From 99a2ceef84f89e7d12d1dfff6e8430a893fe6fc5 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?S=C3=A9bastien=20Miquel?= Date: Tue, 10 Mar 2026 19:27:24 +0100 Subject: [PATCH] Premier passage. --- Exercices 2024.org | 13 +- Exercices 2025.org | 2149 ++++++++++++++++++++++++++++-------- Exercices 2025.pdf | Bin 1609901 -> 1504798 bytes Exercices XENS MP 2025.pdf | Bin 809971 -> 809647 bytes 4 files changed, 1666 insertions(+), 496 deletions(-) diff --git a/Exercices 2024.org b/Exercices 2024.org index 64caf2f..f15e24c 100644 --- a/Exercices 2024.org +++ b/Exercices 2024.org @@ -2,7 +2,7 @@ #+title: Exercices 2024 #+author: Sébastien Miquel #+date: 20-11-2024 -# Time-stamp: <21-06-25 16:21> +# Time-stamp: <26-02-26 13:42> * Meta :noexport: ** Statistiques @@ -6892,12 +6892,12 @@ Soient $n\geq 2$ un entier et $f\colon\R^n\ra\R$ de classe $\mc C^2$ telle que l #+begin_exercice [Mines MP 2024 # 804] Soient $E$ un espace euclidien, $v\in E$ non nul et $f\in\mc{S}^{++}(E)$. - - Montr per qu'il existe une base $(e_1,\ldots,e_n)$ de $E$ formée de vecteurs propres de $f$. - - Montr per que, pour tout $x\in E$ non nul, $\langle f(x),x\rangle\gt 0$. - - Montr per que $g:x\mapsto\frac{1}{2}\langle f(x),x\rangle-\langle v,x\rangle$ est de classe $\mc C^1$. + - Montrer qu'il existe une base $(e_1,\ldots,e_n)$ de $E$ formée de vecteurs propres de $f$. + - Montrer que, pour tout $x\in E$ non nul, $\langle f(x),x\rangle\gt 0$. + - Montrer que $g:x\mapsto\frac{1}{2}\langle f(x),x\rangle-\langle v,x\rangle$ est de classe $\mc C^1$. - Calculer les derivées partielles de $g$ relativement à la base $(e_1,\ldots,e_n)$ et le gradient de $g$. - - Montr per que $g$ admet un unique point critique $c$. - - Montr per que $g$ admet un minimum global en $c$. Existe-t-il d'autres extrema locaux? + - Montrer que $g$ admet un unique point critique $c$. + - Montrer que $g$ admet un minimum global en $c$. Existe-t-il d'autres extrema locaux? #+end_exercice #+begin_exercice [Mines MP 2024 # 805] @@ -6908,6 +6908,7 @@ Montrer que $\phi:x\in\overline{\mc{B}}\mapsto f(x)+\eps(\|x\|^2-1)$ admet un ma - On suppose que $\Delta f=0$. Montrer que $\min\limits_{\overline{\mc{B}}}f=\min\limits_{\mc{S}}f$ et $\max\limits_{\overline{\mc{B}}}f=\max\limits_{\mc{S}}f$. #+end_exercice +# ID:8765 #+begin_exercice [Mines MP 2024 # 806] Déterminer les espaces tangents en $I_n$ aux parties $\text{SL}_n(\R)$ et $\mc{O}_n(\R)$ de $\M_n(\R)$. #+end_exercice diff --git a/Exercices 2025.org b/Exercices 2025.org index a379967..b0ade70 100644 --- a/Exercices 2025.org +++ b/Exercices 2025.org @@ -1,8 +1,8 @@ -# -*- org-export-switch: "xens mp"; -*- +# -*- org-export-switch: "todoes"; -*- #+title: Exercices 2025 #+author: Sébastien Miquel #+date: 22-07-2025 -# Time-stamp: \lt 22-07-25 16:44\gt +# Time-stamp: <03-03-26 14:07> #+OPTIONS: toc:t * Meta :noexport: @@ -14,7 +14,7 @@ #+RESULTS: | ? | ! | todo | unexed | unexed xens | -| 0 | 0 | 0 | 450 | 0 | +| 4 | 9 | 13 | 1477 | 139 | ** Options @@ -35,10 +35,10 @@ *** XENS MP -#+select_tags: xens -#+exclude_tags: autre -#+exclude_types: proof -#+export_file_name: Exercices XENS MP 2025 +# #+select_tags: xens +# #+exclude_tags: autre +# #+exclude_types: proof +# #+export_file_name: Exercices XENS MP 2025 *** Centrale @@ -59,10 +59,10 @@ *** todoes -# #+options: title:nil nopage:t tags:nil toc:nil -# #+select_tags: todo -# #+export_file_name: Exercices 2025 todo -# #+relocate_tags: todo +#+options: title:nil nopage:t tags:nil toc:nil +#+select_tags: todo +#+export_file_name: Exercices 2025 todo +#+relocate_tags: todo *** autre @@ -71,6 +71,12 @@ # #+export_file_name: Exercices XENS 2025 autres # #+relocate_tags: todo +*** autres + +# #+options: title:nil nopage:t tags:nil +# #+select_tags: autres +# #+export_file_name: Exercices 2025 Autres MP +# #+relocate_tags: todo * ENS MP :xens: @@ -79,9 +85,9 @@ # ID:8419 #+begin_exercice [ENS L 2025 # 1] -Soit $n \in \mathbb{N}^*$. Un chemin auto-évitant de longueur $n$ de $\mathbb{Z}^2$ est une suite injective de points $a_0, \ldots, a_n$ de $\mathbb{Z}^2$ telle que $a_0 = (0,0)$ et, pour tout $i$, $\|a_{i+1} a_i\| = 1$ pour la norme euclidienne canonique de $\mathbb{R}^2$. On note $A_n$ le nombre de chemins auto-évitants de longueur $n$. - 1. Montrer que, pour tous $m, n \in \mathbb{N}^*$, $A_{m+n} \leqslant A_m A_n$. - 1. Montrer qu'il existe $\eps > 0$ tel que, pour $n$ assez grand, $(2 + \eps)^n \leqslant A_n \leqslant (3 - \eps)^n$. +Soit $n \in \N^*$. Un chemin auto-évitant de longueur $n$ de $\Z^2$ est une suite injective de points $a_0, \ldots, a_n$ de $\Z^2$ telle que $a_0 = (0,0)$ et, pour tout $i$, $\|a_{i+1} a_i\| = 1$ pour la norme euclidienne canonique de $\R^2$. On note $A_n$ le nombre de chemins auto-évitants de longueur $n$. + 1. Montrer que, pour tous $m, n \in \N^*$, $A_{m+n} \leq A_m A_n$. + 1. Montrer qu'il existe $\eps \gt 0$ tel que, pour $n$ assez grand, $(2 + \eps)^n \leq A_n \leq (3 - \eps)^n$. 1. Montrer que $\big(\sqrt[n]{A_n}\big)$ converge. #+end_exercice #+BEGIN_proof @@ -95,8 +101,8 @@ Soit $n \in \mathbb{N}^*$. Un chemin auto-évitant de longueur $n$ de $\mathbb{Z # ID:8420 #+begin_exercice [ENS SR 2025 # 2] Un sous-ensemble non vide $S$ de $\mathbb Z$ est dit direct si, pour $x,y,s,t\in S$, la condition $x+y=s+t$ implique que $\{x,y\}=\{s,t\}$. - 1. Les ensembles $\{1, 3, 6\}$ et $\{1, 3, 6, 10, 15\}$ sont-ils directs? - 2. Trouvez un ensemble infini direct. + 1. s Les ensembles $\{1, 3, 6\}$ et $\{1, 3, 6, 10, 15\}$ sont-ils directs? + 2. s Trouvez un ensemble infini direct. 3. Montrer qu'il existe $B\gt 0$ telle que pour tout $n \in \mathbb{N}^*$, pour tout ensemble direct $S$ inclus dans $\db{0,n}$, on ait $|S| \leq B n^{1/2}$, 4. Montrer qu'il existe $A\gt 0$ telle que pour tout $n\in\N^*$ il @@ -105,22 +111,29 @@ Un sous-ensemble non vide $S$ de $\mathbb Z$ est dit direct si, pour $x,y,s,t\in Indication : On pourra rajouter des éléments un à un à un ensemble de $\db{0, n}$. - 4. Existe-t-il un ensemble $S$ direct inclus dans $\mathbb{N}$ tel + 4. s Existe-t-il un ensemble $S$ direct inclus dans $\mathbb{N}$ tel que $S + S = \mathbb{N}$ ? - 5. Existe-t-il un ensemble $S$ direct inclus dans $\mathbb{Z}$ tel + 5. s Existe-t-il un ensemble $S$ direct inclus dans $\mathbb{Z}$ tel que $\mathbb{N}$ soit inclus dans S+S? - 6. Existe-t-il un ensemble $S$ direct inclus dans $\mathbb{Z}$ tel + 6. s Existe-t-il un ensemble $S$ direct inclus dans $\mathbb{Z}$ tel que $S + S = \mathbb{Z}$ ? #+end_exercice #+BEGIN_proof 1. 2. - 3. - 4. Si $|S|\lt n^{1/3}$ est direct, on peut ajouter un élément à $S$ - qui laisse $S$ direct. En effet, $S+S$ est de cardinal $\lt - n^{2/3}$, et on cherche un translaté de $S$ qui n'intersecte pas - $S+S$. Chaque fois que ça ne marche pas, il existe un $(s\in S, - ss\in S+S)$ qui coince, et cette association est injective. + 3. Si $S\subset \db{0,n}$ est direct, alors $|S+S|= {|S|\choose 2} + |S|$ et $S+S\subset \db{0, 2n}$, donc $|S| \frac{|S|+1}{2}\leq 2n+1$, donc $|S|^2 \leq 2n+1$, donc $|S|^2\leq 3n$, donc $|S|\leq \sqrt{3n}$. + 4. Si $S\subset \db{0,n}$ est direct et vérifie $|S|\leq + \frac{n^{1/3}}{10}$, on montre qu'on peut ajouter un élément de + $\db{0,n}$ à $S$ qui préserve son caractère direct. + + En effet, $S+S$ est de cardinal $\leq \frac{n^{2/3}}{10}$. + + On cherche un élément $a\in \db{0,n}$ tel que $a+S$ ne rencontre + pas $S+S$. Supposons par l'absurde qu'un tel $a$ n'existe pas. + Alors pour tout $a\in\db{0,n}$ il existe $s\in S$ et $s_2\in S+S$ + tels que $a+s = s_2$. + + Par ailleurs, l'application $a\mapsto (s, s_2)$ est clairement injective, donc $|\db{0,n}| \leq |S| \times |S+S|$. C'est une contradiction. 5. $S$ doit contenir $0$, donc $1$, donc pas $2$, donc $3$, qui donne $3, 4, 6$, donc $5$ ce qui donne une contradiction. 6. Oui, tu mets $\{-10^n+n, 10^n\}$, mais il faut sauter des $n$. si les sommes précédentes peuvent déjà les faire. Construction par récurrence. @@ -516,105 +529,240 @@ Reste, dans $\R^3$, le cas de $\begin{pmatrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & #+END_proof +# ID:8561 #+begin_exercice [ENS P 2025 # 27] Soit $n \geq 2$. On note $\mathcal{R}_n$ l'ensemble des matrices $M$ de $\mathrm{GL}_n(\C)$ telles que $M\overline{M}$ appartient à $\C^*I_n$. On définit une relation d'équivalence $\sim$ sur $\mathcal{M}_n(\C)$ en posant $A\sim B$ s'il existe $M \in GL_n(\C)$ et $\lambda \in \C^*$ tels que $A = \lambda \overline{M}BM^{-1}$. Justifier que $\sim$ induit une relation d'équivalence sur $\mathcal{R}_n$. Déterminer les classes d'équivalence sur $\mathcal{R}_n$. + +Indication : Considérer $f\colon X\mapsto A\ol{X}$. #+end_exercice #+BEGIN_proof -Si on restreint aux matrices réelles, ce sont les matrices telles que $M^2 = \la I_n$. On a comme classe d'équivalence le nombre de valeurs propres d'un signe donné. +Si on restreint aux matrices réelles, ce sont les matrices telles que $M^2 = I_n$. On a comme classes d'équivalence le nombre de valeurs propres d'un signe donné. -Ce sont peut-être les seules classes d'équivalence. +Si $Q\ol{Q} = I_n$, en écrivant $Q = A+iB$, on obtient $AB = BA$, $\a\in\R$ et $A^2 - B^2 = I_n$. Bof bof. + +Plutôt : on montre qu'il n'y a qu'une seule classe d'équivalence, c'est-à-dire que si $A\ol{A} = I_n$ alors $A = \ol{P^{-1}} P$. + +Si $A$ s'écrit comme ça, l'équation $AX = \ol{X}$ est équivalente à $PX \in\R^n$. + +Réciproquement, si $\ol{A} A = I_n$, l'application $f\colon X\mapsto A\ol{X}$ est $\R$-linéaire et $f\circ f = Id_{\C^n}$, donc ses valeurs propres sont $\pm 1$. + +Il s'agit essentiellement de montrer qu'il y a autant de $1$ que de $-1$. Si on note $j$ la multiplication par $i$, on a $f\circ j = -j\circ f$, d'où le résultat. + +On a ainsi trouvé la matrice $P$. #+END_proof +# ID:8562 #+begin_exercice [ENS P 2025 # 28] -On note $G_n$ l'ensemble des sous-espaces vectoriels de $\mathbb{R}^n$. Soit $\Phi: G_n \to \mathbb{R}^n$ $\mathbb{R}$ une application telle que $\forall V, W \in G_n, \ \Phi(V \cap W) + \Phi(V + W) \leqslant \Phi(V) + \Phi(W)$ - -et $\Phi(\{0\}) \ge 0$. Montrer qu'il existe un unique $V_0 \in G_n$ de dimension maximale tel que $\inf_{V \in G_n \setminus \{(0)\}} \frac{\Phi(V)}{\dim V} = \frac{\Phi(\hat{V}_0)}{\dim V_0}$. +On note $G_n$ l'ensemble des sous-espaces vectoriels de $\mathbb{R}^n$. Soit $\Phi: G_n \to \mathbb{R}$ une application telle que $\forall V, W \in G_n, \ \Phi(V \cap W) + \Phi(V + W) \leqslant \Phi(V) + \Phi(W)$ et $\Phi(\{0\}) \ge 0$. Montrer qu'il existe un unique $V_0 \in G_n$ de dimension maximale tel que $\inf\limits_{V \in G_n \setminus \{(0)\}} \frac{\Phi(V)}{\dim V} = \frac{\Phi(V_0)}{\dim V_0}$. #+end_exercice +#+BEGIN_proof +On obtient rien en prenant : un des espaces égal à $\{0\}$, un des +espaces égal à $\R^n$, les deux espaces égaux. + +Existence de la forme inférieure : si des espaces de dimension $1$ avaient des valeurs arbitrairement petites, pour $n = 2$, on obtient une contradiction, qui s'étend sans aucun doute à $n$ quelconque. + +Pourquoi la borne inf, sur les espaces de dimension $1$, dans $\R^2$, est-elle atteinte ? On a $\Phi(V) + \Phi(W)\geq \Phi_0 + \Phi_2$, pour tous $V\neq W$. En particulier, il ne peut pas y en avoir deux $\lt \frac{\Phi_0 + \Phi_2}{2}$ (mais un oui). + +S'il y en a un, alors on note $\Ph_1 \lt \frac{\Phi_0 + \Phi_2}{2}$, et le minimum est ou bien atteint en lui, ou bien en $\Phi_2$. + +S'il n'y en a aucun, les valeurs sont, puisque $\Phi_0\geq 0$, $\geq \frac{\Phi_2}{2}$, donc l'inf est en $\Phi_2$. + +On passe de $n = 2$ à $n = 3$ : Les valeurs sur des espaces de dimension $1$ sont minorées, en prenant trois espaces de dimension $2$ en position générique. De même, les valeurs sur des espaces de dimension $2$ sont minorées en prenant trois espaces de dimension $1$ en position générique. + +Considérons la borne inférieure totale. On l'approche par des espaces +de dimension maximale possible. Si elle est approchée par des espaces +de dimension $1$, forcément elle est atteinte. Sinon, elle est +approchée par des espaces de dimension $2$, s'il y en a plusieurs +$V,W$, on a $\phi(V\cap W) + \phi(V+W)\leq 2 inf$, donc les deux +approchent l'inf, ce qui est une contradiction. +#+END_proof + + +# ID:8563 #+begin_exercice [ENS PLSR 2025 # 29] Soient $G$ un groupe admettant une partie génératrice finie et $H$ un groupe fini. 1. Montrer que l'ensemble $E$ des morphismes de groupes de $G$ vers $H$ est fini. - $\textbf{\textit{b}}) \ \ \text{Soit} \ \psi \ \text{un endomorphisme surjectif du groupe} \ G. \ \ \text{Montrer que} \ \ \text{Ker}(\psi) \subset \ \bigcap \ \ \text{Ker}(\phi)$. - 1. On pose $G = \{M \in \mathcal{M}_2(\mathbb{Z}), \det(M) = 1\}$. - a) Montrer que $G$ est un groupe multiplicatif. - b) Montrer que $G$ est engendré par $S = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, $T = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ et $U = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. - c) Montrer que tout endomorphisme surjectif du groupe $G$ est bijectif. + 2. Soit $\psi$ un endomorphisme surjectif du groupe $G$. Montrer que + $\Ker(\psi) \subset \bigcap_{\phi\in E} \Ker(\phi)$. + 3. s On pose $G = \{M \in \mathcal{M}_2(\mathbb{Z}), \det(M) = 1\}$. + 1. Montrer que $G$ est un groupe multiplicatif. + 2. Montrer que $G$ est engendré par $S = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, $T = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ et $U = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. + 3. Montrer que tout endomorphisme surjectif du groupe $G$ est bijectif. #+end_exercice +#+BEGIN_proof + 1. + 2. On considère $\phi\mapsto \phi\circ \psi$, qui est bijective, car $E$ est fini. + 3. + 1. + 2. Par opérations. + 3. Il s'agit de montrer que tout élément de $G$ peut être non nul, + dans un morphisme dans un groupe fini. + Rmq : PSL2(Z) n'est pas du tout simple. + + On a une relation comme $S T^{-1} S = - U$. + + On a évidemment des morphismes dans $SL_2(Z/p/Z)$. +#+END_proof + + +# ID:8564 #+begin_exercice [ENS PLSR 2025 # 30] Soit $A \in \mathcal{M}_2(\mathbb{Z})$ telle que $\det(A) = 1$ et $\operatorname{tr}(A) = \gamma > 2$. Pour $k \in \mathbb{Z}$ et $U \in \mathcal{M}_{2,1}(\mathbb{Z})$, on pose $(k, U) = \begin{pmatrix} A^k & U \\ 0 & 1 \end{pmatrix}$. 