +4828,7 @@ Soit $A\in\M_n(\C)$ nilpotente. #+begin_exercice [Mines 2023 # 569] Soient $E$ un espace vectoriel de dimension finie et $(u,v)\in\mc{L}(E)^2$. - Montrer que $|\op{rg}(u)-\op{rg}(v)|\leq\op{rg}(u+v) \leq\op{rg}(u)+\op{rg}(v)$. - - Soient $F$ un sous-espace vectoriel de $E$, $G$ et $H$ deux supplementaires de $F$. On note $p$ (resp. $q$) la projection sur $F$ (sur $H$) parallélément a $G$ (a $F$). + - Soient $F$ un sous-espace vectoriel de $E$, $G$ et $H$ deux supplémentaires de $F$. On note $p$ (resp. $q$) la projection sur $F$ (sur $H$) parallélément a $G$ (a $F$). Montrer que $\op{rg}(p+q)=\op{rg}p+\op{rg}q$. #+end_exercice @@ -4675,11 +4843,11 @@ Soit $G$ un sous-groupe fini de $\mathrm{GL}_n(\C)$. Montrer que $\sum_{M\in G}\ #+begin_exercice [Mines 2023 # 572] - Soit $G$ un sous-groupe fini de $\mathrm{GL}_n(\R)$ tel que $\sum_{g\in G}\op{tr}g=0$. Montrer que $\sum_{g\in G}g=0$. - - Soient $G$ un sous-groupe fini de $\mathrm{GL}_n(\R)$ et $V$ un sous-espace vectoriel de $\R^n$ stable par tous les éléments de $G$. Montrer que $V$ admet un supplementaire stable par tous les éléments de $G$. + - Soient $G$ un sous-groupe fini de $\mathrm{GL}_n(\R)$ et $V$ un sous-espace vectoriel de $\R^n$ stable par tous les éléments de $G$. Montrer que $V$ admet un supplémentaire stable par tous les éléments de $G$. #+end_exercice #+begin_exercice [Mines 2023 # 573] -Déterminer les ideaux bilateres de $\M_n(\R)$, c'est-a-dire les sous-groupes additifs stables par multiplication a gauche et a droite par n'importe quel élément de $\M_n(\R)$. +Déterminer les idéaux bilateres de $\M_n(\R)$, c'est-a-dire les sous-groupes additifs stables par multiplication a gauche et a droite par n'importe quel élément de $\M_n(\R)$. #+end_exercice #+begin_exercice [Mines 2023 # 574] @@ -4805,7 +4973,7 @@ Soit $E=\M_n(\R)$. Soient $A\in E$ et $u_A:M\in E\mapsto AM$. Soient $A,B\in\M_n(\mathbb{K})$ diagonalisables. Soit $\Delta$ l'endomorphisme de $\M_n(\mathbb{K})$ défini par $\forall M\in\M_n(\mathbb{K})$, $\Delta(M)=AM+MB$. Montrer que $\Delta$ est diagonalisable et preciser ses valeurs propres. #+begin_exercice [Mines 2023 # 595] -Soit $\sigma$ une permutation de $\db{1,n]\!]$. Pour $M\in\M_n(\mathbb{K})$, on pose $p(M)=M'$ avec : $\forall(i,j)\in[\![1,n}^2,\,m'_{i,j}=m_{i,j}$ si $i=\sigma(j)$ et $m'_{i,j}=0$ sinon. +Soit $\sigma$ une permutation de $\db{1,n}$. Pour $M\in\M_n(\mathbb{K})$, on pose $p(M)=M'$ avec : $\forall(i,j)\in\db{1,n}^2,\,m'_{i,j}=m_{i,j}$ si $i=\sigma(j)$ et $m'_{i,j}=0$ sinon. - Montrer que $p$ est un projecteur. Déterminer son noyau et son image. Soit $A\in\M_n(\mathbb{K})$ non nulle. On définit deux applications $\phi$ et $u_A$ par : @@ -4873,7 +5041,7 @@ Soit $A\in{\cal M}_n({\C})$. On pose $f_A:M\in{\cal M}_n({\C})\mapsto AMA^T\in{\ #+end_exercice #+begin_exercice [Mines 2023 # 605] -Soit $p$ une permutation de $\db{1,n]\!]^2$. On considère l'application $u:{\cal M}_n({\C})\to{\cal M}_n({\C})$ définie par $:u(A)=(A_{p(i,j)})_{(i,j)\in[\![1,n}^2}$. Montrer que $u$ est un endomorphisme de ${\cal M}_n({\C})$. Est-il diagonalisable? +Soit $p$ une permutation de $\db{1,n}^2$. On considère l'application $u:{\cal M}_n({\C})\to{\cal M}_n({\C})$ définie par $:u(A)=(A_{p(i,j)})_{(i,j)\in \db{1,n}^2}$. Montrer que $u$ est un endomorphisme de ${\cal M}_n({\C})$. Est-il diagonalisable? #+end_exercice #+begin_exercice [Mines 2023 # 606] @@ -4909,9 +5077,56 @@ Montrer que $A^2=I_2$ ou qu'il existe $k\in{\N}^*$ tel que $A^{2^k}=-I_2$. #+begin_exercice [Mines 2023 # 611] Soit $u$ un endomorphisme diagonalisable de ${\C}^n$. Montrer que les propositions suivantes sont équivalentes : + + l'endomorphisme $u$ admet $n$ valeurs propres distinctes. + + la famille $(\op{id}, u, \dots, u^{n-1})$ est libre. + + il existe $x\in\C^n$ tel que $(x, u(x),\dots, u^{n-1}(x))$ soit libre. #+end_exercice -!! Page manquante +#+BEGIN_exercice [Mines 2023 # 612] +Soit $A\in GL_n(\C)$. + - Montrer que $A$ est triangulaire supérieure si et seulement si pour tout $k\geq 2$, $A^k$ est triangulaire supérieure. + - Donner un contre-exemple si $A$ est une matrice non inversible. +#+END_exercice + +#+BEGIN_exercice [Mines 2023 # 613] +Soit $E$ un $\C$-ev de dimension finie et $u\in\mc L(E)$. Montrer que $u$ est diagonalisable si et seulement si il existe $n\in\N^*$, $\a_1,\dots, \a_n\in\C$ et $v_1,\dots,v_n\in\mc L(E)$ tels que $\forall k\in\db{0,n},\, u^k = \sum_{i=1}^n \a_i^k v_i$. +#+END_exercice + +#+BEGIN_exercice [Mines 2023 # 614] +Soient $A,B\in\M_n(\R)$. On note $[A,B] = AB - BA$. Soit $C = [A,B]$. On suppose que $[A,C] = O_n$. + - Montrer que pour tout $k\gt 0$, $\tr C^k = 0$. En déduire que $C$ est nilpotente. + - Montrer que pour tout $&\gt 0$, $[B,A^p] = -pC A^{p-1}$. + - On suppose que $A$ est nilpotente. Montrer que $AB$ l'est aussi. +#+END_exercice + +#+BEGIN_exercice [Mines 2023 # 615] +Soit $A\in\M_n(\C)$. Montrer l'équivalence entre + + l'unique valeur propre de $A$ est $1$. + + $\tr A = \tr A^2 = \dots = \tr A^n = n$ +#+END_exercice + +#+BEGIN_exercice [Mines 2023 # 616] + - Soit $D\in\M_n(\C)$. Déterminer l'inverse de $\begin{pmatrix}I_n & D \\ 0 & I_n\end{pmatrix}$. + - Soient $A,B,C$ des matrices diagonalisable telles que $\op{sp} A \cap \op{sp} B = \emptyset$. Montrer que $\begin{pmatrix}A & C \\ 0 & B\end{pmatrix}$ et $\begin{pmatrix}A & 0 \\ 0 & B\end{pmatrix}$ sont semblables. +#+END_exercice + +#+BEGIN_exercice [Mines 2023 # 617] +Soit $A\in\M_n(\C)$. On pose $M_p = \frac{1}{p}\sum_{k=0}^{p-1} A^k$. Donner une CNS pour que la suite $(M_p)_{p\geq 1}$ soit bornée. +#+END_exercice + +#+BEGIN_exercice [Mines 2023 # 618] +Soit $A\in\M_n(\C)$ et $C(A) = \{M\mid AM = MA\}$. + 1. Montrer que $C(A)$ est de dimension $\geq n$. + 2. Montrer que $C(A)$ est de dimension $n$ si et seulement si $\chi_A = \pi_A$. +#+END_exercice + +#+BEGIN_exercice [Mines 2023 # 619] +Soit $M\in\M_n(\R)$ possédant $n$ valeurs propres distinctes. Montrer que $C(M) = \vect I_n,\dots, M^{n-1}$. +#+END_exercice + +#+BEGIN_exercice [Mines 2023 # 620] +Déterminer les $A\in\M_n(\C)$ telles que toute matrice commutant à $A$ soit diagonalisable. +#+END_exercice #+begin_exercice [Mines 2023 # 621] Quelles sont les $A\in{\cal M}_n(\R)$ telles que, pour toute $P\in\mbox{GL}_n(\R)$, $PA$ soit diagonalisable? @@ -5094,7 +5309,7 @@ Soit $E$ un espace euclidien de dimension $4$. Trouver les endomorphismes $f\neq #+end_exercice #+begin_exercice [Mines 2023 # 647] -Soit $M\in{\cal O}_n(\R)$. Pour $k\in\N^*$, on pose $C_k=\frac{1}{k+1}\sum_{j=0}^kM^j$. Étudier la convergence de la suite $(C - {k\in\N}$. +Soit $M\in{\cal O}_n(\R)$. Pour $k\in\N^*$, on pose $C_k=\frac{1}{k+1}\sum_{j=0}^kM^j$. Étudier la convergence de la suite $(C_k)_{k\in\N}$. #+end_exercice #+begin_exercice [Mines 2023 # 648] @@ -5206,7 +5421,7 @@ On considère la forme quadratique $q\colon (x,y,z)\in\R^3\mapsto(x+z)^2+2xy+4yz ** Analyse #+begin_exercice [Mines 2023 # 667] -Soient $E$ un espace vectoriel norme et $A$ une partie de $E$. On considère l'ensemble des parties que l'on peut obtenir en appliquant successivement des passages a l'interieur ou a l'adherence a partir de $A$. +Soient $E$ un espace vectoriel norme et $A$ une partie de $E$. On considère l'ensemble des parties que l'on peut obtenir en appliquant successivement des passages a l'interieur ou a l'adhérence a partir de $A$. - Montrer qu'il y en a au plus $7$. - Donner une partie $A$ telle qu'il y en ait exactement $7$. #+end_exercice @@ -5257,7 +5472,7 @@ Si $f\in E$, on pose $u(f)=\sum_{k=1}^{+\i}\left(-\frac{1}{2}\right)^kf\left( \f #+begin_exercice [Mines 2023 # 674] Soient $L^1(\R)$ l'espace vectoriel des suites sommables et $N:x\mapsto\sum_{n=0}^{+\i}\lvert x_n\rvert$. - Montrer que $N$ est une norme. - - Soit $A$ l'ensemble des suites de $L^1(\R)$ nulle a partir d'un certain rang. Donner l'adherence et l'interieur de $A$. Ind. Remarquer que $A$ est dense dans $L^1(\R)$. + - Soit $A$ l'ensemble des suites de $L^1(\R)$ nulle a partir d'un certain rang. Donner l'adhérence et l'interieur de $A$. Ind. Remarquer que $A$ est dense dans $L^1(\R)$. #+end_exercice #+begin_exercice [Mines 2023 # 675] @@ -5319,11 +5534,11 @@ On pose : $E=\Big{\{}M\in\M_n(\R),\ P(M)=0\Big{\}}$. #+begin_exercice [Mines 2023 # 683] - Soit $P\in\R[X]$ unitaire de degré $n\in\N^*$. Montrer que $P$ est scinde sur $\R$ si et seulement si : $\forall z\in\C$, $|P(z)|\geq|\text{Im}(z)|^n$. - Montrer que l'ensemble des matrices de $\M_n(\R)$ trigonalisables est ferme. - - Quelle est l'adherence de l'ensemble des matrices diagonalisables de $\M_n(\R)$? + - Quelle est l'adhérence de l'ensemble des matrices diagonalisables de $\M_n(\R)$? #+end_exercice #+begin_exercice [Mines 2023 # 684] -Soient $n\geq 2$ et $r\in[1,n-1]$. L'ensemble $\mc{E}$ des matrices carrées de taille $n$ et de rang $r$ est-il ouvert? ferme? Déterminer l'interieur et l'adherence de $\mc{E}$. +Soient $n\geq 2$ et $r\in[1,n-1]$. L'ensemble $\mc{E}$ des matrices carrées de taille $n$ et de rang $r$ est-il ouvert? ferme? Déterminer l'interieur et l'adhérence de $\mc{E}$. #+end_exercice #+begin_exercice [Mines 2023 # 685] @@ -5384,17 +5599,17 @@ On pose $u_n=\sum_{k=1}^n\frac{1}{k}\left(1-\frac{1}{n}\right)^k$ pour tout $n\g #+end_exercice #+begin_exercice [Mines 2023 # 694] -Soit $ f:[0,2]\to\R$ une fonction $C^1$. On pose $ u_n=\frac{1}{n}\sum_{k=1}^nf\left(\frac{k}{n}+\frac{k}{n^2}\right)$ pour $n\geq 1$. +Soit $f:[0,2]\to\R$ une fonction $C^1$. On pose $u_n=\frac{1}{n}\sum_{k=1}^nf\left(\frac{k}{n}+\frac{k}{n^2}\right)$ pour $n\geq 1$. Étudier la convergence de la suite $(u_n)_{n\geq 1}$. #+end_exercice #+begin_exercice [Mines 2023 # 695] -Pour $n\in\N^*$, on pose $ u_n=\sum_{k=1}^n\sin\left(\frac{\sqrt{k}}{n}\right)$. Déterminer un équivalent de $u_n$. +Pour $n\in\N^*$, on pose $u_n=\sum_{k=1}^n\sin\left(\frac{\sqrt{k}}{n}\right)$. Déterminer un équivalent de $u_n$. #+end_exercice #+begin_exercice [Mines 2023 # 696] -Soit $\mc{B}$ le sous-espace de $\C^{\Z}$ forme des suites $(u - {n\in\Z}$ bornées. Soit $T$ l'endomorphisme de $\mc{B}$ qui a $(u - {n\in\Z}$ associe $(u_{n+1})_{n\in\Z}$. +Soit $\mc{B}$ le sous-espace de $\C^{\Z}$ formé des suites $(u_n)_{n\in\Z}$ bornées. Soit $T$ l'endomorphisme de $\mc{B}$ qui a $(u_n)$ associe $(u_{n+1})_{n\in\Z}$. - Montrer que $T$ est linéaire. Déterminer ses valeurs propres et ses sous-espaces propres. - Déterminer les sous-espaces de dimension finie de $\mc{B}$ stables par $T$. #+end_exercice @@ -5402,19 +5617,19 @@ Soit $\mc{B}$ le sous-espace de $\C^{\Z}$ forme des suites $(u - {n\in\Z}$ born #+begin_exercice [Mines 2023 # 697] Étudier les suites définies par $u_1,v_1$ réels et -$\forall n\in\N^*$, $ u_{n+1}=u_n+v_n\arctan\left(\frac{1}{n^2}\right)$ et $ v_{n+1}=v_n-u_n\arctan\left(\frac{1}{n^2}\right)$. +$\forall n\in\N^*$, $u_{n+1}=u_n+v_n\arctan\left(\frac{1}{n^2}\right)$ et $v_{n+1}=v_n-u_n\arctan\left(\frac{1}{n^2}\right)$. #+end_exercice #+begin_exercice [Mines 2023 # 698] -$\ \ - La suite $(d_n)_{n\geq 1}$ est-elle convergente? - La suite $(d_n)_{n\geq 1}$ est-elle bornée? +?? $(d_n)_{n\geq 1}$ est-elle convergente? - La suite $(d_n)_{n\geq 1}$ est-elle bornée? #+end_exercice #+begin_exercice [Mines 2023 # 699] -Soit $(b - {n\in\N}$ une suite strictement positive, croissante et non majorée. - - Montrer que, si $(a - {n\in\N}$ est une suite réelle convergente de limite $\ell$, alors +Soit $(b_n)_{n\in\N}$ une suite strictement positive, croissante et non majorée. + - Montrer que, si $(a_n)_{n\in\N}$ est une suite réelle convergente de limite $\ell$, alors $$\frac{1}{b_n}\sum_{k=0}^{n-1}(b_{k+1}-b_k)a_k\underset{n\to+\i}{ \longrightarrow}\ell.$$ - - Soit $(a - {n\in\N}$ une suite réelle. Montrer que, si la suite $\left(\frac{a_{n+1}-a_n}{b_{n+1}-b_n}\right)_{n\in\N}$ converge vers $\ell\in\R$, alors $\frac{a_n}{b_n}\to\ell$ quand $n\to+\i$. + - Soit $(a_n)_{n\in\N}$ une suite réelle. Montrer que, si la suite $\left(\frac{a_{n+1}-a_n}{b_{n+1}-b_n}\right)_{n\in\N}$ converge vers $\ell\in\R$, alors $\frac{a_n}{b_n}\to\ell$ quand $n\to+\i$. - La réciproque de la propriété précédente est-elle vraie? #+end_exercice @@ -5432,7 +5647,7 @@ Soit $a\in]0,1[$. On définit $(u_n)$ par $u_0=a$ et, pour $n\in\N$, $u_{n+1}=u_ #+end_exercice #+begin_exercice [Mines 2023 # 702] -Soit $(u - {n\in\N}$ définie par $u_0\in]0,\pi/2]$ et $\forall n\in\N,\ u_{n+1}=\sin(u_n)$. Étudier la convergence de $(u_n)$. Déterminer un équivalent de $u_n$. +Soit $(u_n)_{n\in\N}$ définie par $u_0\in]0,\pi/2]$ et $\forall n\in\N,\ u_{n+1}=\sin(u_n)$. Étudier la convergence de $(u_n)$. Déterminer un équivalent de $u_n$. #+end_exercice #+begin_exercice [Mines 2023 # 703] @@ -5456,7 +5671,7 @@ Soit $(u_n)$ une suite réelle définie par $u_0\geq 0$ et, pour tout $n\in\N$, #+begin_exercice [Mines 2023 # 705] Pour $n\geq 2$, on considère l'équation $\sin(x)=\frac{x}{n}$. - Montrer que cette équation admet une unique solution sur $]0,\pi[$ qu'on notera $x_n$. - - Montrer que la suite $(x - {n\geq 2}$ converge. Quelle est sa limite? - Donner un développement asymptotique de $x_n$ a la precision $o\left(\frac{1}{n^3}\right).$ + - Montrer que la suite $(x_n)_{n\geq 2}$ converge. Quelle est sa limite? - Donner un développement asymptotique de $x_n$ a la precision $o\left(\frac{1}{n^3}\right).$ #+end_exercice #+begin_exercice [Mines 2023 # 706] @@ -5474,7 +5689,7 @@ Soit $(u_n)$ la suite définie par $u_0\geq 0$ et, pour tout $n\in\N$, $u_{n+1}= #+begin_exercice [Mines 2023 # 708] Pour tout $P\in\R[X]$, on pose $N(P)=\sup_{t\in[0,1]}|P(t)|.$ Pour tout $n\in\N$, on note $E_n$ l'ensemble des polynômes unitaires de $\R_n[X]$ et $a_n=\inf_{P\in E_n}N(P)$. - Montrer que $a_n\gt 0$; calculer $a_0$ et $a_1$. - - Montrer que $(a - {n\in\N}$ est decroissante et de limite nulle. + - Montrer que $(a_n)_{n\in\N}$ est decroissante et de limite nulle. #+end_exercice #+begin_exercice [Mines 2023 # 709] @@ -5488,8 +5703,8 @@ Soit $(u_n)$ une suite réelle vérifiant : $\forall(m,n)\in\N^2,u_{n+m}\leq u_m #+begin_exercice [Mines 2023 # 711] - Montrer que tout sous-groupe de $(\R,+)$ est de la forme $a\Z$ ($a\in\R$) ou dense dans $\R$. Soit $\theta\in\R^*$ tel que $\frac{\pi}{\theta}\notin\Q$. - Montrer que $A=\big{\{}p\theta+2\pi q,\ (p,q)\in\Z^2\big{\}}$ est dense dans $\R$. - - Expliciter les valeurs d'adherence de la suite $\left(\cos(n\theta)\right)_{n\in\N}$. - - Expliciter les valeurs d'adherence de la suite $\left(\cos(\sqrt{n}\theta)\right)_{n\in\N}$. + - Expliciter les valeurs d'adhérence de la suite $\left(\cos(n\theta)\right)_{n\in\N}$. + - Expliciter les valeurs d'adhérence de la suite $\left(\cos(\sqrt{n}\theta)\right)_{n\in\N}$. #+end_exercice #+begin_exercice [Mines 2023 # 712] @@ -5529,7 +5744,7 @@ Soient $\alpha\gt 0$ et $\beta\in]0,1[$. Nature de la série $\sum\dfrac{(-1)^{\ #+end_exercice #+begin_exercice [Mines 2023 # 720] -- Montrer que $\dfrac{\pi}{4}=\sum_{k=0}^{+\i}\dfrac{(-1)^k}{2k+1}$. + - Montrer que $\dfrac{\pi}{4}=\sum_{k=0}^{+\i}\dfrac{(-1)^k}{2k+1}$. - Nature de la série de terme général $u_n=\ln\left(\tan\left(\sum_{k=0}^n\dfrac{(-1)^k}{2k+1}\right)\right)$? #+end_exercice @@ -5685,12 +5900,17 @@ $\forall x\in\R,\int_0^xe^{t^2}\,dt=F(x)\,e^{x^2}$. #+end_exercice #+begin_exercice [Mines 2023 # 748] -Étudier la fonction $f:x\mapsto\int_x^{x^2}\frac{dt}{t\sqrt{1-t}}$. +Étudier la fonction $f\colon x\mapsto\int_x^{x^2}\frac{dt}{t\sqrt{1-t}}$. #+end_exercice +# ID:7031 #+begin_exercice [Mines 2023 # 749] -Calculer $I=\int_{-1}^1\frac{\cos x}{e^{\frac{1}{x}}+1}\,dx$. +Calculer $I=\int_{-1}^1\frac{\cos x}{e^{\frac{1}{x}}+1}\dx$. #+end_exercice +#+BEGIN_proof +Donne $\sin 1$. On somme avec l'autre partie, pour obtenir $\int_0^1 \frac{\cos x}{e^{1/x} + 1} + \frac{\cos x}{e^{-1/x} + 1}$, qui fait exactement $\cos x$. +#+END_proof + #+begin_exercice [Mines 2023 # 750] Soient $a,b\in\R$ avec $a\lt b$, $f\in\mc C^0([a,b],R)$, $\epsilon\gt 0$. Montrer qu'il existe $P,Q\in\R[X]$ tels que $\forall x\in[a,b]$, $P(x)\leq f(x)\leq Q(x)$ et $\int_a^b(Q-P)\leq\epsilon$. Est-ce toujours vrai si $f$ est uniquement continue par morceaux? @@ -5893,7 +6113,7 @@ Soit $f:x\mapsto\sum_{n=1}^{+\i}\frac{(\sin(nx))^2}{n^2}$. - Justifier la convergence pour $x\in[0,1[$ de $f(x)=\prod_{n=0}^{+\i}\left(\frac{1+x^n}{1+x^{n+1}}\right)^{x^n}$. - Montrer que, pour tout $x\in\,]0,1[$, on a $\ln f(x)=\frac{x-1}{x}\sum_{n=1}^{+\i}x^n\ln(1+x^n)+\ln 