1. Montrer que l'ensemble $G_A = \{(k,U) ; k \in \mathbb{Z}, U \in \mathcal{M}_{2,1}(\mathbb{Z})\}$ est un groupe pour la loi de multiplication matricielle. Est-il abélien? - 1. Montrer l'existence d'un morphisme injectif de groupes de $G_A$ dans le groupe - -$$S = \left\{ \begin{pmatrix} e^t & 0 & x \\ 0 & e^{-t} & y \\ 0 & 0 & 1 \end{pmatrix}, \ (t, x, y) \in \mathbb{R}^3 \right\}.$$ + 1. Montrer l'existence d'un morphisme injectif de groupes de $G_A$ dans le groupe $S = \left\{ \begin{pmatrix} e^t & 0 & x \\ 0 & e^{-t} & y \\ 0 & 0 & 1 \end{pmatrix}, \ (t, x, y) \in \mathbb{R}^3 \right\}$. 1. Soit $D_A$ le sous-groupe de $G_A$ engendré par les éléments $ghg^{-1}h^{-1}$ où $(g,h) \in G_A^2$. Montrer que $D_A = \{(0, (I_2 - A)U), U \in \mathcal{M}_{2,1}(\mathbb{Z})\}$. #+end_exercice +#+BEGIN_proof + 1. + 2. La matrice $A$ est diagonalisable. +#+END_proof + +# ID:8593 #+begin_exercice [ENS L 2025 # 31] - 1. Soient $r \in \N^*$ et $(F_1, \ldots, F_r) \in \C(X)^r$. On pose $M_r = (F_j^{(i-1)})_{1 \leq i,j \leq r} \in \mathcal{M}_r(\C(X))$. Montrer que la famille $(F_1, \ldots, F_r)$ est liée si et seulement si la matrice $M_r$ - n'est pas inversible. - 1. Soient $n \in \N^*$ et $A = (A_{i,j})_{1 \leq i,j \leq n} \in \mathcal{M}_n(\C(X))$. - Pour $p \in \N$, on note $A^{(p)} = (A_{i,j}^{(p)})$ la matrice des dérivées $p^{\text{èmes}}$ des coefficients de A. + 1. Soient $r \in \N^*$ et $(F_1, \ldots, F_r) \in \C(X)^r$. On pose + $M_r = (F_j^{(i-1)})_{1 \leq i,j \leq r} \in + \mathcal{M}_r(\C(X))$. Montrer que la famille $(F_1, \ldots, F_r)$ + est liée si et seulement si la matrice $M_r$ n'est pas inversible. + 1. Soient $n \in \N^*$ et $A = (A_{i,j})_{1 \leq i,j \leq n} \in + \mathcal{M}_n(\C(X))$. Pour $p \in \N$, on note $A^{(p)} = + (A_{i,j}^{(p)})$ la matrice des dérivées $p$-ème des coefficients de $A$. -Montrer que les matrices $A^{(p)}$ pour $p \in \N$ commutent deux à deux si et seulement s'il existe $r \in \N^*, (F_1, \dots, F_r) \in (\C(X))^r$ et des matrices $C_1, \dots, C_r \in \mathcal{M}_n(\C)$ commutant deux à deux telles que $A = F_1C_1 + \cdots + F_rC_r$. + Montrer que les matrices $A^{(p)}$ pour $p \in \N$ commutent deux + à deux si et seulement s'il existe $r \in \N^*, (F_1, \dots, F_r) + \in (\C(X))^r$ et des matrices $C_1, \dots, C_r \in + \mathcal{M}_n(\C)$ commutant deux à deux telles que $A = F_1C_1 + + \cdots + F_rC_r$. #+end_exercice +#+BEGIN_proof + 1. Si la famille est liée, ok. + Réciproquement, si la matrice est non inversible, il existe une + relation sur les colonnes, qui donne une liaison sur les $F_j$. + 2. Si $A = \sum F_i C_i$, les $C_i$ commutant deux à deux, alors les + dérivées $p$-èmes commutent. + + Réciproquement. Soit $Q$ un polynôme et $B = Q A$. Alors $B' = Q + A' + Q' A$. Donc les dérivées de $B$ commutent toutes entre-elles. + + On peut donc supposer que $A\in \M_n(\C[X])$. Alors $A$ n'a qu'un + nombre fini de dérivées. + + On procède par récurrence sur le degré maximal $d$. + + Si $d = 0$, trivial. Si $d = 1$, trivial. + Si $d = 2$, on peut appliquer H.R., sur la dérivée, d'où le résultat. +#+END_proof + + +# ID:8565 #+begin_exercice [ENS SR 2025 # 32] -Soit $S = \begin{pmatrix} (0) & 1 \\ & \ddots & \\ 1 & (0) \end{pmatrix} \in \mathcal{M}_n(\mathbb{R})$. +Soit $S = \big(\delta_{i, n+1-j}\big)\in\M_n(\R)$. 1. Justifier la diagonalisabilité de $S$ et donner ses valeurs propres. 1. Donner une base orthonormale de vecteurs propres de $S$. 1. Caractériser les sous-espaces de $\mathbb{R}^n$ stables par $S$. - - 1. Soient $\omega = \exp(2i\pi/n)$ et $A = \left(\frac{\omega^{jk}}{\sqrt{n}}\right)_{1 \leq j, k \leq n} \in \mathcal{M}_n(\C)$. Calculer les puissances de A. En déduire que $A$ est diagonalisable. - - 1. On suppose $n$ impair. Déterminer les valeurs propres de $A$ et leurs multiplicités. #+end_exercice +# ID:8566 #+begin_exercice [ENS SR 2025 # 33] 1. Soit $M \in \mathcal{M}_n(\C)$ admettant $n$ valeurs propres distinctes. Montrer que si $N \in$ $\mathcal{M}_n(\C)$ est suffisamment proche de $M$, alors $N$ admet $n$ valeurs propres distinctes. 1. Soient $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ et $B \in \mathcal{M}_2(\C)$. À quelle condition la matrice $A + \eps B$ admet-elle deux valeurs propres distinctes pour tout $\eps > 0$ assez petit? 1. Même question en demandant que $A + \eps B$ soit diagonalisable pour tout $\eps > 0$ assez petit. #+end_exercice +# ID:8567 #+begin_exercice [ENS L 2025 # 34] Soient $\mathbb{K}$ un corps et $A \in \mathcal{M}_2(\mathbb{K})$. On suppose que $\chi_A$ est irréductible et qu'il existe $B \in GL_2(\mathbb{K})$ telle que $B^{-1}AB$ commute avec $A$, mais que $B$ ne commute pas avec $A$. Montrer que $B^2$ est scalaire. #+end_exercice #+BEGIN_proof + $B^{-1}AB$ commute avec $A$, donc est un polynôme en $A$, de degré $1$ et est $\neq A$ (car $B$ ne commute pas). (Si le commutant de $A$ était de dimension $3$, l'application $X\mapsto AX$ stabiliserait cet espace de dimension $3$, donc aurait une valeur propre). +Considérons $P_B\colon \K[A]\ra \K[A]$. On a $P_B^2 + \la P_B + \mu \op{Id} = \tilde{0}$. + +Plutôt : $M\in \M_2(\C)\mapsto BM$. Si $B (a + bA) = c + d A$ alors, comme $a+bA$ est forcément inversible, $B$ commute avec $A$. + +Donc $\M_2(\C) = \K[A] + B \K[A]$, et $B(a+bA) = (c+dA)B$, donc $B\K[A] = \K[A] B$, donc $B^2 \K[A] = \K[A]$, donc $B^2\in \K[A]$. + +Mais $B^2 \in \K[B]$, donc $B^2$ est scalaire. #+END_proof +# ID:3473 #+begin_exercice [ENS L 2025 # 35] Soient A, $B$ dans $\mathcal{M}_n(\C)$ telles que $\operatorname{rg}(AB - BA) = 1$. Montrer que $A$ et $B$ sont cotrigonalisables. #+end_exercice +# ID:8572 #+begin_exercice [ENS PLSR 2025 # 36] Soient $A, B \in \mathcal{M}_2(\mathbb{R})$ telles que $\operatorname{rg}(A) = \operatorname{rg}(B) = 1$ et $\operatorname{Im} A = \operatorname{Im} B$. 1. Montrer qu'il existe $P,Q\in \mathrm{GL}_2(\mathbb{R})$ telles que $A=P\begin{pmatrix}1&0\\0&0\end{pmatrix}Q$ et $B=P\begin{pmatrix}\alpha&\beta\\0&0\end{pmatrix}Q$. 1. Pour $P,Q \in GL_2(\mathbb{R})$, on pose $\Psi_{P,Q}: M \mapsto PMQ$. On pose $\tau: M \mapsto M^T$. Soit $\Psi \in \mathcal{L}(\mathcal{M}_2(\mathbb{R}))$ qui conserve le rang. Montrer qu'il existe $P,Q \in \mathrm{GL}_2(\mathbb{R})$ telles que $\Psi = \Psi_{P,Q}$ ou $\Psi = \Psi_{P,Q} \circ \tau$. #+end_exercice +#+BEGIN_proof + 1. On commence par conjuguer $A$ en $\begin{pmatrix}1 & u \\ 0 & 0\end{pmatrix}$, alors $B$ est de la forme $\begin{pmatrix}\a & \b \\ 0 & 0\end{pmatrix}$, puis on agit sur les lignes. + 2. On regarde $\Psi(E_{11})$ et $\Psi(E_{1,2})$. On obtient deux + matrices, qui ont soit la même image, soit le même noyau. Si elles + ont le même noyau, on applique $\tau$, alors elles ont la même image, et quitte à appliquer $P,Q$, on peut supposer $\Psi(E_{11}) = E_{11}$ et $\Psi(E_{1,2}) = \begin{pmatrix}\a & \b \\ 0 & 0\end{pmatrix}$ -#+begin_exercice [ENS PLSR 2025 # 37] -Soient $n, k \in \N^*$, $M = \begin{pmatrix} A & C \\ 0 & B \end{pmatrix}$ avec $A \in \mathcal{M}_n(\C)$, $B \in \mathcal{M}_k(\C)$, $C \in \C$ $\mathcal{M}_{n,k}(\C)$. Montrer que $M$ est diagonalisable si et seulement si $A$ et $B$ sont diagonalisables et il existe $X \in \mathcal{M}_{n,k}(\C)$ telle que $C = AX - XB$. + En fait, on a forcément $\b \neq 0$, donc on peut obtenir $\Psi(E_{1,2}) = \begin{pmatrix}0 & \b \\ 0 & 0\end{pmatrix}$ + + Alors $\Psi(E_{2,1})$ est toujours de rang $1$ avec $E_{11}$, et n'est pas combinaison linéaire des deux précédents, donc $\Psi(E_{2,1}) = \begin{pmatrix}u & 0 \\ v & 0\end{pmatrix}$. En changeant $P,Q$, on peut supposer $u = 0$. Puis on regarde $\Psi(E_{2,2})$. +#+END_proof + + +# ID:8573 +#+begin_exercice [ENS PLSR 2025 # 37] Théorème de similitude de Roth +Soient $n, k \in \N^*$, $M = \begin{pmatrix} A & C \\ 0 & B \end{pmatrix}$ avec $A \in \mathcal{M}_n(\C)$, $B \in \mathcal{M}_k(\C)$, $C \in \mathcal{M}_{n,k}(\C)$. Montrer que $M$ est diagonalisable si et seulement si $A$ et $B$ sont diagonalisables et il existe $X \in \mathcal{M}_{n,k}(\C)$ telle que $C = AX - XB$. #+end_exercice +#+BEGIN_proof +Si $M$ est diagonalisable, par polynôme annulateur, $A,B$ le sont. + $\begin{pmatrix}A & C \\ 0 & B\end{pmatrix}^2 = \begin{pmatrix}A^2 & AC + CB \\ 0 & B^2\end{pmatrix}$ et $\dots^3 = \begin{pmatrix}A^3 & A^2C + ACB+ CB^2 \\ 0& B^3\end{pmatrix}$. + +Si $C = AX - XB$, alors $A^k CB^{n-k} = A^{k+1}XB^{n-k} - A^kX B^{n-k+1}$, et si un polynôme annule à la fois $A$ et $B$, il annulera $M$ (somme télescopique). On peut aussi expliciter une matrice par blocs tel que $\begin{pmatrix}A & AX-XB \\ 0 & B\end{pmatrix}\sim \begin{pmatrix}A & 0 \\ 0 & B\end{pmatrix}$. + +On peut supposer que $A$ et $B$ sont diagonales, auquel cas on peut analyser exactement ce que signifie s'écrire $C = AX-XB$, et vérifier le résultat. +#+END_proof + + +# ID:8594 #+begin_exercice [ENS PLSR 2025 # 38] -Soit $\mathbb K$ un sous-corps de $\mathbb C$. On dit qu'une matrice $M=(m_{i,j})_{1\leqslant i,j\leqslant n}$ de $\mathcal{M}_n(\mathbb{K})$ est de Bourdaud si $\chi_M = \prod (X - m_{i,i})$. +Soit $\mathbb K$ un sous-corps de $\mathbb C$. On dit qu'une matrice $M=(m_{i,j})_{1\leq i,j\leq n}$ de $\mathcal{M}_n(\mathbb{K})$ est de Bourdaud si $\chi_M = \prod (X - m_{i,i})$. 1. Montrer qu'une matrice de $\mathcal{M}_n(\mathbb{K})$ est semblable sur $\mathbb{K}$ à une matrice de Bourdaud si et - seulement si elle est trigonalisable sur $K$. - 1. Montrer qu'une matrice de $S_n(\mathbb{R})$ est de Bourdaud si et + seulement si elle est trigonalisable sur $\K$. + 2. Montrer qu'une matrice de $S_n(\mathbb{R})$ est de Bourdaud si et seulement si elle est diagonale. - 1. Est-il vrai que toute matrice de Bourdaud de + 3. Est-il vrai que toute matrice de Bourdaud de $\mathcal{M}_n(\C)$ est diagonalisable? - 1. On dit que $A$ est normale si $A^TA = AA^T$. Déterminer les + 4. On dit que $A$ est normale si $A^TA = AA^T$. Déterminer les matrices réelles normales et de Bourdaud. #+end_exercice +#+BEGIN_proof + 1. Si on est semblable à une matrice de Bourdaud, on est semblable à une matrice de polynôme caractéristique scindé. + 2. Si $S_n$ est symétrique, et si ses valeurs propres sont sur sa + diagonale, on regarde la plus grande, sur la $i$-ème colonne, on + obtient que $E_i$ est un vecteur propre, etc. + 3. Non, cf $\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}$. + 4. Qu'en est-il des matrices antisymétriques ? Elles ne sont pas de + Bourdaud, car leurs valeurs propres sont imaginaires pures. + Que dire d'une valeur propre d'une matrice normale ? Si $X$ est + vecteur propre, $A^TX$ l'est aussi : $A^T$ préserve les espaces + propres (complexes). + + Mais si la matrice est de Bourdaud, les valeurs propres sont + réelles, donc on est diagonalisable dans une BON, donc symétrique. +#+END_proof + + +# ID:nil #+begin_exercice [ENS SR 2025 # 39] -Soient $n, k \in \N^*$, $M = \begin{pmatrix} A & C \\ 0 & B \end{pmatrix}$ avec $A \in \mathcal{M}_n(\mathbb{R})$, $B \in \mathcal{M}_k(\mathbb{R})$, $C \in \mathcal{M}_{n,k}(\mathbb{R})$. On pose $e^M = \begin{pmatrix} M_1 & \phi_{A,B}(C) \\ M_2 & M_2 \end{pmatrix}$. +Soient $n, k \in \N^*$, $M = \begin{pmatrix} A & C \\ 0 & B \end{pmatrix}$ avec $A \in \mathcal{M}_n(\mathbb{R})$, $B \in \mathcal{M}_k(\mathbb{R})$, $C \in \mathcal{M}_{n,k}(\mathbb{R})$. On pose $e^M = \begin{pmatrix} M_1 & \phi_{A,B}(C) \\ M_2 & M_3 \end{pmatrix}$. 1. Déterminer $M_1, M_2, M_3$. 1. Montrer que $\phi_{A,B}$ est linéaire. @@ -623,335 +771,703 @@ Soient $n, k \in \N^*$, $M = \begin{pmatrix} A & C \\ 0 & B \end{pmatrix}$ avec 1. On suppose que $\phi_{A,B}$ est diagonalisable et que toutes ses valeurs propres sont distinctes. Montrer que $A$ et $B$ sont diagonalisables. #+end_exercice +# ID:8556 #+begin_exercice [ENS SR 2025 # 40] -Si $A, B \in \mathcal{M}_n(\C)$, on pose [A, B] = AB BA. Soit $\mathcal{A} = \{ M \in \mathcal{M}_2(\C), \operatorname{tr}(M) = 0 \}$. +Si $A, B \in \mathcal{M}_n(\C)$, on pose $[A, B] = AB-BA$. Soit $\mathcal{A} = \{ M \in \mathcal{M}_2(\C), \operatorname{tr}(M) = 0 \}$. 1. Montrer que $A$ est un sous-espace vectoriel de $\mathcal{M}_2(\C)$ stable par [,]. - 1. Calculer les [A,B] pour les $A,B \in \{X,Y,H\}$ où $X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ et + 2. Calculer les $[A,B]$ pour les $A,B \in \{X,Y,H\}$ où $X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ et $H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. -$$H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$ + 3. Soit $\rho: A \to \mathcal{M}_n(\C)$ linéaire telle que, pour tous $A, B \in A$, $\rho([A, B]) = [\rho(A), \rho(B)]$. Montrer que $\rho(H)$ admet une valeur propre $\alpha$. - 1. Soit $\rho: A \to \mathcal{M}_n(\C)$ linéaire telle que, pour tous $A, B \in A$, $\rho([A, B]) = [\rho(A), \rho(B)]$. Montrer que $\rho(H)$ admet une valeur propre $\alpha$. + Montrer que $\rho(X)(E_{\alpha}(\rho(H))) \subset \operatorname{Ker}(\rho(H) - (\alpha + 2)I_n)$. -Montrer que $\rho(X)(E_{\alpha}(\rho(H))) \subset \operatorname{Ker}(\rho(H) - (\alpha + 2)I_n)$. - - Montrer que $\rho(Y)$ $(E_{\alpha}(\rho(H))) \subset \text{Ker } (\rho(H) (\alpha 2)I_n)$. - 1. On suppose que, si $V$ est un sous-espace de $\C^n$ stable par tous les $\rho(A)$, pour $A \in \mathcal{A}$, - alors $V = \C^n$ ou $V = \{0\}$. Déterminer les $\rho$ possibles. + Montrer que $\rho(Y)$ $(E_{\alpha}(\rho(H))) \subset \text{Ker } (\rho(H) (\alpha 2)I_n)$. + 1. On suppose que, si $V$ est un sous-espace de $\C^n$ stable par tous les $\rho(A)$, pour $A \in \mathcal{A}$, alors $V = \C^n$ ou $V = \{0\}$. Déterminer les $\rho$ possibles. #+end_exercice +# ID:8557 #+begin_exercice [ENS U 2025 # 41] -Soient $k$ un corps de caractéristique nulle, $E$ un $k$-espace vectoriel de dimension finie et $u \in \mathcal{L}(E)$. On écrit $\pi_u = \prod_i P_i^{n_i}$, le polynôme minimal de $u$, où les $P_i$ sont irréductibles - -distincts et les $n_i$ dans $\N^*$. On pose $f=P_1\times\cdots\times P_r$. On définit une suite en posant $u_0=u$ - - et, pour $n \in \N$, $u_{n+1} = u_n f(u_n) f'(u_n)^{-1}$. +Soient $k$ un corps de caractéristique nulle, $E$ un $k$-espace vectoriel de dimension finie et $u \in \mathcal{L}(E)$. On écrit $\pi_u = \prod_i P_i^{n_i}$, le polynôme minimal de $u$, où les $P_i$ sont irréductibles distincts et les $n_i$ dans $\N^*$. On pose $f=P_1\times\cdots\times P_r$. On définit une suite en posant $u_0=u$ et, pour $n \in \N$, $u_{n+1} = u_n - f(u_n) f'(u_n)^{-1}$. 1. Montrer que $(u_n)$ est bien définie. - 1. Montrer que $(u_n)$ est stationnaire de valeur ultime $d \in \mathcal{L}(E)$ où $d$ est un polynôme en $u$, $u$-d est nilpotent et $d$ est annulé par $f$. + 1. Montrer que $(u_n)$ est stationnaire de valeur ultime $d \in + \mathcal{L}(E)$ où $d$ est un polynôme en $u$, $u-d$ est nilpotent + et $d$ est annulé par $f$. #+end_exercice +# ID:8558 #+begin_exercice [ENS L 2025 # 42] Déterminer le cardinal minimal $p$ d'un sous-groupe $G$ de $\mathrm{GL}_2(\C)$ tel que $\operatorname{Vect}(G) = \mathcal{M}_2(\C)$. Si $G_1$ et $G_2$ conviennent et sont de cardinal $p$, sont-ils conjugués? #+end_exercice +#+BEGIN_proof +Le groupe est nécessairement non commutatif, donc de cardinal $\geq 6$. Le seul groupe de cardinal $6$ est $\mc S_3 \simeq mc D_6$. +#+END_proof + +# ID:8574 #+begin_exercice [ENS L 2025 # 43] -On dit que la propriété $MT(n, \mathbb{K})$ est vraie si, pour tout couple (A, B) de matrices de $\mathcal{M}_n(\mathbb{K})$ telles que, pour tout $\lambda \in \mathbb{K}$, $A + \lambda B$ soit diagonalisable, $A$ et $B$ commutent. +On dit que la propriété $MT(n, \mathbb{K})$ est vraie si, pour tout couple $(A, B)$ de matrices de $\mathcal{M}_n(\mathbb{K})$ telles que, pour tout $\lambda \in \mathbb{K}$, $A + \lambda B$ soit diagonalisable, $A$ et $B$ commutent. 1. Montrer que, si $A$ et $B$ sont dans $\mathcal{M}_n(\mathbb{K})$, diagonalisables et commutent, alors, pour tout $\lambda \in \mathbb{K}$, $A + \lambda B$ est diagonalisable. - - 1. On suppose que $n \ge 2$. La propriété $MT(n, \mathbb{R})$ est-elle vraie? - 1. Montrer que $MT(2,\C)$ est vraie. - 1. On suppose que $n \ge 2$. La propriété $MT(n, \mathbb{F}_2)$ est-elle vraie? - 1. Soit $p \ge 3$ un nombre premier. La propriété $MT(2, \mathbb{F}_p)$ est-elle vraie? + 2. On suppose que $n \ge 2$. La propriété $MT(n, \mathbb{R})$ est-elle vraie? + 3. Montrer que $MT(2,\C)$ est vraie. + 4. On suppose que $n \ge 2$. La propriété $MT(n, \mathbb{F}_2)$ est-elle vraie? + 5. Soit $p \ge 3$ un nombre premier. La propriété $MT(2, \mathbb{F}_p)$ est-elle vraie? #+end_exercice +#+BEGIN_proof + 1. + 2. Non : matrices symétriques. + 3. On suppose $A$ diagonale, de coefficients $\neq$. Si $B$ est non + diagonale : si triangulaire, on peut obtenir deux coefficients + diagonaux égaux, donc non diagonalisable. En général, montrer que + le polynôme caractéristique peut avoir un $\Delta = 0$. + 4. Si on est diagonalisable et inversible, on est l'identité, sinon, + on est une projection. + Soit donc $A$ diagonale avec un bloc de $0$ et un bloc de $1$. Il existe plein de matrices avec lesquels $A$ ne commutent pas, pour lesquelles $A+B$ est inversible. + 5. Soit $A$ diagonale. Existe-t-il $B$ qui ne commute pas telle que $A+\la B$ soit non diagonalisable, ce qui revient à ce que son polynôme ait $\Delta = 0$. +#+END_proof + + +# ID:7698 #+begin_exercice [ENS L 2025 # 44] Quelle est l'image de l'application $f: M \in \mathcal{M}_2(\C) \mapsto \sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)!} M^{2n+1}$ ? #+end_exercice +# ID:8575 #+begin_exercice [ENS SR 2025 # 45] - 1. Soient $A, B \in \mathcal{M}_n(\C)$ telles que AB = BA. Justifier que $e^{A+B} = e^A e^B$. - 1. Soient $A \in \mathcal{M}_n(\C)$ et $P \in GL_n(\C)$. Montrer que $e^{PAP^{-1}} = Pe^AP^{-1}$. - 1. Pour $A \in \mathcal{M}_n(\C)$ convenable, on pose $\log A = \sum_{k=1}^{+\infty} \frac{(-1)^{k-1}}{k} (A I_n)^k$. Pour quelles - -matrices $\log A$ est-il défini? Montrer les égalités $\exp(\log A) = A$ et $\log(\exp A) = A$. Pour chaque égalité, déterminer les matrices $A$ qui la satisfont. - - 1. Montrer que, si $A, B \in \mathcal{M}_n(\C), \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k \underset{k \to +\infty}{\longrightarrow} e^{A+B}$. + 1. Soient $A, B \in \mathcal{M}_n(\C)$ telles que $AB = BA$. + Justifier que $e^{A+B} = e^A e^B$. + 2. Soient $A \in \mathcal{M}_n(\C)$ et $P \in GL_n(\C)$. Montrer que + $e^{PAP^{-1}} = Pe^AP^{-1}$. + 3. Pour $A \in \mathcal{M}_n(\C)$ convenable, on pose $\log A = + \sum_{k=1}^{+\infty} \frac{(-1)^{k-1}}{k} (A I_n)^k$. Pour quelles + matrices $\log A$ est-il défini? Montrer les égalités $\exp(\log + A) = A$ et $\log(\exp A) = A$. Pour chaque égalité, déterminer les + matrices $A$ qui la satisfont. + 1. Montrer que, si $A, B \in \mathcal{M}_n(\C), \left(e^{\frac{A}{k}} + e^{\frac{B}{k}}\right)^k \underset{k \to +\infty}{\longrightarrow} + e^{A+B}$. #+end_exercice +# ID:8576 #+begin_exercice [ENS PLSR 2025 # 46] -Soient $(a_n)_{n\geq 0}\in\C^{\N}$ et $f:z\mapsto\sum_{n=0}^{+\infty}a_nz^n$ de rayon de convergence égal à $+\infty$. - 1. Pour $M \in \mathcal{M}_n(\C)$, justifier la définition de $f^*(M) = \sum_{k=0}^{+\infty} a_k M^k$. - 1. Montrer que $f^*$ est continue. - $\emph{c}$ ) On suppose que $f$ est surjective. Montrer que $f$ induit une surjection de l'ensemble des matrices diagonalisables sur lui-même. - 1. On suppose que, pour tout $\lambda \in \C$, il existe $z \in \C$ tel que $f(z) = \lambda$ et $f'(z) \neq 0$. Montrer que $f^*$ est une surjection de $\mathcal{M}_n(\C)$ sur lui-même. +Soient $(a_n)_{n\geq 0}\in\C^{\N}$ et $f\colon z\mapsto\sum_{n=0}^{+\infty}a_nz^n$ de rayon de convergence égal à $+\infty$. + 1. Pour $M \in \mathcal{M}_n(\C)$, justifier la définition de $f^*(M) + = \sum_{k=0}^{+\infty} a_k M^k$. + 2. Montrer que $f^*$ est continue. + 3. On suppose que $f$ est surjective. Montrer que $f$ induit une + surjection de l'ensemble des matrices diagonalisables sur + lui-même. + 4. On suppose que, pour tout $\lambda \in \C$, il existe $z \in \C$ + tel que $f(z) = \lambda$ et $f'(z) \neq 0$. Montrer que $f^*$ est + une surjection de $\mathcal{M}_n(\C)$ sur lui-même. #+end_exercice +# ID:8595 #+begin_exercice [ENS L 2025 # 47] Soit $d \in \N^*$. On munit $\mathbb{R}^d$ de sa structure euclidienne canonique. Déterminer les $n \in \N^*$ pour lesquels il existe un ensemble $F_n \subset \mathbb{R}^d$ de cardinal $n$ tel que, pour toute partie $G$ de $F_n$, il existe $\omega \in \mathbb{R}^d \setminus \{0\}$ et $b \in \mathbb{R}$ tels que $G = \{x \in F_n, \langle \omega, x \rangle + b > 0\}$. #+end_exercice +#+BEGIN_proof +C'est donc que toute partie $G$ de $F_n$ peut être obtenue par un hyperplan. +Pour $d = 1$, c'est $n = 2$. +Pour $d = 2$, c'est $n = 3$, etc. + + $d+1$ est atteignable, en prenant les vecteurs de la base canonique + $0$. + +Pourquoi est-ce que $d+2$ n'est pas atteignable : car un élément sera dans l'enveloppe convexe des autres. +#+END_proof + +# ID:8729 #+begin_exercice [ENS P 2025 # 48] Pour $\omega \in \mathbb{R}$, on pose $R(\omega) = \begin{pmatrix} 0 & -\omega \\ \omega & 0 \end{pmatrix}$. Soit $\phi : \mathbb{R} \to \mathcal{O}_n(\mathbb{R})$ un morphisme de groupes continu. Montrer qu'il existe $\omega_1, \ldots, \omega_r \in \mathbb{R}$ et $P \in \mathcal{O}_n(\mathbb{R})$ tel que, pour tout $t \in \mathbb{R}$, -$$\phi(t) = P \begin{pmatrix} e^{tR(\omega_1)} & 0 & \cdots & 0 \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & e^{tR(\omega_r)} & 0 \\ 0 & \cdots & 0 & I_{r-2r} \end{pmatrix}.$$ +$$\phi(t) = P \begin{pmatrix} e^{tR(\omega_1)} & 0 & \cdots & 0 \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & e^{tR(\omega_r)} & 0 \\ 0 & \cdots & 0 & I_{r-2r} \end{pmatrix}P^{-1}.$$ #+end_exercice +#+BEGIN_proof +On considère $\Phi(t) = \int_0^t \phi(u)\du$, on a $\Phi(t+r) - \Phi(t) = \int_t^{t+r} \phi(u)\du = \phi(t)\int_0^r \phi(u)\du$, et pour $r$ assez petit, $\int_0^r \phi(u)\du$ est inversible, donc $\phi(t)$ est dérivable, puis c'est une exponentielle. +#+END_proof + +# ID:8577 #+begin_exercice [ENS L 2025 # 49] Soient $u$ et $v$ deux endomorphismes autoadjoints positifs d'un espace euclidien. Montrer que $v \circ u$ est diagonalisable. #+end_exercice +#+BEGIN_proof +Conjuguer par une racine carrée. +#+END_proof +# ID:8741 #+begin_exercice [ENS P 2025 # 50] -Déterminer l'ensemble des symétries linéaires sur $\mathcal{S}_n(\mathbb{R})$ qui fixent un hyperplan et stabilisent l'ensemble $\mathcal{S}_n^{++}(\mathbb{R})$. +Déterminer l'ensemble des symétries linéaires sur +$\mathcal{S}_n(\mathbb{R})$ qui fixent un hyperplan et stabilisent +l'ensemble $\mathcal{S}_n^{++}(\mathbb{R})$. #+end_exercice +#+BEGIN_proof +Si on stabilise $\mc S_n^{++}$, on stabilise sa frontière. +Si $A\in\mc F$ est envoyé sur $B\in\mc F$, pour tout $u$. + +De même, par topologie, sur le bord de $\mc S_n^+$, le rang est +préservé. donc les $X X^T$ sont envoyés sur des $Y Y^T$. + +En particulier, si $s$ est la symétrie parallèlement à $S$, on doit avoir $\rg S\leq 2$. + +De plus $S$ a une valeur propre $\gt 0$ et une $\lt 0$. + +Soit $F$ la somme de ses espaces propres. Alors les matrices de rang +$1$ d'image dans $F$ sont envoyés sur d'autres, d'image dans $F$, donc +$\mc S_F$ est stable par $s$. + +On est ramené au cas $n = 2$. $\mc S_2^{++}$ est l'ensemble décrit par $a,b\gt 0$ et $|c|\lt \sqrt{ab}$. + +On suppose que $S = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$, alors la symétrie doit échanger $a$ et $b$, donc l'hyperplan est $H = \begin{pmatrix}u & v \\ v & u\end{pmatrix}$. + +En général, $S = P D P^T$, et $H = P H P^T$ fonctionnent. C'est la +conjuguée de la symétrie par $U\mapsto P U P^T$. Donc