2$. - En déduire $\forall x\in[0,1[\,,\ln f(x)=\ln 2+\sum_{m=1}^{+\i}\frac{(-1)^m}{m} \frac{x^m}{1+x+\cdots+x^m}$. - - Montrer que $f$ possede une limite finie en $1^-$ et l'expliciter. + - Montrer que $f$ possède une limite finie en $1^-$ et l'expliciter. #+end_exercice #+begin_exercice [Mines 2023 # 791] @@ -6188,7 +6408,7 @@ Soient $\alpha\in\Big{]}0,\dfrac{\pi}{2}\Big{[}$ et $f:t\mapsto\dfrac{1}{1-\sin\ #+end_exercice #+begin_exercice [Mines 2023 # 841] -Soit $(\lambda - {n\in\N}$ une suite croissante de réels strictement positifs. +Soit $(\lambda_n)_{n\in\N}$ une suite croissante de réels strictement positifs. On pose : $f(x)=\sum_{n=0}^{+\i}(-1)^n\exp(-\lambda_nx)$. - Déterminer le domaine de définition de $f$.On suppose dans la suite que $(\lambda_n)$ tend vers $+\i$. @@ -6197,7 +6417,7 @@ On pose : $f(x)=\sum_{n=0}^{+\i}(-1)^n\exp(-\lambda_nx)$. #+end_exercice #+begin_exercice [Mines 2023 # 842] -Soient $a$ et $b$ deux fonctions continues de $\R^+$ dans $\R^+$ et $S$ l'ensemble des solutions de $y'=ay+b$. Montr per l'equivalence entre : +Soient $a$ et $b$ deux fonctions continues de $\R^+$ dans $\R^+$ et $S$ l'ensemble des solutions de $y'=ay+b$. Montrer l'équivalence entre : i) tous les éléments de $S$ sont bornes, ii) $a$ et $b$ sont intégrables. #+end_exercice @@ -6209,7 +6429,7 @@ Déterminer les fonctions $y$ de $\R$ dans $\R$ dérivables et telles que $y'(x) #+begin_exercice [Mines 2023 # 844] Soit $f$ la fonction de $\R$ dans $\R$ telle que $f(0)=0$ et que $\forall x\in\R^*,f(x)=e^{-1/x^2}$. - Montr per que $f$ est de classe $C^{\i}$ sur $\R$. - - La fonction $f$ est-elle solution d'une équation différentielle linéaire homogéné? + - La fonction $f$ est-elle solution d'une équation différentielle linéaire homogène ? #+end_exercice #+begin_exercice [Mines 2023 # 845] @@ -6355,7 +6575,7 @@ Montrer que : $\forall x\in\R^n,f(x)=f(0)+\sum_{i=1}^nx_i\int_0^1 \frac{\partial On pose $E=C^{\i}(\R^n,\R)$ et $D=\Big{\{}\phi\in\mc{L}(E,\R)\ ;\ \forall(f,g)\in E^2,\phi(fg)=f(0) \phi(g)+g(0)\phi(f)\Big{\}}$. - - Montrer que la famille $(\phi - {1\leq i\leq n}$ est libre, avec : $\phi_i:f\mapsto\frac{\partial f}{\partial x_i}(0)$. + - Montrer que la famille $(\phi_i)_{1\leq i\leq n}$ est libre, avec : $\phi_i:f\mapsto\frac{\partial f}{\partial x_i}(0)$. - Montrer que $D$ est de dimension finie. #+end_exercice @@ -6410,8 +6630,8 @@ Soient $a\in\R^n$ et $g:x\in\R^n\mapsto\|f(x)-a\|^2$. #+begin_exercice [Mines 2023 # 877] Soient $U$ un ouvert convexe de $\R^n$ et $f:U\to\R$ une fonction de classe $C^1$. - - Montrre que $f$ est convexe si et seulement si $f(y)-f(x)\geq df_x(y-x)$ pour tous $x,y\in U$. Que donne cette caracterisation dans le cas ou $n=1$? - - Soient $\alpha$ et $\beta$ des réels fixes. On note $E$ l'ensemble des fonctions $f:[0,1]\to\R$ de classe $C^1$ telles que $f(0)=\alpha$ et $f(1)=\beta$. Soit $\Phi:f\in E\mapsto\int_0^1\sqrt{1+f'(x)^2}\,dx$. Montrre que $\Phi$ atteint sa borne inférieure en un unique élément de $E$, que l'on precisera. + - Montrer que $f$ est convexe si et seulement si $f(y)-f(x)\geq df_x(y-x)$ pour tous $x,y\in U$. Que donne cette caracterisation dans le cas ou $n=1$? + - Soient $\alpha$ et $\beta$ des réels fixes. On note $E$ l'ensemble des fonctions $f:[0,1]\to\R$ de classe $C^1$ telles que $f(0)=\alpha$ et $f(1)=\beta$. Soit $\Phi:f\in E\mapsto\int_0^1\sqrt{1+f'(x)^2}\,dx$. Montrer que $\Phi$ atteint sa borne inférieure en un unique élément de $E$, que l'on precisera. #+end_exercice #+begin_exercice [Mines 2023 # 878] @@ -6419,9 +6639,9 @@ Soit $E=\M_n(\R)$ muni de la norme euclidienne canonique. On pose $f:M\in E\mapsto\|M\|^2=\mathrm{tr}(M^TM)$ et $g:M\in E\mapsto\det M-1$. On note $h$ la restriction de $f$ a $\mathrm{SL}_n(\R)$. - Justifier que $f$ et $g$ sont de classe $\mc C^1$ et calculer leur gradient en une matrice $M\in\mathrm{SL}_n(\R)$. - - Montrre que $f$ admet un minimum sur $\mathrm{SL}_n(\R)$. Soit $M_0$ une matrice ou il est ateint. - - Soit $H\in\M_n(\R)$ orthogonale au gradient de $g$ en $M_0$. Montrre qu'il existe un chemin $\gamma$ de classe $\mc C^1$ défini sur un voisinage de $0$ dans $\R$, a valeurs dans $\mathrm{SL}_n(\R)$ tel que $\gamma(0)=M_0$ et $\gamma'(0)=H$. - - Montrre que $(\nabla f_{M_0})^{\perp}=(\nabla g_{M_0})^{\perp}$. + - Montrer que $f$ admet un minimum sur $\mathrm{SL}_n(\R)$. Soit $M_0$ une matrice ou il est ateint. + - Soit $H\in\M_n(\R)$ orthogonale au gradient de $g$ en $M_0$. Montrer qu'il existe un chemin $\gamma$ de classe $\mc C^1$ défini sur un voisinage de $0$ dans $\R$, a valeurs dans $\mathrm{SL}_n(\R)$ tel que $\gamma(0)=M_0$ et $\gamma'(0)=H$. + - Montrer que $(\nabla f_{M_0})^{\perp}=(\nabla g_{M_0})^{\perp}$. - Calculer le minimum de $h$ sur $\mathrm{SL}_n(\R)$. #+end_exercice @@ -6437,7 +6657,7 @@ On tire au hasard un élément $A$ de $P(\db{1,n})$. Calculer la probabilité qu #+end_exercice #+begin_exercice [Mines 2023 # 881] -Soient $m,n\in\N^*$ tel que $m\leq\frac{n}{2}$. On se donne deux urnes contenant chacune des boules numérotées de $1$ a $n$. On tire $m$ boules dans chaque urne et l'on note $X$ le nombre de doublons. Calculer la loi de $X$ puis sa variance. +Soient $m,n\in\N^*$ tel que $m\leq\frac{n}{2}$. On se donne deux urnes contenant chacune des boules numérotées de $1$ à $n$. On tire $m$ boules dans chaque urne et l'on note $X$ le nombre de doublons. Calculer la loi de $X$ puis sa variance. #+end_exercice #+begin_exercice [Mines 2023 # 882] @@ -6514,7 +6734,7 @@ Soient $X_1,X_2$ deux variables aléatoires indépendantes qui suivent la loi g - Calculer $\mathbf{P}(Y=0)$. - Déterminer la loi de $Y$. - Montrer que $Y$ est d'espérance finie et calculer $\mathbf{E}(Y)$. - - Montrer que $Y$ possede un moment d'ordre 2 et calculer $\mathbf{V}(Y)$. + - Montrer que $Y$ possède un moment d'ordre 2 et calculer $\mathbf{V}(Y)$. #+end_exercice #+begin_exercice [Mines 2023 # 895] @@ -6523,7 +6743,7 @@ Soient $X_1,X_2$ deux variables aléatoires indépendantes qui suivent la loi g #+end_exercice #+begin_exercice [Mines 2023 # 896] -Soient $n\in\N^*$, $\sigma$ une variable aléatoire suivant la loi uniforme sur $\mc{S}_n$. Pour $m\in\db{1,n]\!]$, on note $X_m=\min\left\{k\in[\![1,n]\!],\ \sigma(k)\geq m\right\}$ et $Y_m=\max\left\{k\in[\![1,n},\ \sigma(k)\geq m\right\}$. Calculer la loi de $X_m$ et $Y_m$, et leur espérance. +Soient $n\in\N^*$, $\sigma$ une variable aléatoire suivant la loi uniforme sur $\mc{S}_n$. Pour $m\in\db{1,n}$, on note $X_m=\min\left\{k\in \db{1,n},\ \sigma(k)\geq m\right\}$ et $Y_m=\max\left\{k\in\db{1,n},\ \sigma(k)\geq m\right\}$. Calculer la loi de $X_m$ et $Y_m$, et leur espérance. #+end_exercice #+begin_exercice [Mines 2023 # 897] @@ -6531,7 +6751,7 @@ Soient $\lambda\gt 0$ et $X$ une variable aléatoire qui suit la loi de Poisson #+end_exercice #+begin_exercice [Mines 2023 # 898] -Soit $p\in\,]0,1[$. Soit $(X - {k\in\N^*}$ une suite de variables aléatoires i.i.d. vérifiant : +Soit $p\in\,]0,1[$. Soit $(X_k)_{k\in\N^*}$ une suite de variables aléatoires i.i.d. vérifiant : $\mathbf{P}(X_k=1)=p$ et $\mathbf{P}(X_k=-1)=1-p$. Pour tout $n\in\N^*$, on pose $S_n=\sum_{k=1}^nX_k$. Montrer que $p=\dfrac{1}{2}$ si et seulement si : $\forall n\in\N^*,\max_{k\in\Z}\mathbf{P}(S_{2n}=k)=\mathbf{P}( S_{2n}=0)$. #+end_exercice @@ -6578,22 +6798,22 @@ Soit $p\geq 3$ premier. Soit $K=\big{\{}x^2,\ x\in\Z/p\Z\big{\}}$. #+end_exercice #+begin_exercice [Mines 2023 # 904] -Caracteriser les couples $(X,a)$ avec $X$ variable aléatoire discrète complexe et $a\in\C$ tels que $X\sim aX$. +Caractériser les couples $(X,a)$ avec $X$ variable aléatoire discrète complexe et $a\in\C$ tels que $X\sim aX$. #+end_exercice #+begin_exercice [Mines 2023 # 905] Soit $\alpha\gt 1$. On munit $\N^*$ de la loi de probabilité $\mathbf{P}_{\alpha}$ définie par $\mathbf{P}_{\alpha}(\{n\})=\dfrac{1}{\zeta(\alpha)n^{\alpha}}$ pour $n\geq 1$. - Calculer $\mathbf{P}_{\alpha}(m\N^*)$ pour $m\geq 1$. - - On note $(p - {k\geq 1}$ la suite strictement croissante des nombres premiers. Montrere que les $p_k\N^*$ sont mutuellement indépendants. + - On note $(p_k)_{k\geq 1}$ la suite strictement croissante des nombres premiers. Montrer que les $p_k\N^*$ sont mutuellement indépendants. - En déduire la formule d'Euler $\zeta(\alpha)=\prod_{k=1}^{+\i}\left(1-\dfrac{1}{p_k^{\alpha}} \right)^{-1}$. #+end_exercice #+begin_exercice [Mines 2023 # 906] -Soient $X$ et $Y$ deux variables aléatoires discrètes strictement positives, de meme loi et d'espérance finie. Montrere que $\mathbf{E}(X/Y)\geq 1$. Ind. Commencer par le cas où $X$ et $Y$ sont indépendantes. +Soient $X$ et $Y$ deux variables aléatoires discrètes strictement positives, de meme loi et d'espérance finie. Montrer que $\mathbf{E}(X/Y)\geq 1$. Ind. Commencer par le cas où $X$ et $Y$ sont indépendantes. #+end_exercice #+begin_exercice [Mines 2023 # 907] -Soit $(X - {n\in\N^*}$ une suite de variables aléatoires i.i.d. suivant la loi géométrique de paramètre $p\in]0,1[$. +Soit $(X_n)_{n\in\N^*}$ une suite de variables aléatoires i.i.d. suivant la loi géométrique de paramètre $p\in]0,1[$. On pose $Y_n=\min(X_1,\ldots,X_n),\alpha_n=\mathbf{E}(Y_n)$ et $Z_n=\max(X_1,\ldots,X_n),\beta_n=\mathbf{E}(Z_n)$. - Étudier la monotonie des suites $(\alpha_n)$ et $(\beta_n)$. - Exprimer $\alpha_n$ en fonction de $n$. @@ -6611,9 +6831,9 @@ Soient $p\in]0,1[$, $X$ une variable aléatoire suivant la loi géométrique de #+end_exercice #+begin_exercice [Mines 2023 # 910] -Soient $n\in\N^*$, $X_1,\ldots,X_n$ i.i.d. suivant la loi uniforme sur $\db{1,d]\!]$. Pour $j\in\{1,\ldots,n\}$, on pose $Y_j=|\{i\in[\![1,n},\;X_i=j\}|$. +Soient $n\in\N^*$, $X_1,\ldots,X_n$ i.i.d. suivant la loi uniforme sur $\db{1,d}$. Pour $j\in\{1,\ldots,n\}$, on pose $Y_j=|\{i\in\db{1,n},\;X_i=j\}|$. - Déterminer la loi de $Y_j$. - - Soient $i,j\in\db{1,n]\!]$ avec $i\neq j$ et $k,\ell\in[\![1,n}$. Calculer $\mathbf{P}(Y_i=k,Y_j=\ell)$. + - Soient $i,j\in\db{1,n}$ avec $i\neq j$ et $k,\ell\in\db{1,n}$. Calculer $\mathbf{P}(Y_i=k,Y_j=\ell)$. #+end_exercice #+begin_exercice [Mines 2023 # 911] @@ -6622,7 +6842,7 @@ Soit $X$ une variable aléatoire discrète a valeurs dans $\R^{+*}$ telle que $\ Pour tout $t\in\R^+$, on pose : $F_X(t)=\mathbf{E}(e^{-tX})$. - Montrer que $F_X$ est bien définie (a valeurs réelles) et continue. - Montrer la convergence et calculer $\int_0^{+\i}F_X(t)\,dt$. - - Soient $X$ et $Y$ deux variables aléatoires indépendantes suivant la loi géométrique de paramètre $p\in\!]0,1[$. Calculer $\mathbf{E}\left(\dfrac{1}{X+Y}\right)$. + - Soient $X$ et $Y$ deux variables aléatoires indépendantes suivant la loi géométrique de paramètre $p\in\\interval]{0, 1}[$. Calculer $\mathbf{E}\left(\dfrac{1}{X+Y}\right)$. - Généraliser a $m$ variables i.i.d. suivant la loi géométrique de paramètre $p$. #+end_exercice @@ -6636,7 +6856,7 @@ Pour $n\in\Z$, soit $A_n=(\exists k\geq 0,\;S_k=-n)$ et $p_n=\mathbf{P}(A_n)$. #+end_exercice #+begin_exercice [Mines 2023 # 913] -Soient $X$ une variable aléatoire a valeurs dans $\{-1,1\}$, $(X - {k\geq 1}$ une suite i.i.d. de variables aléatoires suivant la loi de $X$. Pour $n\in\N^*$, soit $S_n=\sum_{k=1}^nX_k$. Donner une condition nécessaire et suffisante pour que, pour toute partie finie $A$ de $\Z$, $\sum_{n=1}^{+\i}\mathbf{P}(S_n\in A)\lt +\i$. +Soient $X$ une variable aléatoire a valeurs dans $\{-1,1\}$, $(X_n)_{k\geq 1}$ une suite i.i.d. de variables aléatoires suivant la loi de $X$. Pour $n\in\N^*$, soit $S_n=\sum_{k=1}^nX_k$. Donner une condition nécessaire et suffisante pour que, pour toute partie finie $A$ de $\Z$, $\sum_{n=1}^{+\i}\mathbf{P}(S_n\in A)\lt +\i$. #+end_exercice #+begin_exercice [Mines 2023 # 914] @@ -6654,7 +6874,7 @@ Soit $X$ une variable aléatoire a valeurs dans $\R^+$. #+end_exercice #+begin_exercice [Mines 2023 # 916] -Soit $(X - {i\geq 1}$ une suite de variables aléatoires i.i.d. On suppose que chaque variable aléatoire $X_i+1$ suit la loi géométrique de paramètre $p\in\,]0,1[$. Pour tout $n\in\N^*$, on pose $S_n=\sum_{i=1}^nX_i$. +Soit $(X_i)_{i\geq 1}$ une suite de variables aléatoires i.i.d. On suppose que chaque variable aléatoire $X_i+1$ suit la loi géométrique de paramètre $p\in\,]0,1[$. Pour tout $n\in\N^*$, on pose $S_n=\sum_{i=1}^nX_i$. - Déterminer la loi de $S_n$. - Déterminer $M_n=\max\left\{{\bf P}(S_n=k),\ k\in\N\right\}$ puis un équivalent simple de $M_n$ quand $n$ tend vers $+\i$. #+end_exercice @@ -6693,7 +6913,7 @@ Montrer que : $\mathbf{P}(X\geq\lambda\mathbf{E}(X))\geq\dfrac{(1-\lambda)^2 \ma #+end_exercice #+begin_exercice [Mines 2023 # 922] -Soit $(X - {i\in\N^*}$ une suite de variables aléatoires i.i.d. possedant un moment d'ordre $4$. On pose : $m=\mathbf{E}(X_i)$ et $V_4=\mathbf{E}((X_i-m)^4)$. +Soit $(X_i)_{i\in\N^*}$ une suite de variables aléatoires i.i.d. possedant un moment d'ordre $4$. On pose : $m=\mathbf{E}(X_i)$ et $V_4=\mathbf{E}((X_i-m)^4)$. - Justifier la bonne définition (dans $\R$) de $m$ et $V_4$. Pour $\epsilon\gt 0$, on pose : $A_n^{\epsilon}=\left(\left|\dfrac{1}{n}\sum_{i=1}^n(X_i-m) \right|\geq\epsilon\right)$. @@ -6723,7 +6943,7 @@ Soit $(P)$ le plan de $\R^3$ d'équation $x-2z-y=0$ et $u$ le vecteur $(1,2,1)^T #+begin_exercice [Mines PSI 2023 # 926] Soit $E=\R_n[X]$. On considère les polynômes $E_k=\binom{n}{k}X^k(1-X)^{n-k}$, $0\leq k\leq n$. - - Montrer que $(E - {0\leq k\leq n}$ est une base de $E$. + - Montrer que $(E_k)_{0\leq k\leq n}$ est une base de $E$. - Calculer $\sum_{k=0}^nkE_k$ et $\sum_{k=0}^nk^2E_k$. - Comment aurait-on pu prévoir les résultats obtenus? #+end_exercice @@ -6761,7 +6981,7 @@ Diagonaliser $A=$$\begin{pmatrix}0&\cdots&0&1\\ \vdots&&\vdots&\vdots\\ 0&\cdots #+end_exercice #+begin_exercice [Mines PSI 2023 # 1201] - On considère un de equilibre a $n$ faces. Les lancers se modelisent par une suite $(X - {i\geq 1}$ i.i.d de variables aléatoires suivant la loi uniforme sur $\db{1,n}$. + On considère un dé équilibré à $n$ faces. Les lancers se modélisent par une suite $(X_n)_{i\geq 1}$ i.i.d de variables aléatoires suivant la loi uniforme sur $\db{1,n}$. Pour $k\in\db{1,n}$, on note $T_k=\min\{n\in\N^*,\ |\{X_1,\ldots,X_n\}|=k\}$. - Déterminer la loi de $T_k$. @@ -6802,21 +7022,21 @@ Les variables aléatoires $A,B$ suivent la loi uniforme sur l'ensemble $\mc{P}(\ #+end_exercice #+begin_exercice [Mines PSI 2023 # 1207] -- Soit $(A - {n\in\N^*}$ une suite d'évènements. Montrer que $B:\triangleleft$ II existe un rang a partir duquel $A_n$ est vraie $\mathchar 13334\relax$ est un évènement et que $B=\bigcup\limits_{n\in\N^*}\left(\bigcap\limits_{k\geq n}A_{k }\right)$. - - Soient $(X - {n\in\N^*}$ une suite i.i.d. de variables aléatoires réelles de meme loi que $X$.On suppose $\mathbf{E}(X)=0$ et $\mathbf{E}(X^4)\lt +\i$. On pose $S_n=\sum_{k=1}^nX_k$. +- Soit $(A_n)_{n\in\N^*}$ une suite d'évènements. Montrer que $B:\triangleleft$ II existe un rang a partir duquel $A_n$ est vraie $\mathchar 13334\relax$ est un évènement et que $B=\bigcup\limits_{n\in\N^*}\left(\bigcap\limits_{k\geq n}A_{k }\right)$. + - Soient $(X_n)_{n\in\N^*}$ une suite i.i.d. de variables aléatoires réelles de meme loi que $X$.On suppose $\mathbf{E}(X)=0$ et $\mathbf{E}(X^4)\lt +\i$. On pose $S_n=\sum_{k=1}^nX_k$. - Calculer $\mathbf{E}({S_n}^4)$ en fonction de $n$, $\mathbf{E}(X^2)$ et $\mathbf{E}(X^4)$. - - En déduire que pour tout $\eps\gt 0$, $\lim_{n\to+\i}\mathbf{P}\left(\bigcap_{k\geq n}\left(\left|\frac{S_{ k}}{k}\right|\leq\eps\right)\right)=1$ et que, presque surement, $\lim_{n\to+\i}\frac{S_n}{n}=0$. + - En déduire que pour tout $\eps\gt 0$, $\lim_{n\to+\i}\mathbf{P}\left(\bigcap_{k\geq n}\left(\left|\frac{S_{ k}}{k}\right|\leq\eps\right)\right)=1$ et que, presque sûrement, $\lim_{n\to+\i}\frac{S_n}{n}=0$. #+end_exercice #+begin_exercice [Mines PSI 2023 # 1208] Soit $\alpha\gt 0$. - - Montrer l'existence d'une variable aléatoire $X$ valeurs dans $\N$ de fonction génératrice $ G_X\left(t\right)=\frac{1}{\left(2-t\right)^{\alpha}}$. + - Montrer l'existence d'une variable aléatoire $X$ valeurs dans $\N$ de fonction génératrice $G_X\left(t\right)=\frac{1}{\left(2-t\right)^{\alpha}}$. - Donner un équivalent de $\mathbf{P}\left(X=n\right)$ quand $n\to+\i$. - Pour $\lambda\gt 0$, montrer que $\mathbf{P}\left(X\geq\lambda+\alpha\right)\leq\frac{2\alpha}{\lambda ^2}$. #+end_exercice -* Centrale +* Centrale :cent: ** Algèbre #+begin_exercice [Centrale 2023 # 1209] @@ -6827,7 +7047,7 @@ On considère, pour $n\in\N$, $C_n=\frac{1}{n+1}\binom{2n}{n}$. #+end_exercice #+begin_exercice [Centrale 2023 # 1210] -Pour $n\in\N^*$, on note $\mc{P}(n)$ l'ensemble des nombres premiers inférieurs ou egaux a $n$ et $P_n=\prod_{p\in\mc{P}(n)}p$. +Pour $n\in\N^*$, on note $\mc{P}(n)$ l'ensemble des nombres premiers inférieurs ou égaux a $n$ et $P_n=\prod_{p\in\mc{P}(n)}p$. - Montrer que $\forall n\geq 2$, $\frac{4^n}{2\sqrt{n}}\lt \binom{2n}{n}\lt 4^n$. - Montrer que $\forall n\geq 1$, $\binom{2n+1}{n}\lt 4^n$. - Montrer que $\forall n\in\N$, $P_{2n+1}\lt 4^nP_{n+1}$. @@ -6836,14 +7056,22 @@ Pour $n\in\N^*$, on note $\mc{P}(n)$ l'ensemble des nombres premiers inférieurs #+begin_exercice [Centrale 2023 # 1211] Soit $(G,\cdot)$ un groupe fini commutatif tel que le nombre d'automorphismes de $G$ est $3$. -_a) i)_: Donner la définition d'un automorphisme. Montrer que $\phi:x\mapsto x^{-1}$ est un automorphisme de $G$. + - Donner la définition d'un automorphisme. Montrer que $\phi\colon x\mapsto x^{-1}$ est un automorphisme de $G$. - Montrer que, pour tout $x\in G$, $x^2=e$. - - Montrer que $G$ possede un sous-groupe $V$ d'ordre $4$ et preciser les automorphismes de $V$. + - Montrer que $G$ possède un sous-groupe $V$ d'ordre $4$ et préciser les automorphismes de $V$. + - Montrer que $G = V$. #+end_exercice +#+BEGIN_proof + - + - Les automorphismes forment un groupe. S'il est de cardinal trois, il ne peut pas avoir d'élément d'ordre 2, donc $\phi$ est l'identité. + - Si on prend deux éléments quelconques, ils engendre un groupe de cardinal $4$. Le groupe des automorphismes est bien de cardinal 3. + - Revient à montrer que $G = (\Z/2Z)^n$. +#+END_proof + #+begin_exercice [Centrale 2023 # 1212] Soient $p$ un nombre premier tel que $p\equiv 3$[4] et $C=\{x\in\Z/p\Z,\ \exists y\in\Z/p\Z,\ x=y^2\}$. - - Rappeler l'enonce du petit théorème de Fermat. Montrer que $-1\notin C$. + - Rappeler l'énonce du petit théorème de Fermat. Montrer que $-1\notin C$. On pose $\pi_x=\prod_{y\in C\setminus\{x\}}(x+y)$ pour $x\in C\setminus\{0\}$ et $\pi=\prod_{x\neq y\in C}(x+y)$. - Déterminer le cardinal de $C$. @@ -6860,7 +7088,7 @@ Pour $n\in\N$, on note $M_n=2^n-1$ et $s_n=u^{2^n}+v^{2^n}$. - Soit $q$ un nombre premier. On munit l'ensemble $B=(\Z/q\Z)^2$ des deux lois de composition interne définies par : $(x,y)+(x',y')=(x+x',y+y')$ et $(x,y)\cdot(x',y')=(xx'+3yy',xy'+x^{ '}y)$. - - Montrer que les deux lois précédentes muinssent $B$ d'une structure d'anneau commutatif fini. + - Montrer que les deux lois précédentes munissent $B$ d'une structure d'anneau commutatif fini. - Montrer que, si $3$ n'est pas un carré modulo $q$, alors l'anneau precedent est un corps. - On note $A=\Z+\sqrt{3}\Z$. Montrer que l'application $\pi$ définie par $\pi(a+b\sqrt{3})=(\overline{a},\overline{b})$ est bien définie et est un morphisme surjectif d'anneaux de $A$ dans $B$. - On suppose $n$ premier. Montrer que, si $M_n$ divise $s_{n-2}$ alors $M_n$ est premier. @@ -6869,38 +7097,39 @@ Ind. On pourra raisonner par l'absurde en considérant le plus petit facteur pre #+end_exercice #+begin_exercice [Centrale 2023 # 1214] -Soit $A$ un anneau commutatif. On dit que $A$ est noetherien lorsque tous ses ideaux sont engendres par une partie finie de $A$. - - Les anneaux $\Z$ et $\R[X]$ sont-ils noetheriens? - - Montrer que $A$ est noetherien si et seulement si toute suite croissante d'ideaux est stationnaire. - - Soit $A$ un anneau non commutatif. On dit que $\mc{I}$ est un ideal a gauche de $A$ lorsque $\mc{I}A\subset\mc{I}$ (définition similaire pour un ideal a droite). Soit $A$ noetherien, c'est-a-dire que tous les ideaux, a droite ou a gauche, de $A$ sont de type fini. Montrer que l'inversibilité a gauche equivaut a l'inversibilité a droite, i.e. $\forall a\in A,\Big{(}\exists b\in A,\ ab=1\Longleftrightarrow\exists b\in A,\ ba=1 \Big{)}$. +Soit $A$ un anneau commutatif. On dit que $A$ est noethérien lorsque tous ses idéaux sont engendres par une partie finie de $A$. + - Les anneaux $\Z$ et $\R[X]$ sont-ils noethériens? + - Montrer que $A$ est noethérien si et seulement si toute suite croissante d'idéaux est stationnaire. + - Soit $A$ un anneau non commutatif. On dit que $\mc{I}$ est un ideal a gauche de $A$ lorsque $\mc{I}A\subset\mc{I}$ (définition similaire pour un ideal a droite). Soit $A$ noethérien, c'est-a-dire que tous les idéaux, a droite ou a gauche, de $A$ sont de type fini. Montrer que l'inversibilité a gauche équivaut a l'inversibilité a droite, i.e. $\forall a\in A,\Big{(}\exists b\in A,\ ab=1\Longleftrightarrow\exists b\in A,\ ba=1 \Big{)}$. -Ind. Considèrer $\phi:x\mapsto ax$. +Ind. Considérer $\phi:x\mapsto ax$. #+end_exercice #+begin_exercice [Centrale 2023 # 1215] -- Soit $G$ un groupe commutatif fini. Si $a$ et $b$ sont deux éléments de $G$ d'ordre premiers entre eux, quel est l'ordre de $ab$? + - Soit $G$ un groupe commutatif fini. Si $a$ et $b$ sont deux éléments de $G$ d'ordre premiers entre eux, quel est l'ordre de $ab$? - Soit $G$ un groupe commutatif fini. Montrer qu'il existe un élément de $G$ dont l'ordre est le ppcm des ordres des éléments de $G$. - Soit $p$ un nombre premier. Montrer que le groupe $\mathbb{F}_p^*$ est cyclique. #+end_exercice #+begin_exercice [Centrale 2023 # 1216] Soit $(T_n)_{n\in\N}$ la suite de polynômes réels définie par $T_0(X)=1,\ T_1(X)=X$ et pour $n\in\N,\,T_{n+2}(X)=2XT_{n+1}(X)-T_n(X)$. - - Montrer que, pour $n\in\N$, $\forall\theta\in\R$, $T_n(\cos\theta)=\cos(n\theta)$. - Montrer que $T_n\circ T_m=T_m\circ T_n$ pour $(m,n)\in\N^2$. - - Montrre que, pour $n\geq m$, $2T_nT_m=T_{n+m}+T_{n-m}$. + 1. Montrer que, pour $n\in\N$, $\forall\theta\in\R$, $T_n(\cos\theta)=\cos(n\theta)$. + 2. Montrer que $T_n\circ T_m=T_m\circ T_n$ pour $(m,n)\in\N^2$. + 3. Montrer que, pour $n\geq m$, $2T_nT_m=T_{n+m}+T_{n-m}$. -On considère l'équation différentielle $(E):(1-x^2)P^{' 2}=n^2(1-P^2)$. - - Montrre que, pour $n\in\N$, $T_n$ et $-T_n$ sont solutions de $(E)$ sur $\R$. - - Montrre que tout polynôme solution de $(E)$ est de degré $n$, puis déterminer les polynômes solution de $(E)$ sur $\R$. + On considère l'équation différentielle $(E)\colon (1-x^2)P^{' 2}=n^2(1-P^2)$. + 3. [@4] Montrer que, pour $n\in\N$, $T_n$ et $-T_n$ sont solutions de $(E)$ sur $\R$. + 4. Montrer que tout polynôme solution de $(E)$ est de degré $n$, puis déterminer les polynômes solution de $(E)$ sur $\R$. #+end_exercice #+begin_exercice [Centrale 2023 # 1217] Soient $a_1\lt a_2\lt \cdots\lt a_p$ et $b_1\lt b_2\cdots\lt b_p$ des réels et $M=(e^{a_ib_j})_{1\leq i,j\leq p}$. - Calculer $\det M$ lorsque $b_k=k-1$ pour tout $k$. - - Montrre que $M$ est inversible, puis que $\det M\gt 0$. + - Montrer que $M$ est inversible, puis que $\det M\gt 0$. #+end_exercice #+begin_exercice [Centrale 2023 # 1218] -- Rappeler la définition de l'indicatrice d'Euler, exprimer $\phi(n)$ en fonction de sa decomposition en facteurs premiers. + - Rappeler la définition de l'indicatrice d'Euler, exprimer $\phi(n)$ en fonction de sa decomposition en facteurs premiers. - Pour $n\geq 2$, calculer $\sum_{d|n}\phi(d)$ (la somme étant restreinte aux diviseurs positifs). - En déduire le déterminant de $A$, ou $A_{i,j}=i\wedge j$. #+end_exercice @@ -6909,22 +7138,22 @@ Soient $a_1\lt a_2\lt \cdots\lt a_p$ et $b_1\lt b_2\cdots\lt b_p$ des réels et Soit $f\colon\R\to\R$ une fonction continue telle que, pour toute matrice $A\in\mathrm{GL}_n(\R)$, l'on ait $\left(f(a_{ij})\right)_{1\leq i,j\leq n}\in\mathrm{GL}_n(\mathbb{ R})$. - À l'aide des matrices $U_{x,y}=\left(\begin{array}{cc}x&1\\ y&1\end{array}\right)$, montrer que $f$ est injective. - En utilisant l'ensemble $T=\{(x,y)\in\R^2,\ x\lt y\}$, en déduire que $f$ est strictement monotone. - - On suppose que $f(\R)=\R^{+*}$. Montrre qu'il existe $a\in\R$ tel que, pour tous $x,y\in\R$, il existe $z_{x,y}\in\R$ tel que $f(x)f(y)=f(a)f(z_{x,y})$, et conclure a une absurdite. + - On suppose que $f(\R)=\R^{+*}$. Montrer qu'il existe $a\in\R$ tel que, pour tous $x,y\in\R$, il existe $z_{x,y}\in\R$ tel que $f(x)f(y)=f(a)f(z_{x,y})$, et conclure a une absurdite. - Traiter de meme le cas $f(\R)=\R^{-*}$. #+end_exercice #+begin_exercice [Centrale 2023 # 1220] -- Rappeler la formule de développement d'un déterminant par rapport a une ligne ou une colonne. En déduire, pour $A\in\M_n(\R)$, une relation entre $\mathrm{Com}\,A$, $A$ et $\det A$. + - Rappeler la formule de développement d'un déterminant par rapport a une ligne ou une colonne. En déduire, pour $A\in\M_n(\R)$, une relation entre $\mathrm{Com}\,A$, $A$ et $\det A$. - Soit $A=(a_{i,j})_{1\leq i,j\leq n}$ la matrice de $\M_n(\R)$ définie par : $a_{i,i}=2$, $a_{i,j}=-1$ si $|i-j|=1$ et $a_{i,j}=0$ dans tout autre cas. Calculer le déterminant de $A$. - - Soit $A\in\M_n(\R)$ une matrice dont les coefficients diagonaux sont strictement positifs, dont les autres coefficients sont negatifs et telle que $\sum_{j=1}^na_{i,j}\gt 0$ pour tout $i$. Montrre que $A$ est inversible. + - Soit $A\in\M_n(\R)$ une matrice dont les coefficients diagonaux sont strictement positifs, dont les autres coefficients sont negatifs et telle que $\sum_{j=1}^na_{i,j}\gt 0$ pour tout $i$. Montrer que $A$ est inversible. - Montrer que les coefficients de $A^{-1}$ sont positifs. #+end_exercice #+begin_exercice [Centrale 2023 # 1221] Soient $M\in\mathrm{GL}_n(\R)$, $F:X\mapsto MXM^{-1}$ et $f:(A,B)\mapsto\mathrm{tr}(A)\,\mathrm{tr}(B)-\mathrm{tr}(AB)$. - - Montrre que, pour tous $A$, $B\in\M_n(\R)$, $\mathrm{tr}(AB)=\mathrm{tr}(BA)$. + - Montrer que, pour tous $A$, $B\in\M_n(\R)$, $\mathrm{tr}(AB)=\mathrm{tr}(BA)$. - Trouver les endomorphismes $h$ de $\M_n(\R)$ qui vérifient, pour tous $A$, $B\in\M_n(\R)$, $f(F(A),B)=f(A,h(B))$. - - Dans cette question, on suppose que $n=2$.Soit $h:\left(\begin{array}{cc}a&b\\ c&d\end{array}\in{\cal M}_2(\R)\right)\mapsto\left(\begin{array}{cc}a+c &b+d-a-c\\ c&d-c\end{array}\right).$ Déterminer les endomorphismes $k$ de ${\cal M}_n(\R)$ tels que $f(h(A),B)=f(A,k(B))$ pour tout $(A,B)\in{\cal M}_n(\R)^2.$ Parmi eux, preciser ceux qui sont trigonalisables, diagonalisables. + - Dans cette question, on suppose que $n=2$. Soit $h:\left(\begin{array}{cc}a&b\\ c&d\end{array}\in{\cal M}_2(\R)\right)\mapsto\left(\begin{array}{cc}a+c &b+d-a-c\\ c&d-c\end{array}\right)$. Déterminer les endomorphismes $k$ de ${\cal M}_n(\R)$ tels que $f(h(A),B)=f(A,k(B))$ pour tout $(A,B)\in{\cal M}_n(\R)^2$. Parmi eux, preciser ceux qui sont trigonalisables, diagonalisables. #+end_exercice #+begin_exercice [Centrale 2023 # 1222] @@ -6934,7 +7163,7 @@ Soient $M\in\mathrm{GL}_n(\R)$, $F:X\mapsto MXM^{-1}$ et $f:(A,B)\mapsto\mathrm{ #+end_exercice #+begin_exercice [Centrale 2023 # 1223] -Soit $A=\left(\begin{array}{cccc}0&\cdots&0&a_1\\ \vdots&&\iddots&\vdots\\ \vdots&\iddots&&\vdots\\ a_n&0&\ldots&0\end{array}\right)\in{\cal M}_n(\R)$ +Soit $A=\left(\begin{array}{cccc}0&\cdots&0&a_1\\ \vdots&&\ddots&\vdots\\ \vdots&\ddots&&\vdots\\ a_n&0&\ldots&0\end{array}\right)\in{\cal M}_n(\R)$ - Donner la définition du polynôme minimal $\pi_A$. Donner une condition nécessaire et suffisante pour que $A$ soit diagonalisable. - Calculer $\det(A)$ et $A^2$. - Montrer que $A$ est diagonalisable si et seulement si ${\rm Ker}(A)={\rm Ker}(A^2)$. Donner une condition sur les $a_1,\ldots,a_n$ pour que $A$ soit diagonalisable. @@ -6945,7 +7174,7 @@ On se place dans ${\cal M}_n(\C)$. - Montrer que toute matrice est trigonalisable sur $\C$. - Soient $\alpha_1,\ldots,\alpha_n\in\C$ et $D={\rm Diag}\,(\alpha_1,\ldots,\alpha_n)$. Montrer qu'il existe un polynôme $f$ tel que pour tout $i\in\db{1,n}$, $f(\alpha_i)^2=\alpha_i$. En déduire que $f(D)^2=D$. -On considère la suite $(c - {k}$ définie par $c_0=1$ et, pour tout $k\in\N$, $c_{k+1}=\sum_{i=0}^kc_ic_{k-i}$ et le polynôme $\phi=\sum_{k=0}^{n-1}c_kX^{k+1}$. + On considère la suite $(c_k)_{k\in\N}$ définie par $c_0=1$ et, pour tout $k\in\N$, $c_{k+1}=\sum_{i=0}^kc_ic_{k-i}$ et le polynôme $\phi=\sum_{k=0}^{n-1}c_kX^{k+1}$. - Déterminer le reste de la division euclidienne de $\phi^2$ par $X^n$. - Trouver un polynôme $g$ tel que, pour toute matrice nilpotente $N\in{\cal M}_n(\C)$, on ait $g(N)^2=I_n+N$. - Soit $A$ une matrice inversible. Montrer qu'il existe $R\in\C[A]$ telle que $R^2=A$. @@ -6965,14 +7194,14 @@ Soit $A\in{\cal M}_d({\C})$. On note $\rho(A)=\max_{\lambda\in{\rm Sp}(A)}|\lamb - Donner un exemple de matrice dans ${\cal M}_2({\C})$ telle que $(u_n)$ ne converge pas. On suppose maintenant que $A$ a au moins deux valeurs propres distinctes. - - Soit $z\in{\C}$ tel que $|z|=1$. Montrer que $1$ est valeur d'adherence de $(z^n)$. Montrer que $\rho(A)$ est valeur d'adherence de $u_n$. + - Soit $z\in{\C}$ tel que $|z|=1$. Montrer que $1$ est valeur d'adhérence de $(z^n)$. Montrer que $\rho(A)$ est valeur d'adhérence de $u_n$. #+end_exercice #+begin_exercice [Centrale 2023 # 1227] Soit $E$ un espace-vectoriel de dimension finie $n$ et $f$ un endomorphisme de $E$. Pour toute partie $A\subset{\cal L}(E)$, on note ${\cal C}(A)=\{u\in{\cal L}(E)\;;\;\forall v\in A,\;u\circ v=v\circ u\}$. L'objectif de l'exercice est d'étudier ${\cal B}(f)={\cal C}({\cal C}(\{f\}))$. - Montrer que ${\cal B}(f)$ est une ${\mathbb{K}}$-algèbre contenant ${\mathbb{K}}[f]$. - On suppose $f$ nilpotente d'indice $n$. Montrer que ${\cal B}(f)={\mathbb{K}}[f]$. - - Soient $G_1,G_2$ deux sous-espaces vectoriels supplementaires stables par un $f\in{\cal L}(E)$. On pose $f_i=f_{|G_i}$. On suppose que $\pi_{f_1}\wedge\pi_{f_2}=1$. Montrer que ${\cal B}(f)={\mathbb{K}}[f]$. + - Soient $G_1,G_2$ deux sous-espaces vectoriels supplémentaires stables par un $f\in{\cal L}(E)$. On pose $f_i=f_{|G_i}$. On suppose que $\pi_{f_1}\wedge\pi_{f_2}=1$. Montrer que ${\cal B}(f)={\mathbb{K}}[f]$. #+end_exercice #+begin_exercice [Centrale 2023 # 1228] @@ -6981,19 +7210,17 @@ Soient $E$ un espace vectoriel euclidien de dimension $n\in{\N}$, $a\in E$ un ve - Montrer que, pour $x\in E$, $p(x)=x-\langle a,x\rangle a$. - Soit $\Omega=\{x\in E,\;\langle a,x\rangle\geq 0\;\;\mbox{et}\;\;\langle x,\sigma(x) \rangle\leq 0\}$. -Montrer les equivalences suivantes, pour $x\in E$ : +Montrer les équivalences suivantes, pour $x\in E$ : - $x\in\Omega$ si et seulement si $\langle a,x\rangle\leq\|p(x)\|$, - $x\in\Omega$ si et seulement si $\forall y\in\Omega$, $\langle x,y\rangle\leq 0$. #+end_exercice #+begin_exercice [Centrale 2023 # 1229] Soit $E$ un espace euclidien. Soit $s\in{\cal L}(E)$. - - Rappeler l'identite du parallelogramme et les identites de polarisation. - - Montrer l'equivalence suivante : - -i) $\exists c\in{\R},\;\forall(x,y)\in E^2,\;\langle s(x),s(y)\rangle=c \langle x,y\rangle,$ - -ii) $\forall(x,y)\in E^2,\,\langle x,y\rangle=0\;\Rightarrow\;\langle s(x),s(y) \rangle=0$. + - Rappeler l'identité du parallélogramme et les identités de polarisation. + - Montrer l'équivalence suivante : + + $\exists c\in\R,\forall (x,y)\in E^2,\quad \langle s(x),s(y)\rangle=c \langle x,y\rangle$ + + $\forall(x,y)\in E^2,\,\langle x,y\rangle=0\;\Rightarrow\;\langle s(x),s(y) \rangle=0$. #+end_exercice #+begin_exercice [Centrale 2023 # 1230] @@ -7015,15 +7242,15 @@ Soient $A\in\mc{S}_n^{++}(\R)$ et $b\in\R^n$. #+end_exercice #+begin_exercice [Centrale 2023 # 1233] -Soient $n\in\N$ et $\alpha\gt 0$. On note $S_{\alpha}=\{M\in\mc{S}_n^+(\R),\ \det M\geq\alpha\}$. Le but de cet exercice est de s'interesser, pour $A\in\mc{S}_n^+(\R)$, a la quantite $m_{\alpha}(A)=\inf_{M\in S_{\alpha}}\op{tr}(AM)$. +Soient $n\in\N$ et $\alpha\gt 0$. On note $S_{\alpha}=\{M\in\mc{S}_n^+(\R),\ \det M\geq\alpha\}$. Le but de cet exercice est de s’intéresser, pour $A\in\mc{S}_n^+(\R)$, a la quantité $m_{\alpha}(A)=\inf_{M\in S_{\alpha}}\op{tr}(AM)$. - Montrer que les valeurs propres d'une matrice symétrique réelle sont réelles. Rappeler le théorème spectral. Justifier l'existence de $m_{\alpha}(I_n)$ puis la calculer. - - Soit $A\in\mc{S}_n^+(\R)$. Justifier l'existence de $R\in\mc{S}_n^+(\R)$ telle que $A=R^2$. Prouver l'unicite puis calculer $m_{\alpha}(A)$. + - Soit $A\in\mc{S}_n^+(\R)$. Justifier l'existence de $R\in\mc{S}_n^+(\R)$ telle que $A=R^2$. Prouver l'unicité puis calculer $m_{\alpha}(A)$. - Que se passe-t-il lorsque $\alpha=0$? #+end_exercice #+begin_exercice [Centrale 2023 # 1234] Soient $d\in\N^*$, $A\in\mc{S}_n(\R)$ a