c'est la seule +qui fonctionne. + +En général, si on prend $X$ orthogonal à $\Im S$ alors $X X^T + tS$ ne peut pas être de rang $1$, donc $X X^T$ est dans l'hyperplan. + +Si on prend $\begin{pmatrix}1 & 0 & \a \\ 0 & 1 & 0 \\ \a & 0 & u \end{pmatrix}$. Elle est transformée en $\begin{pmatrix}1 + \phi(\a) & 0 & \a \\ 0 & 1 - \phi(\a) & 0 \\ \a & 0 & u \end{pmatrix}$. Si on prend $u$ énorme, la première est toujours def positive, alors que la seconde non, donc $S$ est exactement l'échange des deux coordonnées $m_{11}$ et $m_{22}$, mais cela ne fonctionne pas. +#+END_proof + + +# ID:8742 #+begin_exercice [ENS P 2025 # 51] Soit $H = (H_{i,j})_{1 \leq i,j \leq n} \in \mathcal{S}_n^{++}(\mathbb{R})$. On suppose que, pour tous $i \neq j, H_{i,j} \leq 0$. Si $(i,j) \in \db{1,n}^2$, on dit que $i$ et $j$ sont connectés s'il existe $m \in \N^*, k_1, \ldots, k_m \in \db{1,n}$ tels que $k_1 = i, k_m = j$ et, pour tout $\ell \in \db{1,m-1}, H_{k_\ell,k_{\ell+1}} \neq 0$. -Montrer que $i$ et $j$ sont connectés si et seulement si $H_{i,j}^{-1} > 0$, où $H_{i,j}^{-1}$ est le coefficient d'indice (i,j) de $H^{-1}$. +Montrer que $i$ et $j$ sont connectés si et seulement si $H_{i,j}^{-1} > 0$, où $H_{i,j}^{-1}$ est le coefficient d'indice $(i,j)$ de $H^{-1}$. #+end_exercice +#+BEGIN_proof +Si on n'est pas connectés, on peut partitionner en deux +sous-ensembles, $H$ est diagonale par blocs, et le $H_{ij}^{-1}$ sera +nul. +On a $H_{i,j}^{-1} = \langle E_i, H^{-1} E_j\rangle = \langle H^{-T} +E_i, E_j\rangle$, et $H^{-T} = \op{Com}(H)$. Sachant que $\det H\gt +0$, on veut montrer que les coefficients de la comatrice sont $\gt 0$. + +Sur la diagonale, c'est classique. + +Quitte à multiplier à droite/gauche par une matrice diagonale +$\sqrt{D}$, on peut supposer $H = I_n - S$, avec $S$ avec des petites +valeurs propres, donc $H^{-1} = \sum_{k=0}^{+\i} S^k$. +#+END_proof + + +# ID:8750 #+begin_exercice [ENS PLSR 2025 # 52] -On considère $n \in \N^*$ et $(A, B) \in \mathcal{A}_{2n}(\mathbb{R})^2$. On pose $C$ = AB et on s'intéresse aux valeurs propres réelles de $C$. +On considère $n \in \mathbb{N}^*$ et $(A, B) \in \mathcal{A}_{2n}(\mathbb{R})^2$. On pose $C = AB$ et on s'intéresse aux valeurs propres réelles de $C$. 1. Donner un exemple de $n$, $A$ et $B$ tels que $\chi_C$ n'admette aucune racine réelle. - - 1. On suppose $A$ inversible. On note $\phi: (\C^{2n})^2 \to \C$ définie par $\phi(X,Y) = X^T A^{-1} Y$. Montrer que les sous-espaces caractéristiques $F_{\lambda}(C)$ de $C$ sont deux à deux $\phi$ -orthogonaux, $i$.e. pour tous $\lambda$ et $\mu$ distinctes dans $\operatorname{Sp} C$, $\forall (X,Y) \in F_{\lambda}(C) \times F_{\mu}(C)$, $\phi(X,Y) = 0$. - - 1. Que peut-on en déduire? + 2. On suppose $A$ inversible. On note $\phi\colon (\mathbb{C}^{2n})^2 \to \mathbb{C}$ définie par $\phi(X,Y) = X^T A^{-1} Y$. Montrer que les sous-espaces caractéristiques $F_{\lambda}(C)$ de $C$ sont deux à deux $\phi$-orthogonaux, i.e. pour tous $\lambda$ et $\mu$ distincts dans $\operatorname{Sp}(C)$, $\forall (X,Y) \in F_{\lambda}(C) \times F_{\mu}(C)$, $\phi(X,Y) = 0$. + 3. Que peut-on en déduire ? #+end_exercice +#+BEGIN_proof + 1. Pour $n = 1$, il n'y a qu'une seule matrice antisymétrique. + Il faut donc $n = 2$, dimension 4, $A,B$ inversibles. Une avec deux blocs $2\times 2$, l'autre avec le bloc du milieu, et $1,-1$ extrémaux : + + $\begin{pmatrix}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{pmatrix} \begin{pmatrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}=\begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}$. On obtient une nouvelle matrice antisymétrique, qui n'a dont pas de racines réelles. + 2. On a $\phi(CX, Y) = \phi(X, CY)$. Écrire Bézout entre $(X-\la)^k$ et $(Y-\mu)^k$, puis $\phi(X,Y) = \phi(X, (PU + QV)Y) = 0$. + 3. On sait que l'espace se décompose comme somme de ces espaces caractéristiques. En particulier, aucun ne contient un vecteur isotrope. Sur $\R$, sur chacun de ces espaces caractéristique la forme $\phi$ est antisymétrique, donc la dimension est paire. +#+END_proof + + +# ID:nil #+begin_exercice [ENS PLSR 2025 # 53] On munit $\mathbb{R}^3$ de sa structure canonique d'espace euclidien orienté, et on note $\mathbf{B}$ sa base canonique. 1. Montrer que, pour tout $u \in \mathbb{R}^3$, il existe un unique endomorphisme $z_u$ de $\mathbb{R}^3$ tel que $\forall (x,y) \in (\mathbb{R}^3)^2$, $\det_{\mathbf{B}}(u,x,y) = \langle z_u(x),y \rangle$, et montrer qu'alors $z_u^{} = -z_u$. - 1. Soient $u \in \mathbb{R}^3$ unitaire et $\theta \in \mathbb{R}$. On munit le plan $\{u\}^{\perp}$ de l'orientation dont les bases directes sont les bases (x,y) de $\{u\}^{\perp}$ telles que (x,y,u) soit une base directe de $\mathbb{R}^3$. On note $r_{u,\theta}$ la rotation de $\mathbb{R}^3$ fixant $u$ et induisant sur $\{u\}^{\perp}$ la rotation d'angle de mesure $\theta$. On note enfin $p_u$ la projection orthogonale sur $\mathbb{R}u$. Exprimer $\operatorname{tr}(r_{u,\theta})$ en fonction de $\theta$, et montrer que $r_{u,\theta} = (\cos\theta)$. id $+(1-\cos\theta)$. $p_u + (\sin\theta)$. $z_u$. + 1. Soient $u \in \mathbb{R}^3$ unitaire et $\theta \in \mathbb{R}$. On munit le plan $\{u\}^{\perp}$ de l'orientation dont les bases directes sont les bases $(x,y)$ de $\{u\}^{\perp}$ telles que $(x,y,u)$ soit une base directe de $\mathbb{R}^3$. On note $r_{u,\theta}$ la rotation de $\mathbb{R}^3$ fixant $u$ et induisant sur $\{u\}^{\perp}$ la rotation d'angle de mesure $\theta$. On note enfin $p_u$ la projection orthogonale sur $\mathbb{R}u$. Exprimer $\operatorname{tr}(r_{u,\theta})$ en fonction de $\theta$, et montrer que $r_{u,\theta} = (\cos\theta)$. id $+(1-\cos\theta)$. $p_u + (\sin\theta)$. $z_u$. 1. Soient $u$,v des vecteurs unitaires de $\mathbb{R}^3$. On pose $\tau=\arccos\left(\langle u,v\rangle\right)$. Soit $(\phi,\psi)\in\mathbb{R}^2$. Justifier que $r_{u,\phi}\circ r_{v,\psi}$ est une rotation, et montrer qu'elle s'écrit $r_{w,\theta}$ pour un vecteur unitaire $w$ et un réel $\theta$ vérifiant $|\cos(\theta/2)|=|\cos(\phi/2)\cos(\psi/2)-\cos(\tau)\sin(\phi/2)\sin(\psi/2)|$. #+end_exercice +# ID:8736 #+begin_exercice [ENS PLSR 2025 # 54] - 1. Soient $A, B \in \mathcal{M}_n(\mathbb{R})$, diagonalisables et telles que AB = BA. Montrer qu'il existe $P \in GL_n(\mathbb{R})$ telle que $PAP^{-1}$ et $PBP^{-1}$ soient diagonales. - 1. Montrer que l'application $\Phi: (S,O) \in \mathcal{S}_n^{++}(\mathbb{R}) \times \mathcal{O}_n(\mathbb{R}) \mapsto SO \in \mathrm{GL}_n(\mathbb{R})$ est bien définie et bijective, et que $\Phi^{-1}$ est continue. - 1. Soit $M \in \mathrm{GL}_n(\mathbb{R})$. Montrer qu'il existe une unique suite de matrices $(M_k)_{k \in \N}$ telle que $M_0 = M$ et $\forall k \in \N, \ M_{k+1} = \frac{M_k}{2}(I_n + (M_k^T M_k)^{-1})$, et étudier sa convergence. + 1. Soient $A, B \in \mathcal{M}_n(\mathbb{R})$, diagonalisables et + telles que $AB = BA$. Montrer qu'il existe $P \in + GL_n(\mathbb{R})$ telle que $PAP^{-1}$ et $PBP^{-1}$ soient + diagonales. + 2. Montrer que l'application $\Phi\colon (S,O) \in + \mathcal{S}_n^{++}(\mathbb{R}) \times \mathcal{O}_n(\mathbb{R}) + \mapsto SO \in \mathrm{GL}_n(\mathbb{R})$ est bien définie et + bijective, et que $\Phi^{-1}$ est continue. + 3. Soit $M \in \mathrm{GL}_n(\mathbb{R})$. Montrer qu'il existe une unique suite de matrices $(M_k)_{k \in \N}$ telle que $M_0 = M$ et $\forall k \in \N, \ M_{k+1} = \frac{M_k}{2}(I_n + (M_k^T M_k)^{-1})$, et étudier sa convergence. #+end_exercice +#+BEGIN_proof + 1. Codiagonalisabilité. + 2. Soit $U\in GL_n(\R)$, on prend la racine carrée de $U U^T$ def positive. + 3. Il $M^T_kM_k$ est def positive, donc $-1$ n'est pas valeur propre. + Par ailleurs si $M_k = S_kO_k$, $M_{k+1} = \frac{S_k O_k}{2} + O_k^{-1}\big(I_n + S_k^T S_k\big)O_k = \frac{S_k}{2} \big(I_n + + (S_k^T S_k)^{-1}\big)O_k$. Si $S_k$ est diagonale, on voit ce + qu'il se passe : $\frac{u_k}{2} + \frac{1}{2 u_k}$, tend vers $1$. + En général, on montre que la plus grand/petite valeur propre se + rapproche de $1$, avec la question 1. +#+END_proof + + +# ID:8737 #+begin_exercice [ENS PLSR 2025 # 55] -On pose $V=\{1,2,\ldots,n\}$ et $F=\mathcal{P}_2(V)$ l'ensemble des paires de $V$. Soient $E\subset F$ et $n_i=|\{j\in V,\ \{i,j\}\in E\}|$ pour $i\in V$. On définit la matrice $L=(\ell_{i,j})\in \mathcal{M}_n(\mathbb{R})$ par $\ell_{i,j}=n_i$ si i=j,-1 si $\{i,j\}\in E$ et 0 sinon. On note $\lambda_1\leqslant\cdots\leqslant\lambda_n$ les valeurs propres (avec multiplicité) de $L$. +On pose $V=\{1,2,\ldots,n\}$ et $F=\mathcal{P}_2(V)$ l'ensemble des paires de $V$. Soient $E\subset F$ et $n_i=|\{j\in V,\ \{i,j\}\in E\}|$ pour $i\in V$. On définit la matrice $L=(\ell_{i,j})\in \mathcal{M}_n(\mathbb{R})$ par $\ell_{i,j}=n_i$ si $i=j$, et $\l_{i,j} = -1$ si $\{i,j\}\in E$ et 0 sinon. On note $\lambda_1\leqslant\cdots\leqslant\lambda_n$ les valeurs propres (avec multiplicité) de $L$. 1. Montrer que $\lambda_1 = 0$. - 1. Montrer que $\lambda_2 = \min_{X \in \{(1,\dots,1)\}^{\perp} \setminus \{0\}} \frac{X^T L X}{X^T X}$. - 1. Pour $S \subset V$, on note $\partial S = \{\{i,j\}, \ \{i,j\} \in E \text{ avec } i \in S \text{ et } j \notin S\}$. + 2. Montrer que $\lambda_2 = \min_{X \in \{(1,\dots,1)\}^{\perp} \setminus \{0\}} \frac{X^T L X}{X^T X}$. + 3. Pour $S \subset V$, on note $\partial S = \{\{i,j\}, \ \{i,j\} \in E \text{ avec } i \in S \text{ et } j \notin S\}$. Montrer que $\frac{\lambda_2}{2} \leqslant \min_{\substack{S \subset V \\ 0 < |S| \leqslant \frac{n}{2}}} \frac{|\partial S|}{|S|}$. #+end_exercice +#+BEGIN_proof + 1. La somme des colonnes fait $0$. + 2. $L$ est symétrique. + 3. Si on prend un tel $S$, on note $s = |S|$. On prend $X$ avec des $n-s$ pour $k\in S$ et $-s$ si $k\not\in S$. + On trouve $X^T L X = n^2 |\partial S|$ et $X^T X = n s (n-s)$. +#+END_proof + + + +# ID:8774 #+begin_exercice [ENS P 2025 # 56] Pour $A, B \in \mathcal{S}_n(\mathbb{R})$ on note $A \geq B$ lorsque $A - B \in \mathcal{S}_n^+(\mathbb{R})$. Si $A \in \mathcal{S}_n^{++}(\mathbb{R})$, on écrit $A = P \operatorname{Diag}(\lambda_1, \dots, \lambda_n) P^{-1}$ avec $P \in \mathcal{O}_n(\mathbb{R})$ et les $\lambda_i > 0$, et on pose, pour $r \in \mathbb{R}$, $A^r = P \operatorname{Diag}(\lambda_1^r, \dots, \lambda_n^r) P^{-1}$ ; cette définition ne dépend pas de l'écriture de $A$ sous la forme précédente. 1. Montrer que $M \mapsto M^{-1}$ est décroissante sur $\mathcal{S}_n^{++}(\mathbb{R})$. - 1. Est-ce que $M \mapsto M^2$ est croissante sur $S_n^{++}(\mathbb{R})$ ? - 1. Montrer que $M\mapsto M^r$ est convexe sur $\mathcal{S}_n^{++}(\mathbb{R})$ lorsque $r\in[-1,0]$. Ceci signifie que, pour tous $A,B\in\mathcal{S}_n^{++}(\mathbb{R})$ et tout $t\in[0,1[$, $(tA+(1-t)B)^r\leqslant tA^r+(1-t)B^r$. + 2. Est-ce que $M \mapsto M^2$ est croissante sur $S_n^{++}(\mathbb{R})$ ? + 3. sV2 Soit $r\in \interval]{-1, 0}[$. Montrer que pour $x\gt 0$, $x^r = \frac{\sin(-r\pi)}{\pi} \int_0^{+\infty} t^r (x + t)^{-1} \dt$ + 4. Montrer que $M\mapsto M^r$ est convexe sur $\mathcal{S}_n^{++}(\mathbb{R})$ lorsque $r\in[-1,0]$. Ceci signifie que, pour tous $A,B\in\mathcal{S}_n^{++}(\mathbb{R})$ et tout $t\in[0,1[$, $(tA+(1-t)B)^r\leqslant tA^r+(1-t)B^r$. #+end_exercice +#+BEGIN_proof + 1. Soit $A\geq B$, il existe $P$ inversible tel que $A = P P^T$ et $B + = P D P^T$, avec $D$ à coefficients $\lt 1$, on est bon. + Alternative : $B$ admet une racine carrée, et $A\geq B\ssi B^{-1/2} A B^{-1/2}\geq I_n \ssi B^{1/2} A^{-1}B^{1/2}\leq I_n \ssi A^{-1} \leq B^{-1}$. + 2. Prenons $A = \begin{pmatrix}2 & a \\ a & 1\end{pmatrix}$ et $B = \begin{pmatrix}2' & b \\ b & 1'\end{pmatrix}$ + Si $\langle AX, X\rangle\leq \langle BX, X\rangle$, mais $B^2$ penche trop. Ce sera encore le cas de la plus grande valeur propre. + + $B-A = \begin{pmatrix}2'-2 & b-a \\ b-a & 1'-1\end{pmatrix}$, on veut $(2'-2)(1'-1) \geq (b-a)^2$ + + Alors $A^2 = \begin{pmatrix} 2 + a^2 & 3a \\ 3a & 1+a^2\end{pmatrix}$. + + $B^2 - A^2 = \begin{pmatrix}2' - 2 + b^2 - a^2 & 0 \\ 0 & 0\end{pmatrix}$, et ce coefficient en haut à gauche n'a pas de raison d'être positif. + + En pratique : on peut prendre $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ et $B = \begin{pmatrix} 3 & 1 \\ 1 & 1 \end{pmatrix}$. + 3. Le cas $r = -1$ se fait par coréduction. + + L'expression intégrale donnée reste valable pour une matrice, et la convexité de $M\mapsto M^r$ se ramène à la convexité des autres. + + $M^r = \frac{\sin(-r\pi)}{\pi} \int_0^{+\infty} t^r (M + tI_n)^{-1} \dt$ + + Alternative : il suffit de le faire pour $r = -1/2$, peut-être. +#+END_proof + + +# ID:8751 #+begin_exercice [ENS PLSR 2025 # 57] -On dit d'une norme $\mathcal{N}$ sur $\mathcal{M}_d(\mathbb{R})$ qu'elle est invariante orthogonalement lorsque $\forall M \in \mathcal{M}_d(\mathbb{R}), \ \forall (P,Q) \in \mathcal{O}_d(\mathbb{R})^2, \ \mathcal{N}(M) = \mathcal{N}(PMQ)$. +On dit d'une norme $\mathcal{N}$ sur $\mathcal{M}_d(\R)$ qu'elle est invariante orthogonalement lorsque $\forall M \in \mathcal{M}_d(\R), \forall (P,Q) \in \mathcal{O}_d(\R)^2, \mathcal{N}(M) = \mathcal{N}(PMQ)$. 1. Donner un exemple d'une telle norme. - 1. Soit $A \in \mathcal{M}_d(\mathbb{R})$, montrer qu'il existe $(P,Q) \in \mathcal{O}_d(\mathbb{R})^2$ tel que $A = PDQ$ où $D = \text{Diag}(\sigma_1, \ldots, \sigma_r, 0, \ldots, 0)$ avec $\forall i \in [1, r], \ \sigma_i > 0$. - 1. On se donne une norme $\mathcal N$ invariante orthogonalement sur $\mathcal M_d(\mathbb R)$. - On note $T(A) = \sup\{\|AX\|, \|X\| = 1\}$ où $\|\|$ désigne la norme euclidienne canonique. Montrer qu'il existe $\alpha > 0$ tel que $\forall A \in \mathcal{M}_d(\mathbb{R}), \operatorname{rg}(A) = 1 \Rightarrow T(A) = \alpha \mathcal{N}(A)$. + 1. Soit $A \in \mathcal{M}_d(\R)$, montrer qu'il existe $(P,Q) \in + \mathcal{O}_d(\R)^2$ tel que $A = PDQ$ où $D = + \operatorname{Diag}(\sigma_1, \dots, \sigma_r, 0, \dots, 0)$ avec + $\forall i \in \db{1,r}, \sigma_i \gt 0$. + 1. On se donne une norme $\mathcal{N}$ invariante orthogonalement sur + $\mathcal{M}_d(\R)$. On note $T(A) = \sup\{\|AX\|,\, \|X\| = 1\}$ + où $\|\cdot\|$ désigne la norme euclidienne canonique. Montrer + qu'il existe $\alpha > 0$ tel que $\forall A \in + \mathcal{M}_d(\R), \operatorname{rg}(A) = 1 \Rightarrow T(A) = + \alpha \mathcal{N}(A)$. #+end_exercice +#+BEGIN_proof + 1. Norme subordonnée 2, ou somme des carrés des coefficients. + 2. Si $A A^T = B B^T$, et inversible, alors $B^{-1}A$ est orthogonale. + En général, l'énoncé revient à trouver une BON (l'image de $Q$) telle que $\langle A e_i, A e_i\rangle = \sigma_i$. + 3. C'est immédiat. +#+END_proof + +# ID:8752 #+begin_exercice [ENS SR 2025 # 58] On munit $\mathbb{R}^n$ de sa structure euclidienne canonique et $\mathcal{M}_n(\mathbb{R})$ de la norme d'opérateur qui lui est subordonnée. 1. Soit $A \in \mathcal{S}_n(\mathbb{R})$. - - Que dire des valeurs propres et des espaces propres de $\it A$ ? - On note $\lambda_1, \ldots, \lambda_r$ les valeurs propres distinctes de A. - - $\text{Soient} \ x \in \mathbb{R}^n \setminus \{0\}, \ \alpha \in \mathbb{R} \ \text{et} \ y = Ax \alpha x. \ \text{Montrer que} \ \min_{1 \leqslant i \leqslant r} |\lambda_i \alpha| \leqslant \frac{\|y\|}{\|x\|}$. - 1. Soient $A \in \mathcal{M}_n(\mathbb{R})$ diagonalisable, $P \in \operatorname{GL}_n(\mathbb{R})$ telle que $P^{-1}AP$ soit diagonale, $\lambda_1, \ldots, \lambda_r$ les valeurs propres distinctes de A. Soient enfin $B \in \mathcal{M}_n(\mathbb{R})$ et $\alpha$ une valeur propre de $A$ + $B$. Montrer que $\min_{1 \le i \le r} |\lambda_i \alpha| \le ||P||_{\operatorname{op}} ||P^{-1}||_{\operatorname{op}} ||B||_{\operatorname{op}}$. + - Que dire des valeurs propres et des espaces propres de $\it A$ ? On note $\lambda_1, \ldots, \lambda_r$ les valeurs propres distinctes de A. + - Soient $x \in \mathbb{R}^n \setminus \{0\}, \, \a \in \mathbb{R}$ et $y = Ax - \a x$. Montrer que $\min\limits_{1 \leqslant i \leqslant r} |\lambda_i - \a| \leqslant \frac{\|y\|}{\|x\|}$. + 1. Soient $A \in \mathcal{M}_n(\mathbb{R})$ diagonalisable, $P \in \operatorname{GL}_n(\mathbb{R})$ telle que $P^{-1}AP$ soit diagonale, $\lambda_1, \ldots, \lambda_r$ les valeurs propres distinctes de A. Soient enfin $B \in \mathcal{M}_n(\mathbb{R})$ et $\alpha$ une valeur propre de $A + B$. Montrer que $\min\limits_{1 \le i \le r} |\lambda_i- \alpha| \le ||P||_{\operatorname{op}} ||P^{-1}||_{\operatorname{op}} ||B||_{\operatorname{op}}$. #+end_exercice +#+BEGIN_proof + 1. + 1. + 2. Écrire dans une BON. + 2. Discuter selon si $\a$ est une valeur propre de $A$. Sinon, on a $(\a I_n - A) v = Bv$, donc $v = (\a I_n - A)^{-1} B v$. +#+END_proof + +# ID:4285 #+begin_exercice [ENS nil 2025 # 59] -Soient $S \in \mathcal{S}_n^{++}(\mathbb{R})$ et $A \in \mathcal{A}_n(\mathbb{R})$. Montrer que SA est diagonalisable sur $\C$. +Soient $S \in \mathcal{S}_n^{++}(\mathbb{R})$ et $A \in \mathcal{A}_n(\mathbb{R})$. Montrer que $SA$ est diagonalisable sur $\C$. #+end_exercice +# ID:8753 #+begin_exercice [ENS P 2025 # 60] -Soit $n \in \N^*$. On appelle forme quadratique sur $\mathbb{R}^n$ toute application $q : \mathbb{R}^n \to \mathbb{R}$ telle qu'il existe $(a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_n(\mathbb{R})$ telle que $q(x) = \sum_{1 \leqslant i,j \leqslant n} a_{i,j} x_i x_j$ pour tout $x = \sum_{1 \leqslant i,j \leqslant n} a_{i,j} x_i x_j$ - $(x_1, \ldots, x_n) \in \mathbb{R}^n$. Soit $G$ un sous-groupe fini de $GL_n(\mathbb{R})$ tels que $\{0\}$ et $\mathbb{R}^n$ sont les seuls sous-espaces de $\mathbb{R}^n$ stables par tous les éléments de $G$. Montrer que les formes quadratiques invariantes par $G$ constituent une droite vectorielle. +Soit $n \in \N^*$. On appelle forme quadratique sur $\mathbb{R}^n$ toute application $q : \mathbb{R}^n \to \mathbb{R}$ telle qu'il existe $(a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_n(\mathbb{R})$ telle que $q(x) = \sum_{1 \leqslant i,j \leqslant n} a_{i,j} x_i x_j$ pour tout $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$. Soit $G$ un sous-groupe fini de $GL_n(\mathbb{R})$ tels que $\{0\}$ et $\mathbb{R}^n$ sont les seuls sous-espaces de $\mathbb{R}^n$ stables par tous les éléments de $G$. Montrer que les formes quadratiques invariantes par $G$ constituent une droite vectorielle. #+end_exercice +#+BEGIN_proof + $q(x) = X^T A X$. La forme est invariante par $g$ si $G^T A G = A$. +Si on prend $A = \sum_{g\in G} g^T g$, c'est invariant, et de trace $\gt 0$. + +Unicité : Si une matrice orthogonale commute avec une matrice $A$, +alors elle commute avec sa partie symétrique, qui aura un espace +propre non trivial, à moins que ce ne soit $\la I_n$, alors c'est la +même forme quadratique. + +En général, si $g^T B g = B$ et $g^T A g = A$, avec $A,B$ symétrique, ou encore $g^T B = Bg^{-1}$, alors l'image de $B$ est stable par $g^T$, donc l'orthogonal de l'image est stable par $g$, et si $B$ est inversible, on est ramené au cas précédent par coréduction. +#+END_proof + + +# ID:8754 #+begin_exercice [ENS SR 2025 # 61] -Soit $n \ge 2$. On munit $\mathbb{R}^n$ de sa structure euclidienne canonique. Soit $H \in \mathcal{S}_n(\mathbb{R})$. On pose $\nabla_H : (x,y) \in (\mathbb{R}^n)^2 \mapsto x^T H y$ et $Q_H : x \in \mathbb{R}^n \mapsto x^T H x$. +Soit $n \ge 2$. On munit $\mathbb{R}^n$ de sa structure euclidienne canonique. Soit $H \in \mathcal{S}_n(\mathbb{R})$. On pose $\nabla_H : (x,y) \in (\mathbb{R}^n)^2 \mapsto x^T H y$ et $Q_H\colon x \in \mathbb{R}^n \mapsto x^T H x$. 1. Soit $H \in \mathcal{S}_n(\mathbb{R})$. Exprimer la norme d'opérateur de $H$ à l'aide de $Q_H$. - 1. Soient $m, n \in \N^*$. On munit $\mathbb{R}^n$ et $\mathbb{R}^m$ de leur structure euclidienne canonique. Si $A \in \mathcal{M}_{m,n}(\mathbb{R})$, comment déterminer la norme d'opérateur de $A$ pour ces normes? + 2. Soient $m, n \in \N^*$. On munit $\mathbb{R}^n$ et $\mathbb{R}^m$ de leur structure euclidienne canonique. Si $A \in \mathcal{M}_{m,n}(\mathbb{R})$, comment déterminer la norme d'opérateur de $A$ pour ces normes? - 1. Soient $J$, $K$ deux ensembles finis non vides, $(a_{j,k})_{(j,k)\in J\times K}\in (\mathbb{R}^+)^{J\times K}$. On suppose qu'il existe $C_1$ et $C_2$ tels que : $\forall j\in J, \sum_{k\in K}a_{j,k}\leqslant C_1$ et $\forall k\in K, \sum_{j\in J}a_{j,k}\leqslant C_2$. On + 3. Soient $J$, $K$ deux ensembles finis non vides, $(a_{j,k})_{(j,k)\in J\times K}\in (\mathbb{R}^+)^{J\times K}$. On suppose qu'il existe $C_1$ et $C_2$ tels que : $\forall j\in J, \sum_{k\in K}a_{j,k}\leqslant C_1$ et $\forall k\in K, \sum_{j\in J}a_{j,k}\leqslant C_2$. -ordonne $J$ et $K$ et on note $A$ la matrice des $a_{j,k}$. Montrer que $||A||_{\text{op}} \leqslant \sqrt{C_1 C_2}$. - - 1. Pour $n \in \N^*$, $J$ = $K$ = [1, n], on pose, pour $1 \leqslant j, k \leqslant n$, $a_{j,k}^n = \frac{1}{(j-k)^2}$ si $j \neq k$, et $a_{j,k}^n=0$ sinon. On note enfin $A^n=\left(a_{j,k}^n\right)_{1\leqslant j,k\leqslant n}\in\mathcal{M}_n(\mathbb{R})$. Déterminer la limite de $(\|A^n\|_{\rm op})_{n\geq 1}$. + On ordonne $J$ et $K$ et on note $A$ la matrice des $a_{j,k}$. Montrer que $||A||_{\text{op}} \leqslant \sqrt{C_1 C_2}$. + 4. Pour $n \in \N^*$, $J = K = \db{1,n}$, on pose, pour $1 \leqslant j, k \leqslant n$, $a_{j,k}^n = \frac{1}{(j-k)^2}$ si $j \neq k$, et $a_{j,k}^n=0$ sinon. On note enfin $A^n=\left(a_{j,k}^n\right)_{1\leqslant j,k\leqslant n}\in\mathcal{M}_n(\mathbb{R})$. Déterminer la limite de $(\|A^n\|_{\rm op})_{n\geq 1}$. #+end_exercice +#+BEGIN_proof + 1. + 2. C'est la plus grande valeur propre de $A^T A$. + 3. Cauchy-Schwarz donne $\lN Ax\rN^2 \leq C_1 C_2 \lN x\rN^2$. + 4. On peut majorer la somme par ligne/colonne par $2\frac{\pi^2}{6}$, la question précédente donne une majoration. + On peut minorer par $u^T A^n u$, où $u = (1,\dots, 1)$. +#+END_proof + + +# ID:8775 #+begin_exercice [ENS PLSR 2025 # 62] -L'espace $\mathbb{R}^n$ est muni de sa norme euclidienne canonique et $\mathcal{M}_n(\mathbb{R})$ de la norme subordonnée notée $\| \|_{op}$. Si $M \in GL_n(\mathbb{R})$, on définit le conditionnement de $M$ comme le réel cond $(M) = ||M||_{\text{op}} ||M^{-1}||_{\text{op}}$. - 1. Calculer cond(M) dans le cas où $M$ est symétrique définie positive. - 1. Montrer que, pour toute matrice $M \in GL_n(\mathbb{R})$, $cond(M) \geq 1$ et $cond(M^T) = cond(M)$. - 1. Que dire des matrices $M \in \mathrm{GL}_n(\mathbb{R})$ telles que $\mathrm{cond}(M) = 1$ ? - 1. Pour $A$ et $B$ dans $S_n^{++}$, montrer que $Cond(A+B) \leq max(Cond(A), Cond(B))$. +L'espace $\mathbb{R}^n$ est muni de sa norme euclidienne canonique et $\mathcal{M}_n(\mathbb{R})$ de la norme subordonnée notée $\| \|_{op}$. Si $M \in GL_n(\mathbb{R})$, on définit le conditionnement de $M$ comme le réel $\op{cond}(M) = ||M||_{\text{op}} ||M^{-1}||_{\text{op}}$. + 1. Calculer $\op{cond}(M)$ dans le cas où $M$ est symétrique définie positive. + 2. Montrer que, pour toute matrice $M \in GL_n(\mathbb{R})$, $\op{cond}(M) \geq 1$ et $\op{cond}(M^T) = \op{cond}(M)$. + 3. Que dire des matrices $M \in \mathrm{GL}_n(\mathbb{R})$ telles que $\op{cond}(M) = 1$ ? + 4. Pour $A$ et $B$ dans $S_n^{++}$, montrer que $\op{cond}(A+B) \leq max(\op{cond}(A), \op{cond}(B))$. #+end_exercice +#+BEGIN_proof + 1. La plus grand valeur propre, divisée par la plus petite. + 2. Trivial. + 3. $\la \mc O_n(\R)$. + 4. On a l'interprétation en termes des valeurs propres. La plus grande valeur propre de $A+B$ est $\leq \la_A + \la_B$, et sa plus petite est $\geq \mu_A + \mu_B$. +#+END_proof + +# ID:nil #+begin_exercice [ENS SR 2025 # 63] -On note $E$ l'ensemble des matrices de $S_n^+(\mathbb{R})$ de rang 1. - 1. Soit $A \in \mathcal{M}_n(\mathbb{R})$. Montrer que $A \in E$ si et seulement s'il existe $U \in \mathcal{M}_{n,1}(\mathbb{R})$ tel que - $A = UU^T$. Soit $a \in \mathcal{C}^0(\mathbb{R}^+, E)$. - 1. Montrer l'équivalence entre les deux assertions suivantes : - $(\alpha)$ il existe $u: \mathbb{R}^+ \to \mathcal{M}_{n,1}(\mathbb{R})$ continue telle que $\forall t \in \mathbb{R}^+, a(t) = u(t)u(t)^T$ ; - $(\beta)$ il existe $z: \mathbb{R}^+ \to \mathcal{M}_{n,1}(\mathbb{R})$ continue telle que $\forall t \in \mathbb{R}^+, z(t)^T a(t) z(t) > 0$. - 1. Soient $0 \le b \le c$. On suppose qu'il existe $(i,j) \in [1,n]^2$ avec $i \ne j$ tel que, pour tout $t \in [b,c], a_{i,i}(t) > 0$ et $a_{j,j}(t) > 0$. Montrer qu'il existe $z : [b,c] \to \mathcal{M}_{n,1}(\mathbb{R})$ continue telle que $\forall t \in [b, c], z(t)^T a(t)z(t) > 0$ et, en outre, $z(b) = e_i, z(c) = \pm e_i$ (les $e_k$ sont les - vecteurs de la base canonique). - 1. En considérant l'ensemble des $d \ge 0$ tels qu'existe $z : [0,d] \to \mathcal{M}_{n,1}(\mathbb{R})$ continue vérifiant $\forall t \in [0, d], z(t)^T a(t) z(t) > 0$ et $z(d) = \pm e_i$, montrer que a vérifie la propriété $(\alpha)$. +On note $E$ l'ensemble des matrices de $S_n^+(\R)$ de rang 1. + 1. Soit $A \in \M_n(\R)$. Montrer que $A \in E$ si et seulement s'il + existe $U \in \M_{n,1}(\R)$ tel que $A = UU^T$. + 2. E Soit $a \in \mc C^0(\R^+, E)$. + 3. Montrer l'équivalence entre les deux assertions suivantes : + + il existe $u\colon \R^+ \ra \M_{n,1}(\R)$ continue telle que $\forall t \in \R^+, a(t) = u(t)u(t)^T$ ; + + il existe $z: \R^+ \ra \M_{n,1}(\R)$ continue telle que $\forall t \in \R^+,\, z(t)^T a(t) z(t) \gt 0$. + 4. Soient $0 \le b \le c$. On suppose qu'il existe $(i,j) \in [1,n]^2$ avec $i \ne j$ tel que, pour tout $t \in [b,c], a_{i,i}(t) \gt 0$ et $a_{j,j}(t) \gt 0$. Montrer qu'il existe $z : [b,c] \ra \M_{n,1}(\R)$ continue telle que $\forall t \in [b, c], z(t)^T a(t)z(t) \gt 0$ et, en outre, $z(b) = e_i, z(c) = \pm e_i$ (les $e_k$ sont les vecteurs de la base canonique). + 5. En considérant l'ensemble des $d \ge 0$ tels qu'existe $z : [0,d] \ra \M_{n,1}(\R)$ continue vérifiant $\forall t \in [0, d], z(t)^T a(t) z(t) \gt 0$ et $z(d) = \pm e_i$, montrer que a vérifie la propriété $(i)$. #+end_exercice +#+BEGIN_proof + 1. + 2. + 3. On peut définir $u$ comme un vecteur directeur unitaire de l'image de $a$, multipliée par la norme de $a$. + Le résultat est faux : la deuxième condition ne donne pas la continuité de la norme de $A$. +#+END_proof + + +# ID:8755 #+begin_exercice [ENS SR 2025 # 64] Soient $n \ge 2$, $a: [0,1] \to \mathcal{S}_n^+(\mathbb{R})$ continue et $A = \int_0^1 a(t) dt$. 