coefficients dans $\{0,1\}$ et de trace nulle. On suppose que $A^2+A-(d-1)I_n=J_n$ ou $J_n$ est la matrice dont tous les coefficients valent $1$. - - Montrer que chaque ligne de $A$ contient $d$ coefficients egaux a $1$. + - Montrer que chaque ligne de $A$ contient $d$ coefficients égaux a $1$. - Montrer que $AU=dU$ ou $U=(1\,\cdots\,1)^T$. En déduire que $n=d^2+1$. - Montrer que la multiplicité de $d$ est égale a $1$. - Montrer que les autres valeurs propres de $M$ sont racines de $X^2+X-d+1=0$. @@ -7049,33 +7276,35 @@ On considère la relation binaire pour $(A,B)\in(\mc{S}_n(\R))^2A\preceq B\Leftr #+end_exercice #+begin_exercice [Centrale 2023 # 1237] -Si $S\in\mc{S}_n(\R)$, on note $\lambda_1(S)\leq\cdots\leq\lambda_n(S)$ le spectre ordonne de $S$. On munit $\R^n$ du produit scalaire canonique note $\langle\,\ \rangle$ et on note $S^{n-1}$ la spere unite. - - Montrer que, si $S\in\mc{S}_n(\R)$, $\lambda_1(S)=\min\{\langle Sx,x\rangle\ ;\ x\in S^{n-1}\}$. - - Si $d\in\db{1,n}$, soit $\mc{V}_d$ l'ensemble des sous-espaces vectoriels de dimension $d$ de $\R^n$. Montrer que, si $k\in\db{1,n}$ et $S\in\mc{S}_n(\R)$, +Si $S\in\mc{S}_n(\R)$, on note $\lambda_1(S)\leq\cdots\leq\lambda_n(S)$ le spectre ordonné de $S$. On munit $\R^n$ du produit scalaire canonique et on note $S^{n-1}$ la sphère unite. + - Montrer que, si $S\in\mc{S}_n(\R)$, $\lambda_1(S)=\min\{\langle Sx,x\rangle,\, x\in S^{n-1}\}$. + - Si $d\in\db{1,n}$, soit $\mc{V}_d$ l'ensemble des sous-espaces vectoriels de dimension $d$ de $\R^n$. -$\lambda_k(S)=\min\limits_{V\in\mc{V}_k}\max\{\langle Sx,x\rangle\ ;\ x\in V \cap S^{n-1}\}=\max\limits_{V\in\mc{V}_{n-k+1}}\min\{\langle Sx,x\rangle \ ;\ x\in V\cap S^{n-1}\}$. + Montrer que, si $k\in\db{1,n}$ et $S\in\mc{S}_n(\R)$, + + $$\lambda_k(S)=\min\limits_{V\in\mc{V}_k}\max\{\langle Sx,x\rangle\ ;\ x\in V \cap S^{n-1}\}=\max\limits_{V\in\mc{V}_{n-k+1}}\min\{\langle Sx,x\rangle \ ;\ x\in V\cap S^{n-1}\}.$$ - Si $(i,j)\in\db{1,n}^2$, $i+j\leq n+1$ et $(S,S')\in\mc{S}_n(\R)^2$, montrer que -$\lambda_{i+j-1}(S+S')\leq\lambda_i(S)+\lambda_j(S')$. + $\lambda_{i+j-1}(S+S')\leq\lambda_i(S)+\lambda_j(S')$. #+end_exercice ** Analyse #+begin_exercice [Centrale 2023 # 1238] -Soient $(E,\|\ \|)$ un espace norme, $F$ un sous-espace vectoriel ferme strict de $E$ et $\delta\in]0,1[$. Montrer qu'il existe un vecteur unitaire $u$ de $E$ tel que $d(u,F)\geq\delta$. +Soient $(E,\|\ \|)$ un espace norme, $F$ un sous-espace vectoriel fermé strict de $E$ et $\delta\in]0,1[$. Montrer qu'il existe un vecteur unitaire $u$ de $E$ tel que $d(u,F)\geq\delta$. #+end_exercice #+begin_exercice [Centrale 2023 # 1239] -Soient $(E,N)$ et $(E',N')$ deux espaces vectoriels normes. +Soient $(E,N)$ et $(E',N')$ deux espaces vectoriels normés. -Soit $d\in\N$. Pour $P(X)=p_0+p_1X+\cdots+p_dX^d\in\R_d[X]$ on pose $\|P\|=\max(|p_0|,...,|p_d|)$. - - Vérifier que l'application $\|\ \|$ est une norme sur $\R_d[X]$. +Soit $d\in\N$. Pour $P(X)=p_0+p_1X+\cdots+p_dX^d\in\R_d[X]$ on pose $\lN P\rN=\max(|p_0|,...,|p_d|)$. + - Vérifier que l'application $\lN \cdot\rN$ est une norme sur $\R_d[X]$. + - + - Soit $(y_n)_{n\in\N}$ une suite d'éléments de $E$, convergeant vers $\ell\in E$. -_b) i)_ Soit $(y_n)_{n\in\N}$ une suite d'éléments de $E$, convergeant vers $\ell\in E$. - -Montrer que l'ensemble $Y=\{y_n,\ n\in\N\}\cup\{\ell\}$ est compact. - - Soit $f:E\to E'$ continue telle que, pour tout compact $K$ de $E'$, $f^{-1}(K)$ est un compact de $E$. Montrer que, si $F$ est un ferme de $E$, alors $f(F)$ est un ferme de $E'$. - - Soit $P\in\R_d[X]$ un polynôme unitaire. Montrer que, si $x\in\R$ est une racine de $P$ telle que $|x|\gt 1$, alors $|x|\leq\|P\|+1$. En déduire que l'ensemble des polynômes unitaires et scindes de $\R_d[X]$ est ferme dans $\R_d[X]$. + Montrer que l'ensemble $Y=\{y_n,\ n\in\N\}\cup\{\ell\}$ est compact. + - Soit $f\colon E\to E'$ continue telle que, pour tout compact $K$ de $E'$, $f^{-1}(K)$ est un compact de $E$. Montrer que, si $F$ est un fermé de $E$, alors $f(F)$ est un fermé de $E'$. + - Soit $P\in\R_d[X]$ un polynôme unitaire. Montrer que, si $x\in\R$ est une racine de $P$ telle que $|x|\gt 1$, alors $|x|\leq \lN P\rN +1$. En déduire que l'ensemble des polynômes unitaires et scindes de $\R_d[X]$ est ferme dans $\R_d[X]$. #+end_exercice #+begin_exercice [Centrale 2023 # 1240] @@ -7090,18 +7319,32 @@ Soit $(E,\|\ \|)$ un espace vectoriel norme. Pour $A\subset E$ non vide et $x\in - On suppose $E$ de dimension infinie et on admet que les sous-espaces vectoriels de dimension finie sont fermes. Montrer que la sphere unite n'est pas un compact de $E$. #+end_exercice +# ID:7075 #+begin_exercice [Centrale 2023 # 1242] Soit $\phi$ la fonction définie sur $[0,1]$ par $\phi(0)=0$ et $\phi(t)=-t\ln(t)$ pour $t\in]0,1]$. Soit $n\in\N^*$. On pose $S_n$ l'ensemble des vecteurs $p=(p_1,...,p_n)\in\R^n$ tels que $p_1+...+p_n=1$ et $p_i\geq 0$ pour tout $1\leq i\leq n$. On pose enfin $H_n(p)=\sum_{i=1}^n\phi(p_i)$ pour $p\in S_n$. - - - Donner la définition d'une partie compacte d'un espace vectoriel norme, et en donner une caracterisation en dimension finie. - - Montrer que $S_n$ est une partie compacte et convexe de $\R^n$. - - - Montrer que $H_n$ est continue. - - Montrer que $H_n$ atteint sur $S_n$ un maximum en un unique point $p_0$, et expliciter $p_0$. + - + - Donner la définition d'une partie compacte d'un espace vectoriel normé, et en donner une caractérisation en dimension finie. + - Montrer que $S_n$ est une partie compacte et convexe de $\R^n$. + - + - Montrer que $H_n$ est continue. + - Montrer que $H_n$ atteint sur $S_n$ un maximum en un unique point $p_0$, et expliciter $p_0$. -Soit $v=(v_1,...,v_n)\in\R^n$. On pose $f_v(p)=H_n(p)+\sum_{i=1}^np_iv_i$ pour $p\in S_n$. + - Soit $v=(v_1,...,v_n)\in\R^n$. On pose $f_v(p)=H_n(p)+\sum_{i=1}^np_iv_i$ pour $p\in S_n$. -On pose $f_v^*=\sup_{p\in S_n}f_v(p)$ et $E_v=\{p\in S_n,\ f_v(p)=f_v^*\}$. - - Montrer que $E_v$ est non vide. Déterminer $f_v^*$ et $E_v$. + On pose $f_v^*=\sup_{p\in S_n}f_v(p)$ et $E_v=\{p\in S_n,\ f_v(p)=f_v^*\}$. + + Montrer que $E_v$ est non vide. Déterminer $f_v^*$ et $E_v$. #+end_exercice +#+BEGIN_proof + - + - + - + - + - + - C'est des extrema liées, mais attention aux bords : le théorème s'applique quand on met une contrainte sur un ouvert. Ici il faut exclure le cas où l'extrema est atteint en un $p_i$ nul, ce qui se traite par récurrence. + - Idem, on peut exclure le bord car la dérivée de $\phi$ en $0$ est $+\i$, donc s'éloigner du bord est toujours bénéfique. +#+END_proof + #+begin_exercice [Centrale 2023 # 1243] Soient $(E,\|\ \|)$, $(E',\|\ \|)$ deva espaces vectoriels normes de dimension finie, $A$ un ferme non vide de $E$, $B$ une partie non vide de $E'$. Soit $f:A\to B$ continue bijective telle que l'image réciproque par $f$ de toute partie bornée de $B$ est bornée. Montrer que $f^{-1}$ est continue. @@ -7116,24 +7359,25 @@ Un espace norme réel est dit separable lorsqu'il contient une partie denombrabl #+begin_exercice [Centrale 2023 # 1245] Soit $E$ l'espace des fonctions polynomiales de $\R$ dans $\R$. Pour tout $f\in E$, on note $\phi(f)$ la primitive de $f$ d'intégrale nulle sur l'intervalle $[0,1]$. - - Justifier la définition de $\phi$ puis etablir qu'il s'agit d'une application linéaire sur $E$. + - Justifier la définition de $\phi$ puis établir qu'il s'agit d'une application linéaire sur $E$. -On munit $E$ de la norme $\|\ \|_{\i}$ sur $[0,1]$. + On munit $E$ de la norme $\lN\cdot\rN_{\i}$ sur $[0,1]$. -On note $\|\phi\|_{\mathrm{op}}=\sup\left\{\frac{\|\phi(f)\|_{\i,[0,1]}}{\|f \|_{\i,[0,1]}},\ f\in E\setminus\{0_E\}\right\}$. - - Montrer que $\|\phi\|_{\mathrm{op}}$ est correctement définie et en trouver un majorant. - Soient $f\in E$ et $G$ la primitive de $F=\phi(f)$ nulle en $0$. Etablir que, pour tout $x\gt 0$, + On note $\lN \phi\rN_{\mathrm{op}}=\sup\left\{\frac{\lN\phi(f)\rN_{\i,[0,1]}}{\lN f\rN_{\i,[0,1]}},\ f\in E\setminus\{0_E\}\right\}$. + - Montrer que $\lN\phi\rN_{op}$ est correctement définie et en trouver un majorant. + - Soient $f\in E$ et $G$ la primitive de $F=\phi(f)$ nulle en $0$. Établir que, pour tout $x\gt 0$, -$$G(x)=xF(x)-\int_0^xtf(t)dt=(x-1)F(x)-\int_x^1(1-t)f(t)\mathrm{d }t.$$ - - Déterminer la norme $\|\phi\|_{\mathrm{op}}$. + $$G(x)=xF(x)-\int_0^xtf(t)dt=(x-1)F(x)-\int_x^1(1-t)f(t)\mathrm{d }t.$$ + - Déterminer la norme $\lN\phi\rN_{\mathrm{op}}$. #+end_exercice #+begin_exercice [Centrale 2023 # 1246] Soit $A\in\M_n(\R)$, on pose $f_A(x)=(A+xI_n)^{-1}A$ pour $x$ réel convenable. - - Montrer que la fonction $f_A$ est définie au voisinage epointe de $0$. + - Montrer que la fonction $f_A$ est définie au voisinage épointé de $0$. - Étudier le comportement de la fonction $f_A$ en $0$ dans le cas ou $A$ est inversible, puis dans le cas ou $A$ est nilpotente. - Soit $u\in\mc{L}(\R^n)$. Montrer l'existence de $p\in\N^*$ tel que $\mathrm{Im}(u^p)\oplus\mathrm{Ker}(u^p)=\R^n$. -En déduire l'existence de deux supplementaires $F$ et $G$ dans $\R^n$, stables par $u$, tels que $u$ induit sur $F$ un automorphisme et induit sur $G$ un endomorphisme nilpotent. +En déduire l'existence de deux supplémentaires $F$ et $G$ dans $\R^n$, stables par $u$, tels que $u$ induit sur $F$ un automorphisme et induit sur $G$ un endomorphisme nilpotent. - Caracteriser les matrices $A$ pour lesquelles $f_A$ a une limite en $0$. #+end_exercice @@ -7144,11 +7388,11 @@ Soient $(a_n)$ une suite a termes réels positifs et $(b_n)$ une suite a termes #+end_exercice #+begin_exercice [Centrale 2023 # 1248] -- Rappeler la regle de d'Alembert pour une série numerique a termes positifs. + - Rappeler la règle de d'Alembert pour une série numérique à termes positifs. - On considère une suite croissante $(q_n)_{n\geq 1}$ d'entiers $\geq 2$. - - Quel est le rayon de convergence de la série entière $\sum\frac{z^n}{q_1...q_n}$? - - Montrer que si la suite $(q_n)$ est stationnaire alors le réel $ x=\sum_{n=1}^{+\i}\frac{1}{q_1...q_n}$ appartient a $\Q\cap]0,1]$. - - On admet réciproquement que si $(q_n)$ tend vers $+\i$ alors $x\notin\Q$. Montrer que les réels $e$, $\mathrm{ch}(\sqrt{2})$ et $e^{\sqrt{2}}$ sont irrationnels. + - Quel est le rayon de convergence de la série entière $\sum\frac{z^n}{q_1...q_n}$? + - Montrer que si la suite $(q_n)$ est stationnaire alors le réel $x=\sum_{n=1}^{+\i}\frac{1}{q_1...q_n}$ appartient a $\Q\cap]0,1]$. + - On admet réciproquement que si $(q_n)$ tend vers $+\i$ alors $x\notin\Q$. Montrer que les réels $e$, $\mathrm{ch}(\sqrt{2})$ et $e^{\sqrt{2}}$ sont irrationnels. - Montrer la réciproque admise ci-dessus. #+end_exercice @@ -7253,7 +7497,7 @@ Pour $n\geq 1$, on note $t_n$ le nombre de $\sigma\in\mc{S}_n$ telles que $\sigm #+end_exercice #+begin_exercice [Centrale 2023 # 1263] -Pour $n\in\N$, on note $\mc{P}_n$ l'ensemble des polynômes $P$ a coefficients dans $\{0,1,2\}$ tels que $P(2)=n$, et $a_n=|\mc{P}_n|$. On note $s_n$ la somme des chiffres de l'écriture binaire de l'entier $n$, et pour $k\in\db{0,7]\!]$, on pose $b_{n,k}=|\{i\in[\![0,n},\ s_i-s_{n-i}\equiv k\ [8]\}|$. +Pour $n\in\N$, on note $\mc{P}_n$ l'ensemble des polynômes $P$ a coefficients dans $\{0,1,2\}$ tels que $P(2)=n$, et $a_n=|\mc{P}_n|$. On note $s_n$ la somme des chiffres de l'écriture binaire de l'entier $n$, et pour $k\in\db{0,7}$, on pose $b_{n,k}=|\{i\in\db{0,n},\ s_i-s_{n-i}\equiv k\ [8]\}|$. - Calculer $a_0$, $a_1$, $a_2$ et $a_3$. - Montrer que $\mc{P}_n$ est fini. - Montrer que, pour $n\in\N$, $a_{2n+1}=a_n$ et que, pour $n\in\N^*$, $a_{2n}=a_n+a_{n-1}$. @@ -7285,7 +7529,7 @@ On dit qu'une suite réelle $(a_n)_{n\in\N}$ vérifie la propriété $(P)$ si :1 #+end_exercice #+begin_exercice [Centrale 2023 # 1265] -Soient $(a_n)_{n\geq 1}$ une suite de carré sommable et $f:t\mapsto\sum_{n=1}^{+\i}\frac{a_n}{n-t}$. +Soient $(a_n)_{n\geq 1}$ une suite de carré sommable et $f\colon t\mapsto\sum_{n=1}^{+\i}\frac{a_n}{n-t}$. - Preciser le domaine de définition de $f$. - Montrer que $f$ est développable en série entière autour de $0$. - Montrer que si $f$ est identiquement nulle sur $[-1/2,1/2]$ alors la suite $(a_n)$ est nulle. @@ -7300,14 +7544,14 @@ Montrer que $f$ est développable en série entière en $0$. #+end_exercice #+begin_exercice [Centrale 2023 # 1267] -On admet le théorème suivant :_Pour $S$ une série entière de rayon de convergence infini dont la somme ne s'annule pas sur $\C$, il existe une série entière $L$ de rayon de convergence infini telle que $\forall z\in\C,\exp(L(z))=\ S(z)$. - - - Rappeler tous les modes de convergence d'une série entière sur son disque ouvert de convergence. - - Soient $F(z)=\sum_{n=0}^{+\i}a_nz^n$ de rayon de convergence infini et $ G(z)=\text{Re}(F(z))$. +On admet le théorème suivant : Pour $S$ une série entière de rayon de convergence infini dont la somme ne s'annule pas sur $\C$, il existe une série entière $L$ de rayon de convergence infini telle que $\forall z\in\C,\exp(L(z))=\ S(z)$. + - Rappeler tous les modes de convergence d'une série entière sur son disque ouvert de convergence. + - Soient $F(z)=\sum_{n=0}^{+\i}a_nz^n$ de rayon de convergence infini et $G(z)=\op{Re}(F(z))$. -Pour $n\in\N^*$, montrer que $\int_0^{2\pi}F(re^{it})dt=2\pi a_nR^n$, puis que + Pour $n\in\N^*$, montrer que $\int_0^{2\pi}F(re^{it})dt=2\pi a_nR^n$, puis que -$\int_0^{2\pi}G(Re^{it})e^{-int}dt=\pi a_nR^n$ et $\int_0^{2\pi}G(Re^{it})dt=2\pi\text{Re}(a_0)$. - - Montrer que, s'il existe $p$ et $q$ réels strictement positifs tels que $\forall z\in\C,\ |G(z)|\leq p|z|+q$, alors $F$ est un polynôme de degré au plus $1$. + $\int_0^{2\pi}G(Re^{it})e^{-int}dt=\pi a_nR^n$ et $\int_0^{2\pi}G(Re^{it})dt=2\pi\text{Re}(a_0)$. + - Montrer que, s'il existe $p$ et $q$ réels strictement positifs tels que $\forall z\in\C,\, |G(z)|\leq p|z|+q$, alors $F$ est un polynôme de degré au plus $1$. - Montrer que l'application $z\mapsto z\exp(z)$ est une surjection de $\C$ sur lui-meme. #+end_exercice @@ -7318,7 +7562,7 @@ Soient $E$ un $\C$-espace vectoriel de dimension finie et $u\in\mc{L}(E)$. Montr #+begin_exercice [Centrale 2023 # 1269.] - - Rappeler la définition de fonction continue par morceaux sur un intervalle $I$ de $\R$. - _Pour $n\in\N^*$, on définit une fonction $f_n$ sur $\R^+$ par $f_n(x)=\frac{1}{n}\left(1-\frac{x}{n}\right)$ si $x\in[0,n]$ et $f_n(x)=0$ sinon. -Montrer que la suite $(f - {n\in\N^*}$ converge simplement sur $\R^+$ vers une fonction $f$ a preciser et que $\int_{\R^+}f_n\not\longrightarrow\int_{\R_+}f$ quand $n\ra+\i$. +Montrer que la suite $(f_n)_{n\in\N^*}$ converge simplement sur $\R^+$ vers une fonction $f$ a preciser et que $\int_{\R^+}f_n\not\longrightarrow\int_{\R_+}f$ quand $n\ra+\i$. - Rappeler le théorème de convergence dominée. Le démontrer sous l'hypothèse supplémentaire d'une convergence uniforme sur tout segment. @@ -7388,7 +7632,7 @@ On note $D(t)$ la somme de cette série. - En déduire que $d_n=n!\sum_{k=0}^n\frac{(-1)^k}{k!}$. - Calculer la limite lorsque $n$ tend vers $+\i$ de la probabilité $p_n$ qu'un élément de $\mc{S}_n$ soit un derangement. -_c) i)_Pour $n$ et $p$ entiers naturels, on note $s_n(p)$ le nombre de surjections de $[\![1,n]$ sur $[1,p]$. +_c) i)_Pour $n$ et $p$ entiers naturels, on note $s_n(p)$ le nombre de surjections de $\db{1,n}$ sur $\db{1,p}$. Montrer que $p^n=\sum_{k=0}^n\binom{p}{k}s_n(k)$. - Soit $(x,y)\in\R^2$. Montrer que la famille $\left(s_n(p)\frac{x^p}{p!}\,\frac{y^n}{n!}\right)_{(n,p)\in \N^2}$ est sommable. diff --git a/Exercices 2023.pdf b/Exercices 2023.pdf index e271cf331636bf3f1eabc6bb36323bac26702ac9..ff5d97243f45909c23a0ee96a3973d89c7279ee0 100644 GIT binary patch delta 724616 zcmV(*K;FNz#YF4oMUW?#aV-WZu~1k60d4G%9AYu8h_QNxfA!E?%JSFbEl})IL#eV?{h7C)qXXq)t$KXHmH^v(Ic3r zqEge<-!GU#nj28FM1}=5zX@9kMv&pV+#Z5t$WU|i9)d+Z&6dM1L#<5&hBzi^OzAff zo7GIqkO*o~`cLvD*mmIyVu7L4S&?M|D@IL0IY&>#iiS z740UA6v$xJcUewW#eJ9M8ECcdvOHnc%D&5T;wn9~PxP*9mq=JpVjhcQ6ql!rV`PtG zo@rR@am-XrLD2%cZ3 zNYH+V;2Eo5l7C}gH0Yber>1+wJB#LMgTSYkxKW(wUF%+~Pf-1>hmLafbzl9Rj)ISR zPQ~ha_>TZA4IjRmE;g(2Y}$z}mFcl%f5{l2=}a&^wmfb1YuaPWXRCX8?87H@hH>z+ z*o!ia=uzfZuJayHo;IM=>d{@w040qU!|z9o+g&$*8h?v(^JP zHN2n_32fqXTv~=jFsx|bsV)&ofhA%Gp2+r=?K$8^BTj(<3>O{4+#Cb@ZNDv5g^b;j zQ;kfc19d<#Qu(7oT3enRexPgdUAYC@vF_|0_#}sa2tU5=0`HO_ifb|#%MzlspN%EY zue&pV>VIK=L_)}_r+(*dnxkP5mG%4Tmst{8*3Y83Q+_P}pqT0WX6FBt!}pl6MH%ib 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