1. Montrer que $A$ appartient à $S_n^+(\mathbb{R})$. - 1. Donner une condition nécessaire et suffisante pour que A=0. Exprimer Ker(A). - 1. Montrer que $M=\left(\frac{1}{1+i+j}\right)_{1\leqslant i,j\leqslant n}$ est dans $\mathcal{S}_n^{++}(\mathbb{R})$. - 1. On suppose a à valeurs dans l'ensemble des matrices de projecteurs orthogonaux. Donner une condition pour que $A$ soit une matrice de projecteur orthogonal. - 1. Soit $\Gamma: x \in \mathbb{R}^{+*} \mapsto \int_0^{+\infty} e^{-t} t^{x-1} dt$. Soient $0 < \alpha < \beta$. - Montrer que $\begin{pmatrix} \Gamma(2\alpha) & \Gamma(\alpha+\beta) \\ \Gamma(\alpha+\beta) & \Gamma(2\beta) \end{pmatrix}$ est dans $\mathcal{S}_2^{++}(\mathbb{R})$. - 1. En déduire que $\ln(\Gamma)$ est convexe + 2. Donner une condition nécessaire et suffisante pour que $A=0$. Exprimer $\Ker(A)$. + 3. Montrer que $M=\left(\frac{1}{1+i+j}\right)_{1\leqslant i,j\leqslant n}$ est dans $\mathcal{S}_n^{++}(\mathbb{R})$. + 4. On suppose $a$ à valeurs dans l'ensemble des matrices de projecteurs orthogonaux. Donner une condition pour que $A$ soit une matrice de projecteur orthogonal. + 5. Soit $\Gamma: x \in \mathbb{R}^{+*} \mapsto \int_0^{+\infty} e^{-t} t^{x-1} dt$. Soient $0 < \alpha < \beta$. + Montrer que $\begin{pmatrix} \Gamma(2\alpha) & \Gamma(\alpha+\beta) \\ \Gamma(\alpha+\beta) & \Gamma(2\beta) \end{pmatrix}$ est dans $\mathcal{S}_2^{++}(\mathbb{R})$. + 6. En déduire que $\ln(\Gamma)$ est convexe #+end_exercice +#+BEGIN_proof + 1. + 2. $\Ker (A) = \bigcap \Ker a(t)$. + 3. C'est une matrice de Gram. + 4. Pour $x$ dans l'image, il faut que $x$ soit dans toutes les images. Pour $x$ dans le noyau, il faut qu'il soit dans tous les noyaux, donc $a$ est constante. + 5. C'est une matrice symétrique, de trace strictement positive, et il reste à montrer que $\Gamma(2\a) \Gamma(2\b)\gt \Gamma(\a+\b)^2$, ce qui est Cauchy-Schwarz. + 6. On a la convexité pour la moitié. +#+END_proof + +# ID:8734 #+begin_exercice [ENS P 2025 # 65] Soit $(O_n)_{n\geq 0}$ une suite d'ouverts non majorés de $\mathbb{R}^{+*}$. Montrer qu'il existe $x\in\mathbb{R}^{+*}$ tels que, pour tout $n\in\N$, l'ensemble $O_n\cap x \N$ soit infini. #+end_exercice +# ID:8756 #+begin_exercice [ENS L 2025 # 66] -Soit $E$ un ensemble non vide. Soit $d: E^2 \to \mathbb{R}$ vérifiant, pour tous $x, y, z \in E$ : +Soit $E$ un ensemble non vide. Soit $d\colon E^2 \ra \R$ vérifiant, pour tous $x, y, z \in E$ : + $d(x,y) = d(y,x)$, + $d(x,y) = 0 \Leftrightarrow x = y$, - + $d(x,y) \leqslant \max(d(x,z), d(z,y))$. - Ainsi $d$ est une distance sur $E$. - Pour $x \in E$ et $r \in \mathbb{R}^+$, on note $B(x,r) = \{y \in E, d(x,y) \leq r\}$ la boule fermée de centre $x$ et de rayon $r$. On suppose que, pour tout $x \in E$ et tous $r$, r' vérifiant 0 < $r$ < r', on a $B(x,r) \subsetneq B(x,r')$. Enfin, on suppose qu'il existe une suite d'éléments de $E$ dense dans (E,d). - Montrer qu'il existe une suite $(z_n)_{n\geq 0}$ d'éléments de $E$ et une suite $(r_n)_{n\geq 0}$ d'éléments de $\mathbb{R}^{+*}$ telles que : $\forall n\in\N, B(z_{n+1},r_{n+1})\subset B(z_n,r_n)$ et $\bigcap_{n\in\N}B(z_n,r_n)=\emptyset$. -#+end_exercice + + $d(x,y) \leq \max(d(x,z), d(z,y))$. +Ainsi $d$ est une distance sur $E$. +Pour $x \in E$ et $r \in \R^+$, on note $B(x,r) = \{y \in E, d(x,y) \leq r\}$ la boule fermée de centre $x$ et de rayon $r$. On suppose que, pour tout $x \in E$ et tous $r, r'$ vérifiant $0 \lt r \lt r'$, on a $B(x,r) \subsetneq B(x,r')$. Enfin, on suppose qu'il existe une suite d'éléments de $E$ dense dans $(E,d)$. + +Montrer qu'il existe une suite $(z_n)_{n\geq 0}$ d'éléments de $E$ et une suite $(r_n)_{n\geq 0}$ d'éléments de $\R_+^*$ telles que : $\forall n\in\N, B(z_{n+1},r_{n+1})\subset B(z_n,r_n)$ et $\bigcap_{n\in\N}B(z_n,r_n)=\emptyset$. +#+end_exercice +#+BEGIN_proof +Si $z\in B(x,r)$, alors $B(x,r)\subset B(z,r)$, donc elles sont égales. + +L'idée est de ne pas faire tendre $r_n$ vers $0$ mais vers $1$ par exemple. + +On prend $B(z_0, 2)$. On peut se débarrasser de $z_0$ en diminuant un peu le rayon : prendre un élément qui n'est pas dans $B(z_0, 2-\eps)$. + +Plus généralement, si $B(a, r)$ contient un $z_i$, on veut s'en +débarrasser, sans trop diminuer le rayon, et bien comme $B(a,r) = +B(z_i,r)$, c'est bon. +#+END_proof + + +# ID:8733 #+begin_exercice [ENS PLSR 2025 # 67] On note $E$ l'ensemble des fonctions lipschitziennes 1-périodiques de $\mathbb R$ dans $\mathbb R$. 1. Pour $\alpha \in ]0,1]$ et $f \in E$, on pose -$$||f||_{\alpha} = \sup_{x \in \mathbb{R}} |f(x)| + \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^{\alpha}}.$$ + $$||f||_{\alpha} = \sup_{x \in \mathbb{R}} |f(x)| + \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^{\alpha}}.$$ -Démontrer que $\| \|_{\alpha}$ est une norme sur $E$. + Démontrer que $\| \|_{\alpha}$ est une norme sur $E$. 1. On note $F = E \cap \mathcal{C}^1(\mathbb{R}, \mathbb{R})$. Démontrer que $F$ est un fermé de $E$ pour la norme $\| \cdot \|_1$. #+end_exercice +#+BEGIN_proof + 1. Clair. + 2. Si $f_n \ra_{\lN \cdot\rN_1} f$, alors la convergence est uniforme, donc $f$ est continue. -#+begin_exercice [ENS P 2025 # 68] -Soient $E$ l'espace des suites réelles $(x_n)_{n\geq 0}$ nulles à partir d'un certain rang, et $T\in\mathcal{L}(E)$. On suppose $T$ continu pour la norme $\|\ \|_1$ et pour la norme $\|\ \|_{\infty}$. Montrer que $T$ est continu pour la norme $\|\ \|_2$. + Le sup des taux d'accroissements est la même chose que le sup de $|f'|$. +#+END_proof + + +#+begin_exercice [ENS P 2025 # 68] :todo: +Soient $E$ l'espace des suites réelles $(x_n)_{n\geq 0}$ nulles à partir d'un certain rang, et $T\in\mathcal{L}(E)$. On suppose $T$ continu pour la norme $\|\ \|_1$ et pour la norme $\|\ \|_{\infty}$. Montrer que $T$ est continue pour la norme $\|\ \|_2$. #+end_exercice +#+BEGIN_proof :todo: +On considère une suite de suites $(u^n)$ dont la norme $2$ tend vers $0$. En particulier, la norme infinie tend vers $0$, donc $T(u^n)$ tend, en norme infinie, vers $0$. +La fonction $T$ est lipschitzienne pour les deux normes. + +Si on suppose $\lN u^n\rN_1 \ra 0$, alors $\lN T(u^n)\rN_1\ra 0$, ce qui implique $\lN T(u^n)\rN_2 \ra 0$. + +Si on suppose $\lN u^n\rN_1$ bornée. On a $\lN T(u^n)\rN_2^2 \leq \lN +T(u^n)\rN_{\i} \lN T(u^n)\rN_1\ra 0$. + +En général : on découpe la suite pour regrouper en des groupes de +termes dont la norme $1$ vaut $1$. (Il y a un nombre fini de groupes à +chaque fois car les suites sont nulles APCR). + +#+END_proof + + +# ID:8732 #+begin_exercice [ENS SR 2025 # 69] Soit $E = C^0([0, 1], \mathbb{R})$. 1. La forme linéaire $\phi: f \mapsto f(0)$ est-elle continue pour $\|\cdot\|_{\infty}$ ? pour $\|\cdot\|_{1}$ ? Dans chaque cas calculer l'adhérence de $\operatorname{Ker} \phi$. - 1. Soit $\phi: f \mapsto \int_0^1 f(x) \cos(2\pi x) dx$. Montrer que $\phi$ est continue pour $\| \cdot \|_1$ et calculer sa norme subordonnée. + 1. Soit $\phi\colon f \mapsto \int_0^1 f(x) \cos(2\pi x) dx$. Montrer que $\phi$ est continue pour $\| \cdot \|_1$ et calculer sa norme subordonnée. #+end_exercice +#+BEGIN_proof + 1. Oui, non, $E$. + 2. On a $\int_0^1 f(x)\cos (2\pi x)\dx\leq \int_0^1 |f(x)|\dx$, et on peut s'en approcher. +#+END_proof + +# ID:8731 #+begin_exercice [ENS L 2025 # 70] -Soit $E = C^0([0,1], \mathbb{R})$. - -$$\text{Si } a=(a_n)_{n\geq 0}\in [0,1]^{\N} \text{, on pose, pour } f,g\in E, \langle f,g\rangle_a=\sum_{n=0}^{+\infty}\frac{f(a_n)\,g(a_n)}{2^n}.$$ +Soit $E = C^0([0,1], \mathbb{R})$. Si $a=(a_n)_{n\geq 0}\in [0,1]^{\N}$, on pose, pour $f,g\in E$, $\quad\displaystyle \langle f,g\rangle_a=\sum_{n=0}^{+\infty}\frac{f(a_n)\,g(a_n)}{2^n}$. 1. Donner une condition nécessaire et suffisante pour que $\langle \; , \; \rangle_a$ soit un produit scalaire sur $E$. On note alors $\|\; \|_a$ la norme associée. - 1. Si $a,b \in [0,1]^{\N}$ vérifient les hypothèses de a), donner une condition nécessaire et suffisante pour que $\| \|_a$ et $\| \|_b$ soient équivalentes. + 2. Si $a,b \in [0,1]^{\N}$ vérifient les hypothèses de a), donner une condition nécessaire et suffisante pour que $\| \|_a$ et $\| \|_b$ soient équivalentes. #+end_exercice +#+BEGIN_proof + 1. Densité. + 2. Il faut qu'elles prennent les mêmes valeurs (d'un point de vue ensembliste). Si $a_0$ n'est pas pris, on peut prendre des fonctions arbitrairement pics en $a_0$, dont la valeur pour $\lN \rN_2$ est arbitrairement petite. +#+END_proof -#+begin_exercice [ENS nil 2025 # 71] + +# ID:8739 +#+begin_exercice [ENS 2025 # 71] Soient $n \ge 2$ et $f \in \mathcal{C}^0(\mathbb{R}^n, \mathbb{R})$ telle que, pour tout $x \in \mathbb{R}$, $f^{-1}(\{x\})$ est compact. 1. Montrer que $f$ admet un extremum global. + 2. Montrer que $f(x)$ possède une limite quand $\lN x\rN\ra +\i$. #+end_exercice +#+BEGIN_proof + 1. Sinon, on a une suite de presque maximaux, et presque minimaux qui tendent vers l'infini, qu'on peut relier par de longs arcs de cercles, et qui contredisent la compacité. + 2. Idem : s'il y a deux valeurs d'adhérence distinctes… +#+END_proof + +# ID:8738 #+begin_exercice [ENS P 2025 # 72] Soient $(E, \langle , \rangle)$ un espace préhilbertien de dimension infinie et $K$ une partie bornée de $E$ dont la frontière est compacte. Montrer que $K$ est d'intérieur vide dans $E$. -Peut-on généraliser le résultat à n'importe quel espace vectoriel normé de dimension infinie? +Peut-on généraliser le résultat à n'importe quel espace vectoriel normé de dimension infinie ? #+end_exercice +#+BEGIN_proof +Si $K$ est d'intérieur non vide, on prend un point dedans, on essaie +de le relier à une boule de rayon $R$. À chaque point de la frontière, +on associe le point de la boule. C'est continu, et surjectif. +Dans un espace préhilbertien Dans un espace préhilbertien, on peut +prendre les droites dirigées par une famille libre orthonormée, qui +touchent la frontière, de manière éloignée du point d'origine, donc +leur distance les uns aux autres est grande. +#+END_proof + + +# ID:8740 #+begin_exercice [ENS P 2025 # 73] -Pour $x$,y réels et $\eps>0$, on dit que $x\approx_{\eps}y$ s'il existe $k\in\mathbb{Z}$ tel que |x-y-k|< - - $\eps$. Soient $\lambda_1, \lambda_2$ deux réels non nuls. Montrer que $\frac{\lambda_1}{\lambda_2} \notin \mathbb{Q}$ si et seulement si, pour tout $(a_1, a_2) \in [0, 1]^2$ et tout $\eps > 0$, il existe $x \in \mathbb{R}$ tel que $x\lambda_1 \approx_{\eps} a_1$ et $x\lambda_2 \approx_{\eps} a_2$. +Pour $x,y$ réels et $\eps>0$, on dit que $x\approx_{\eps}y$ s'il existe $k\in\mathbb{Z}$ tel que $|x-y-k|<\eps$. Soient $\lambda_1, \lambda_2$ deux réels non nuls. Montrer que $\frac{\lambda_1}{\lambda_2} \notin \mathbb{Q}$ si et seulement si, pour tout $(a_1, a_2) \in [0, 1]^2$ et tout $\eps > 0$, il existe $x \in \mathbb{R}$ tel que $x\lambda_1 \approx_{\eps} a_1$ et $x\lambda_2 \approx_{\eps} a_2$. #+end_exercice +#+BEGIN_proof +Si $\frac{\la_1}{\la_2}\in\Q$, on se ramène à $\la_1 = 1$ et +$\la_2\in\Q$, alors la fonction est périodique, géométriquement… +Réciproquement, on regarde les $(x, x\la)$, pour $\la\not\in\Q$. On prend $x = k + a_1 \pm \eps$, alors $x\la = k\la + a_1 \la \pm \eps'$ on veut $(k+a_1) \la \simeq a_2 [1]$, c'est-à-dire $k\la \simeq a_2 - a_1 \la [1]$, ce qui s'obtient par densité des parties fractionnaires $\{k \la\}$. +#+END_proof + + +# ID:8578 #+begin_exercice [ENS P 2025 # 74] Soient $E$ un espace vectoriel normé de dimension finie $n \ge 2$ et $C$ une partie non vide, convexe et bornée de $E$. Montrer que la frontière de $C$ est connexe par arcs. #+end_exercice +#+BEGIN_proof +Pas trop de difficulté je pense. Fixer un point dans $C$. Pour deux +points sur la frontière, on est ramené à $n = 2$, prendre +l'application $\theta\mapsto s(\theta)$, où $s(\theta)$ est +l'intersection du rayon issu de $x$ et de la frontière. Il faut +justifier la continuité. S'il y avait un défaut de continuité, alors la frontière contiendrait un segment, dont la droite passe par $x$. +#+END_proof -#+begin_exercice [ENS PLSR 2025 # 75] -Soient $E$ un espace vectoriel normé et $f: E \to E$ une application telle que f(0) = 0 et $\forall x, y \in E, ||f(x) f(y)|| = ||x y||$. -On pose, pour $x, y \in E$, ||f(x) - f(y)|| = ||x - y||. $\left\| \frac{f(x) + f(y)}{2} - f\left(\frac{x + y}{2}\right) \right\|$. +# ID:8791 +#+begin_exercice [ENS PLSR 2025 # 75] Mazur-Ulam +Soient $E$ un espace vectoriel normé et $f\colon E \to E$ une application telle que $f(0) = 0$ et $\forall x, y \in E, ||f(x)- f(y)|| = ||x- y||$. - 1. Montrer que $\forall x, y \in E, df(x, y) \leq \frac{1}{2} \|x y\|$. - 1. Montrer que $f$ est linéaire si et seulement si df est identiquement nulle. - 1. Trouver une fonction vérifiant les propriétés de la fonction $f$, non linéaire et non surjective. - d) On suppose que $f$ est surjective. Montrer que $f$ est linéaire. +On pose, pour $x, y \in E$, $df(x,y) = \left\| \frac{f(x) + f(y)}{2} - f\left(\frac{x + y}{2}\right) \right\|$. + + 1. Montrer que $\forall x, y \in E, df(x, y) \leq \frac{1}{2} \|x - y\|$. + 2. Montrer que $f$ est linéaire si et seulement si $df$ est identiquement nulle. + 3. Trouver une fonction vérifiant les propriétés de la fonction $f$, + non linéaire et non surjective. + 4. On suppose que $f$ est surjective. + 1. sV2 Construire une isométrie $g$ telle que $dg(x,0) = 2 df(x,0)$. + 2. Montrer que $f$ est linéaire. #+end_exercice +#+BEGIN_proof + 1. L'image $f(\frac{x+y}{2})$ est à une distance $\leq \frac{\lN x-y\rN}{2}$ de $f(x)$ et de $f(y)$, donc à une distance $\leq \frac{\lN x-y\rN}{2}$ de leur moyenne. + 2. Si $df$ est nulle, alors $f(2x) = f(x)$, puis par densité, $f(\la x) = \la f(x)$. + 3. Pour non surjective, un shift sur les polynômes. -#+begin_exercice [ENS P 2025 # 76] -On munit $E = \mathcal{C}^0([0,1],\mathbb{R})$ des normes $\| \|_2$ et $\| \|_{\infty}$. Soit $(n_k)_{k\geq 0}$ une suite strictement croissante d'entiers naturels. Soit $F = \text{Vect}(x \mapsto x^{n_k}, k \geq 0)$. À quelle condition $F$ est-il dense dans $E$ pour la norme $\| \|_2$ ? pour la norme $\| \|_{\infty}$ ? + Pour non linéaire, le shift sur les polynômes + on met $\lN P\rN_{\i}$ comme coefficient constant. + 4. Si la norme est strictement convexe, c'est trivial. + + On prend $b = 0$ et on suppose $f(a) = a$. + + $f$ réalise une bijection sur l'ensemble des points $z$ tel que $\lN z\rN = \frac{\lN a\rN}{2}$ et $\lN z-a\rN = \frac{\lN a\rN}{2}$. + + On suppose que $u = \frac{a}{2}$ est envoyé ailleurs que sur $u$ : $f(u) = u + v$. + + Soit $s\colon u +h \mapsto u-h$. + + Alors $\lN s(f(u)) - f(u)\rN$ est deux fois plus grand, et c'est + la même chose que $\lN f^{-1}(s(f(u))) - u\rN$, et encore que $\lN + f^{-1}(s(f(s(u)))) - u\rN$. Donc on peut trouver une isométrie qui + préserve $a,0$ et qui double $df(u)$, ce qui est contradictoire. +#+END_proof + + +# ID:8779 +#+begin_exercice [ENS P 2025 # 76] Muntz-Szasz +Soit $E = \mathcal{C}^0([0,1],\mathbb{R})$. Soit $(n_k)_{k\geq 0}$ une suite strictement croissante d'entiers naturels. Soit $F = \text{Vect}(x \mapsto x^{n_k}, k \geq 0)$. À quelle condition $F$ est-il dense dans $E$ pour la norme $\lN \cdot\rN_2$ ? pour la norme $\lN \cdot\rN_{\i}$ ? #+end_exercice +#+BEGIN_proof +Puis c'est équivalent à ce que pour tout $i$, la distance, pour la norme $2$, de $F_k$ à $x^i$ tende vers $0$. Si $P$ est un polynôme $\lN P \rN_2 = \int_0^1 P(t)^2 \dt = \sum \frac{a_i a_k}{n_k+n_i+1}$. On suppose $a_0 = 1$, peut-on rendre ça arbitrairement petit… -#+BEGIN_exercice [ENS L 2025 # 77] -Soit $f \in \mathcal{C}^0(\mathbb{R}, \mathbb{R})$. On note $D = \{\ell 2^{-k} + 2^{-k}[0, 1] ; (k, \ell) \in \mathbb{Z}^2\}$. Pour tout - intervalle $I$ de $D$, on note $\log(I)$ la longueur de $I$ et on pose $M_I(f) = \frac{1}{\log(I)} \int_I f$. On - pose $||f|| = \sup \left\{ \frac{1}{\log(I)} \int_{\Gamma} |f M_I(f)| \; ; \; I \in D \right\}$. - 1. On suppose ||f|| finie. Soit $m \in \N^*$, $(I, J) \in D^2$ avec $I \subset J$ tels que $\log(J) = 2^m \log(I)$. Démontrer que $|M_I(f) M_I(f)| \le 2m||f||$ - $2^m \log(I)$. Démontrer que $|M_I(f) M_J(f)| \le 2m ||f||$. +On trouve si et seulement si $\sum \frac{1}{n_k}$ diverge. + +Pour $\lN \cdot\rN_{\i}$, il faut $n_0 = 0$. Il faut aussi que $\sum \frac{1}{n_k}$ diverge. Réciproquement, en utilisant $\lN f-g\rN_{\i}\leq \lN f'-g'\rN_2$, on peut conclure. +#+END_proof + + + +#+BEGIN_exercice [ENS L 2025 # 77] :todo: +Soit $f \in \mathcal{C}^0(\mathbb{R}, \mathbb{R})$. On note $D = \{\ell 2^{-k} + 2^{-k}[0, 1] ; (k, \ell) \in \mathbb{Z}^2\}$. Pour tout intervalle $I$ de $D$, on note $\op{long}(I)$ la longueur de $I$ et on pose $M_I(f) = \frac{1}{\op{long}(I)} \int_I f$. On + pose $||f|| = \sup \left\{ \frac{1}{\op{long}(I)} \int_{\Gamma} |f M_I(f)| \; ; \; I \in D \right\}$. + 1. On suppose $||f||$ finie. Soit $m \in \N^*$, $(I, J) \in D^2$ avec $I \subset J$ tels que $\op{long}(J) = 2^m \op{long}(I)$. Démontrer que $|M_I(f)- M_J(f)| \le 2m||f||$. 1. On suppose que ||f|| = 1 et $M_{[0,1]}(f) = 0$. - On note $F_k = \{I \in D : I \subset [0,1], M_I(f) > 5k \text{ et } I \text{ maximal pour cette propriété}\}$. On pose $\Omega_k = \bigcup_{I \in F_k} I \text{ et et } \log(\Omega_k) = \sum_{I \in F_k} \log(F_I)$. + On note $F_k = \{I \in D : I \subset [0,1], M_I(f) > 5k \text{ et } I \text{ maximal pour cette propriété}\}$. On pose $\Omega_k = \bigcup_{I \in F_k} I \text{ et et } \op{long}(\Omega_k) = \sum_{I \in F_k} \op{long}(F_I)$. -Montrer que, pour $k \geq 1$, $\log(\Omega_k) \leqslant \frac{1}{3} \log(\Omega_{k-1})$. +Montrer que, pour $k \geq 1$, $\op{long}(\Omega_k) \leqslant \frac{1}{3} \op{long}(\Omega_{k-1})$. #+END_exercice +# ID:8780 #+begin_exercice [ENS PLSR 2025 # 78] On munit les espaces $\ell^1_{\mathbb{Z}}(\mathbb{R})$ et $\ell^2_{\mathbb{Z}}(\mathbb{R})$ de leurs normes usuelles $\| \cdot \|_1$ et $\| \cdot \|_2$. On pose $H = \left\{ u \in \mathbb{R}^{\mathbb{Z}} \; ; \; \sum_{n \in \mathbb{Z}} u_n^2 (1 + n^2) < +\infty \right\}$. 1. Définir un produit scalaire sur $H$. Écrire l'inégalité de Cauchy-Schwarz. - 1. Quelles inclusions a-t-on entre $\ell^1_{\mathbb{Z}}(\mathbb{R})$, $\ell^2_{\mathbb{Z}}(\mathbb{R})$ et H? Montrer que ces inclusions sont continues (i.e. les injections canoniques sont continues). - 1. Soit $u \in \mathbb{R}^{\mathbb{Z}}$. Montrer que $u \in H$ si et seulement si l'application $\mu_u : H \to H$ définie par $\forall v \in H, \ \mu_u(v) = u * v \text{ avec } (u * v)_n = \sum_{i=1}^n u_i v_i \text{ est bien définie et continue.}$ + 1. Quelles inclusions a-t-on entre $\ell^1_{\mathbb{Z}}(\mathbb{R})$, $\ell^2_{\mathbb{Z}}(\mathbb{R})$ et $H$ ? Montrer que ces inclusions sont continues (i.e. les injections canoniques sont continues). + 1. Soit $u \in \mathbb{R}^{\mathbb{Z}}$. Montrer que $u \in H$ si et seulement si l'application $\mu_u : H \to H$ définie par$\forall v \in H, \ \mu_u(v) = u * v \text{ avec } (u * v)_n = \sum_{i=1}^n u_i v_i \text{ est bien définie et continue.}$ #+end_exercice -#+begin_exercice [ENS P 2025 # 79] -On note $\ell^1$ l'ensemble des suites sommables de $\mathbb{R}^{\N}$. On munit $\ell^1$ de la norme - définie, pour $u = (u_n)_{n \geq 0}$, par $||u||_1 = \sum_{n=0}^{+\infty} |u_n|$. - Soient $(u^k)_{k\in\N}$ une suite d'éléments de $\ell^1$ et $u\in\ell^1$. Montrer l'équivalence entre : +#+begin_exercice [ENS P 2025 # 79] :todo: +On note $\ell^1$ l'ensemble des suites sommables de $\mathbb{R}^{\N}$. On munit $\ell^1$ de la norme définie, pour $u = (u_n)_{n \geq 0}$, par $||u||_1 = \sum_{n=0}^{+\infty} |u_n|$. +Soient $(u^k)_{k\in\N}$ une suite d'éléments de $\ell^1$ et $u\in\ell^1$. Montrer l'équivalence entre : + la suite $(u^{\overline{k}})_{k\in\N}$ converge vers $u$ pour la norme $\| \|_1$, + pour toute suite $(\phi_n)_{n\in\N}$ bornée, $\sum_{n=0}^{+\infty} \phi_n u_n^k \underset{k\to+\infty}{\longrightarrow} \sum_{n=0}^{+\infty} \phi_n u_n$. #+end_exercice +#+BEGIN_proof + $i)\Rightarrow ii)$ est trivial + Réciproquement, on veut prendre $\phi_n = \sign (u_n^k - u_n)$, s'il est fixé à partir d'un certain rang. Si ce n'est pas le cas, alors forcément $u_n = 0$ et la somme de ces termes tend vers $0$ par CVD ?? !! Non. +#+END_proof + +# ID:8787 #+begin_exercice [ENS L 2025 # 80] On note $S=\{z\in\C,\,|z|=1\}$ et $\Gamma=\{\gamma\in\mathcal{C}^0([0,1],S)\,\,;\,\,\gamma(0)=\gamma(1)=1\}$. 1. Soit $\gamma\in\Gamma$, montrer qu'il existe $\theta:[0,1]\to\mathbb{R}$ continue telle que $\forall t,\,\,\gamma(t)=e^{i2\pi\theta(t)}$. - 1. On prend $\gamma_0, \gamma_1 \in \Gamma$. On note $F$ la propriété : « il existe $h \in \mathcal{C}([0,1]^2,S)$ tel que $\forall x \in [0,1], \ h(x,\cdot) \in \Gamma, \ h(0,\cdot) = \gamma_0$ et $h(1,\cdot) = \gamma_1$ ». On pose $\gamma_0 = 1$ et $\gamma_1 : t \mapsto e^{2i\pi t}$. Montrer que $\gamma_0$ et $\gamma_1$ ne vérifient pas $F$. - 1. On note $D$ le disque fermé unité de $\C$. Existe-t-il $f \in \mathcal{C}^0(D,S)$ telle que $f|_S = \mathrm{id}$ ? + 2. On prend $\gamma_0, \gamma_1 \in \Gamma$. On note $F$ la propriété : « il existe $h \in \mathcal{C}([0,1]^2,S)$ tel que $\forall x \in [0,1], \ h(x,\cdot) \in \Gamma, \ h(0,\cdot) = \gamma_0$ et $h(1,\cdot) = \gamma_1$ ». On pose $\gamma_0 = 1$ et $\gamma_1 : t \mapsto e^{2i\pi t}$. Montrer que $\gamma_0$ et $\gamma_1$ ne vérifient pas $F$. + 3. On note $D$ le disque fermé unité de $\C$. Existe-t-il $f \in \mathcal{C}^0(D,S)$ telle que $f|_S = \mathrm{id}$ ? #+end_exercice +#+BEGIN_proof + 1. relèvement continue + 2. $\gamma \mapsto \theta(1)- \theta(0)$ est bien définie et cette association est continue. + 3. Non, cf 2), pour $\gamma_r = f(re^{i \cdot})$ +#+END_proof -#+BEGIN_exercice [ENS PLSR 2025 # 81] + +#+BEGIN_exercice [ENS PLSR 2025 # 81] :todo: 1. Soit $f \in \mathcal{C}^2(\mathbb{R}, \mathbb{R})$ telle qu'il existe $x^* \in \mathbb{R}$ vérifiant $f(x^*) = 0$ et $f'(x^*) \neq 0$. - On définit par récurrence une suite $(x_k)$ avec $x_0 \in \mathbb{R}$ et $x_{k+1} = x_k \frac{f(x_k)}{f'(x_k)}$. - Montrer qu'il existe $\eps > 0$ tel que, pour $x_0 \in [x^* \eps, x^* + \eps]$, la suite $(x_k)$ est bien définie et converge vers $x^*$. - 1. Avec $f: x \mapsto e^x 1$, quelles sont les valeurs de $x_0 \in \mathbb{R}$ pour lesquelles la suite $(x_k)$ - précédente est stationnaire? c) On revient au cas général et on suppose f'' > 0 et f' ne s'annule pas sur $\mathbb{R}$. Pour quelles valeurs de $x_0 \in \mathbb{R}$ la suite $(x_k)$ est-elle stationnaire? + On définit par récurrence une suite $(x_k)$ avec $x_0 \in \mathbb{R}$ et $x_{k+1} = x_k \frac{f(x_k)}{f'(x_k)}$. + + Montrer qu'il existe $\eps > 0$ tel que, pour $x_0 \in [x^* \eps, x^* + \eps]$, la suite $(x_k)$ est bien définie et converge vers $x^*$. + 2. Avec $f\colon x \mapsto e^x- 1$, quelles sont les valeurs de $x_0 \in \mathbb{R}$ pour lesquelles la suite $(x_k)$ précédente est stationnaire? + 3. On revient au cas général et on suppose $f'' > 0$ et $f'$ ne s'annule pas sur $\mathbb{R}$. Pour quelles valeurs de $x_0 \in \mathbb{R}$ la suite $(x_k)$ est-elle stationnaire? #+END_exercice -#+begin_exercice [ENS L 2025 # 82] -Soit $f \in \mathcal{C}^0([a,b],[a,b])$. On suppose dans les questions a) et b) que $f$ n'a pas de +#+begin_exercice [ENS L 2025 # 82] :todo: +Soit $f \in \mathcal{C}^0([a,b],[a,b])$. On suppose dans les questions 1) et 2) que $f$ n'a pas de point de période 2, c'est-à-dire que $\forall x \in [a,b], f(x) \neq x \Rightarrow (f \circ f)(x) \neq x$. - 1. Soit $c \in [a, b]$ tel que f(c) > $c$. Montrer que pour tout $k \in \N^*$, $f^k(c) > c$. + 1. Soit $c \in [a, b]$ tel que $f(c) >c$. Montrer que pour tout $k \in \N^*$, $f^k(c) > c$. 1. Soit $x_0 \in [a,b]$, on pose pour tout $n$, $x_{n+1} = f(x_n)$. Démontrer que la suite $(x_n)$ converge. 1. Démontrer que la suite $(x_n)$ converge pour tout choix de $x_0$ si et seulement si $f$ n'a pas de point de période 2. #+end_exercice +# ID:nil # Classique #+begin_exercice [ENS PLSR 2025 # 83] 1. Déterminer la nature des séries $\sum \frac{\sin n}{n}, \sum \frac{\sin^2 n}{n}, \sum \frac{|\sin n|}{n}$. 1. Soit $x \in \mathbb{R} \setminus \mathbb{Q}$ et $Q \in \N^*$. Montrer qu'il existe $p \in \mathbb{Z}$ et $q \in [1, Q]$ tels que $|qx p| \leqslant \frac{1}{Q}$. @@ -960,203 +1476,382 @@ Soit $f \in \mathcal{C}^0([a,b],[a,b])$. On suppose dans les questions a) et b) 1. On admet que $\pi$ est irrationnel. Déterminer la nature de la série $\sum \frac{1}{n \sin(n)}$. #+end_exercice +# ID:8592 #+begin_exercice [ENS P 2025 # 84] Soit $(a_n)$ une suite de réels décroissante de limite nulle. Pour $P\subset \N$, on note $A(P)=\sum_{n\in P}a_n$. On suppose $A(\N)=A_\infty\in \mathbb{R}$. Montrer que - $$\{A(P),\ P\in\mathcal{P}(\N)\}=[0,A_{\infty}] \text{ si et seulement si } \forall n\in\N,\ a_n\leqslant \sum_{k=n+1}^{+\infty}a_k.$$ #+end_exercice +#+BEGIN_proof +Simple. +#+END_proof -#+begin_exercice [ENS L 2025 # 85] - 1. Pour quels réels *s* la somme $\sum_{n,m\in\N^*} \frac{|n-m|^s}{nm(n^2-m^2)^2}$ est-elle finie? - $\begin{array}{l} \textbf{\textit{b})} \ \ \text{Pour } n=(n_1,n_2)\in \mathbb{Z}^2 \text{, on note } |n|=\sqrt{n_1^2+n_2^2}. \\ \\ \text{Pour quels réels } s \ \text{la somme} \ \sum_{(n,m)\in (\mathbb{Z}^2\backslash \{0\})^2} \frac{|n-m|^s}{|n||m|(1+(|n|-|m|)^2)} \ \text{est-elle finie}? \\ \end{array}$ + +#+begin_exercice [ENS L 2025 # 85] :todo: + 1. Pour quels réels $s$ la somme $\sum_{n\neq m\in\N^*} \frac{|n-m|^s}{nm(n^2-m^2)^2}$ est-elle finie? + 2. E Pour $n=(n_1,n_2)\in \mathbb{Z}^2$ on, on note $|n|=\sqrt{n_1^2+n_2^2}$. + 3. Pour quels réels $s$ la somme $\sum_{(n,m)\in (\mathbb{Z}^2\backslash \{0\})^2} \frac{|n-m|^s}{|n||m|(1+(|n|-|m|)^2)}$ est-elle finie ? #+end_exercice +#+BEGIN_proof :todo: + 1. On peut prendre $n\gt m$. Poser $u = n-m$ et sommer sur $u$ et $m$ : $\sum_{u,m\geq 1}\frac{u^s}{m (m+u) u^2 (2 m+ u)}$. Pour $u$ fixé, on trouve facilement un équivalent de la somme. + 2. + 3. Sommer sur $m$, puis sur les $n$ tel que $|n-m|\in [k, k+4]$. Ce + sont les $n$ dans un anneau $C_k$. On somme $\sum_{n\in C_{k,m}} \frac{1}{|n|} \frac{1}{1 + (|n|- |m|)^2}$. On peut par ailleurs supposer que $|n|\geq |m|$. + Il semble plus simple de prendre la norme infinie en haut, les $n\in C_{k,m}$ sont clairs, ils sont sur les quatre bords d'un carré. Pour le bord de droite, … +#+END_proof + + +# ID:8788 #+begin_exercice [ENS PLSR 2025 # 86] On note $S$ l'ensemble des suites croissantes à termes dans $\N \setminus \{0,1\}$. - 1. Pour $a \in S$, montrer que $\phi(a) = \sum_{k=0}^{+\infty} \left( \prod_{k=0}^{n} \frac{1}{a_k} \right)$ appartient à ]0,1]. - 1. Montrer que $\phi$ définit une bijection de $S$ sur ]0,1]. + 1. Pour $a \in S$, montrer que $\phi(a) = \sum_{k=0}^{+\infty} \left( \prod_{k=0}^{n} \frac{1}{a_k} \right)$ appartient à $]0,1]$. + 1. Montrer que $\phi$ définit une bijection de $S$ sur $]0,1]$. 1. Donner une condition nécessaire et suffisante sur $a \in S$ pour que $\phi(a) \in \mathbb{Q}$. #+end_exercice +#+BEGIN_proof + 1. Sans difficulté. + 2. Sans difficulté. + 3. Si la suite est constante APCR, on est bien rationnel. -#+begin_exercice [ENS L 2025 # 87] + Réciproquement, si on suppose que la suite converge vers $\frac{p}{q}$, prendre un rang tel que $\frac{1}{a_{k+1}} + \frac{1}{(a_{k+1})^2}+\dots\lt \frac{1}{q}$, on obtient une contradiction. +#+END_proof + + +#+begin_exercice [ENS L 2025 # 87] :todo: Soit $f: \N \to \mathbb{R}^{+*}$ décroissante de limite nulle. Soit $\phi: \N \to \N$ croissante. On suppose que, pour tout $\alpha \in \mathbb{R}^{+*}$, il existe une unique suite $(n_i)_{i \in \N}$ telle que $\alpha = \sum_{i=0}^{+\infty} f(n_i)$ et, pour tout $i \in \N$, $n_{i+1} \geq \phi(n_i)$. Montrer que $\phi(0)=0$ et, pour tout $n\in\N^*$, $f(n-1)=\sum_{i=0}^{+\infty}f\left(\phi^i(n)\right)$, où $\phi^i$ désigne l'itérée $i$-ème de $\phi$ pour la composition des applications. #+end_exercice +# ID:8744 #+begin_exercice [ENS P 2025 # 88] -Soit $f: \mathbb{R} \to \mathbb{R}$. Montrer l'équivalence entre les conditions suivantes : - + f(x) = O(x); - + $\sum_{r \neq els} f(a_n)$ converge absolument pour toute série $\sum_{r \neq els} a_n$ absolument convergente à termes +Soit $f\colon \mathbb{R} \to \mathbb{R}$. Montrer l'équivalence entre les conditions suivantes : + + $f(x) = O(x)$; + + $\sum f(a_n)$ converge absolument pour toute série $\sum a_n$ absolument convergente à termes réels + $\sum f(a_n)$ converge pour toute série $\sum a_n$ absolument convergente à termes réels. #+end_exercice +#+BEGIN_proof + $i)\Rightarrow ii) \Rightarrow iii)$. +Réciproquement, assez simple, si $f(x)$ n'est pas un $O(x)$, on peut faire diverger les choses. +#+END_proof + + +# ID:86 #+begin_exercice [ENS P 2025 # 89] -Soit $f: \mathbb{R} \to \mathbb{R}$ telle que $\sum f(a_n)$ converge pour toute série convergente $\sum a_n$ à termes réels. Montrer qu'il existe un réel $\lambda$ tel que $f(x) = \lambda x$ pour $x$ voisin de 0. +Soit $f\colon \mathbb{R} \to \mathbb{R}$ telle que $\sum f(a_n)$ converge pour toute série convergente $\sum a_n$ à termes réels. Montrer qu'il existe un réel $\lambda$ tel que $f(x) = \lambda x$ pour $x$ voisin de $0$. #+end_exercice +# ID:8743 #+begin_exercice [ENS SR 2025 # 90] - 1. Soient $a, b \in \mathbb{R}$ avec a < $b$ et $f : [a, b] \to [a, b]$. + 1. Soient $a, b \in \mathbb{R}$ avec $a\lt b$ et $f\colon [a, b] \to [a, b]$. 1. Si $f$ est continue, montrer que $f$ possède un point fixe. 2. Si $f$ est croissante, montrer que $f$ possède un point fixe. - 1. Soit $f\colon \R \ra \R$ monotone. Montrer que l'ensemble dis(f) des points de discontinuité de $f$ est au plus dénombrable. - 1. Construire $f\colon \R \ra \R$ monotone dont l'ensemble des points de discontinuité est $\Q$. + 2. Soit $f\colon \R \ra \R$ monotone. Montrer que l'ensemble $\op{dis}(f)$ des points de discontinuité de $f$ est au plus dénombrable. + 3. Construire $f\colon \R \ra \R$ monotone dont l'ensemble des points de discontinuité est $\Q$. #+end_exercice +#+BEGIN_proof + 1. + 2. + 3. Prendre $f\colon x\mapsto \sum_{u_n \mid u_n \geq x} \frac{1}{2^n}$ +#+END_proof +# ID:nil #+begin_exercice [ENS P 2025 # 91] -Trouver les $f:[0,1] \ra \R$ continues telles que $\forall x \in [0,1], \ f(x) = \sum_{n=0}^{+\i} \frac{f(x^n)}{2^n}$. +Trouver les $f\colon [0,1] \ra \R$ continues telles que $\forall x \in [0,1], \ f(x) = \sum_{n=0}^{+\i} \frac{f(x^n)}{2^n}$. #+end_exercice -#+begin_exercice [ENS PLSR 2025 # 92] +#+begin_exercice [ENS PLSR 2025 # 92] :todo: Soit $f$ une fonction de $\R$ dans $\R \cup \{+\i\}$ non identiquement égale à $+\i$. Pour $y \in \R$, on pose $f^*(y) = \sup\{xy - f(x) ; x \in \R\}$. 1. Montrer que $\{x \in \R, f^*(x) \lt +\i\}$ est un intervalle (éventuellement vide) sur lequel $f^*$ est convexe. - 1. Montrer que, si $f$ est dérivable et convexe sur $\R$, alors $f^{} = f$. - 1. On suppose que $f$ est de classe $C^2$ sur $\R$, que f''\gt 0 sur $\R$ et que $\frac{f(x)}{|x|}\underset{|x|\ra+\i}{\longrightarrow}+\i$. Montrer que $f^*$ est dérivable sur $\R$ et que : $\forall (x,y)\in\R^2,\,y=f'(x)\Leftrightarrow x=(f^*)'(y)$. + 1. Montrer que, si $f$ est dérivable et convexe sur $\R$, alors $f^{*} = f$. + 1. On suppose que $f$ est de classe $C^2$ sur $\R$, que $f''\gt 0$ sur $\R$ et que $\frac{f(x)}{|x|}\underset{|x|\ra+\i}{\longrightarrow}+\i$. Montrer que $f^*$ est dérivable sur $\R$ et que : $\forall (x,y)\in\R^2,\,y=f'(x)\Leftrightarrow x=(f^*)'(y)$. #+end_exercice +# ID:8781 #+begin_exercice [ENS SR 2025 # 93] Pour $f:[0,1]\ra\R$, on pose $B_n(f)(x)=\sum_{k=0}^nf\Big(\frac{k}{n}\Big)\binom{n}{k}x^k(1-x)^{n-k}$. 1. Calculer $B_n(u_1)$ et $B_n(u_2)$ où $u_n: x \mapsto x^n$. -$\textbf{\textit{b}) Montrer que, pour tout } x \in [0,1], \sum_{k=0}^n \left| x \frac{k}{n} \right| \binom{n}{k} x^k (1-x)^{n-k} \leq \sqrt{\frac{x(1-x)}{n}}$. + 2. Montrer que, pour tout $x \in [0,1], \,\sum_{k=0}^n \left| x \frac{k}{n} \right| \binom{n}{k} x^k (1-x)^{n-k} \leq \sqrt{\frac{x(1-x)}{n}}$. 1. En déduire que si $f$ est $M$-lipschitzienne, alors $|B_n(f)(x) f(x)| \leq \frac{M}{2\sqrt{n}}$ pour tout $x$. #+end_exercice +# ID:8789 #+begin_exercice [ENS L 2025 # 94] Trouver toutes les fonctions $f\colon \R \ra \R$ telles que : - - $f$ est croissante, à valeurs dans [0, 1], $f$ est continue à droite, - - - $f(x) \xrightarrow[x \ra -\i]{} 0$, $f(x) \xrightarrow[x \ra +\i]{} 1$, $\forall k \in \N^*$, $\exists b_k \in \R$, $\forall x \in \R$, $f(x)^k = f(x + b_k)$. + + $f$ est croissante, à valeurs dans $[0, 1]$, $f$ est continue à droite, + + $f(x) \xrightarrow[x \ra -\i]{} 0$, $f(x) \xrightarrow[x \ra +\i]{} 1$, $\forall k \in \N^*$, $\exists b_k \in \R$, $\forall x \in \R$, $f(x)^k = f(x + b_k)$. #+end_exercice +#+BEGIN_proof +Si $f$ prend la valeur $1$, ou $0$, c'est facile. -#+begin_exercice [ENS PLSR 2025 # 95] - 1. Soient $a, b \in \R$ avec $|b| \lt \pi$. +Si $f(x)^2 = f(x-1)$, alors $f(x)^4 = f(x-2)$, et la stricte croissance de $f$ (nécessaire) donnera $b_k = \ln_2(k)$. -Montrer qu'il existe $z \in \C$ tel que $z + e^z = a + ib$. +Peut-on l'étendre aux puissances fractionnaires ? Oui : $f(x)^{1/q} = f(x+\ln_2(q))$, donc $f(x)^{p/q} = f(x +\ln_2(q) - \ln_2(p))$. On trouve $f = e^{- e^{x}}$, à des constantes près. +#+END_proof + +# Relier à 6558 +#+begin_exercice [ENS PLSR 2025 # 95] :todo: + 1. Soient $a, b \in \R$ avec $|b| \lt \pi$. Montrer qu'il existe $z \in \C$ tel que $z + e^z = a + ib$. 1. Montrer que l'application $z \mapsto z e^z$ est surjective de $\C$ sur $\C$. #+end_exercice +# ID:8746 #+begin_exercice [ENS P 2025 # 96] -Soient $\sigma \gt 0$ et $f\colon \R \ra \R$ une fonction continue telle que: $\forall x, y \in \R, |f(x)+f(y)-f(x+y)| \leq \sigma$. Montrer que $f$ est la somme d'une fonction linéaire $\ell: \R \ra \R$ et d'une fonction bornée par $\sigma$. +Soient $\sigma \gt 0$ et $f\colon \R \ra \R$ une fonction continue telle que: $\forall x, y \in \R, |f(x)+f(y)-f(x+y)| \leq \sigma$. Montrer que $f$ est la somme d'une fonction linéaire $\ell\colon \R \ra \R$ et d'une fonction bornée par $\sigma$. #+end_exercice +#+BEGIN_proof +Trouver la partie linéaire comme la limite de $\frac{f(n\tau)}{n}$, +qui est indépendante de $\tau$. +#+END_proof -#+begin_exercice [ENS L 2025 # 97] -Une partie $E$ de [0, 1] est dite négligeable si, pour tout $\eps \gt 0$, il existe une suite $(I_n)_{n\geq 0}$ d'intervalles de [0,1] dont la réunion contient $X$ et dont la somme des longueurs est majorée par $\eps$. Soit $f$ une fonction dérivable de [0,1] dans $\R$. On suppose qu'il existe une partie négligeable $E$ de [0,1] telle que, pour tout $x \in [0,1] \setminus E$, on ait $f'(x) \ge 0$. Montrer que $f$ est croissante. + +#+begin_exercice [ENS L 2025 # 97] :todo: +Une partie $E$ de $[0, 1]$ est dite négligeable si, pour tout $\eps \gt 0$, il existe une suite $(I_n)_{n\geq 0}$ d'intervalles de $[0,1]$ dont la réunion contient $X$ et dont la somme des longueurs est majorée par $\eps$. Soit $f$ une fonction dérivable de $[0,1]$ dans $\R$. On suppose qu'il existe une partie négligeable $E$ de $[0,1]$ telle que, pour tout $x \in [0,1] \setminus E$, on ait $f'(x) \ge 0$. Montrer que $f$ est croissante. #+end_exercice +#+BEGIN_proof +#+END_proof + + +# ID:8747 #+BEGIN_exercice [ENS P 2025 # 98] -Soient $n \in \N^*$, $(P_k)_{k \in \db{1,n}}$ et $(Q_k)_{k \in \db{1,n}}$ deux familles de polynômes réels, $f$ la fonction de $\mathbb{P}$ dens $\mathbb{P}$ telle que, pour tout $x \in \mathbb{P}$, $f(x) = \sum_{n=1}^n P_n(x) e^{Q_k(x)}$. Montrer que si - -fonction de $\mathbb R$ dans $\mathbb R$ telle que, pour tout $x\in\mathbb R$, $f(x)=\sum_{k=1}P_k(x)\,e^{Q_k(x)}$. Montrer que, si - $f$ n'est pas identiquement nulle, alors $f$ ne possède qu'un nombre fini de zéros. +Soient $n \in \N^*$, $(P_k)_{k \in \db{1,n}}$ et $(Q_k)_{k \in \db{1,n}}$ deux familles de polynômes +réels, $f$ la fonction de $\R$ dans $\R$ telle que, pour tout $x \in \R$, $f(x) = \sum_{k=1}^n P_k(x) +e^{Q_k(x)}$. Montrer que, si $f$ n'est pas identiquement nulle, alors +$f$ ne possède qu'un nombre fini de zéros. #+END_exercice +#+BEGIN_proof +On peut supposer les polynômes disjoints, et même regrouper ceux qui +ne diffèrent qu'en leur terme constant. Alors, en $\pm \i$, on a un +équivalent. Reste à montrer que sur un segment une telle fonction ne +s'annule qu'un nombre fini de fois. + +Si on factorise par $e^{Q_1(x)}$, on tombe sur une récurrence. +#+END_proof + +# ID:8748 #+begin_exercice [ENS P 2025 # 99] -Soit $n$ un entier impair supérieur ou égal à 3. Déterminer les fonctions continues $f$ de [0,1] dans $\mathbb R$ telles que, pour tout $k\in [1,n-1]$, $\int_0^1 (f(x^{1/k}))^{n-k} \,dx = \frac{k}{n}$. +Soit $n\geq 3$ un entier impair. Déterminer les fonctions continues $f$ de $[0,1]$ dans $\mathbb R$ telles que, pour tout $k\in\db{1,n-1}$, $\int_0^1 (f(x^{1/k}))^{n-k} \dx = \frac{k}{n}$. #+end_exercice +#+BEGIN_proof +On pose $u = x^{1/k}$, alors $\du = \frac{1}{k} x^{1/k - 1}\dx$ +donc $\int_0^1 (f(x^{1/k}))^{n-k} \dx = k\int_0^1 f(u)^{n-k} u^{k-1}\du$ +Donc $\int_0^1 f(u)^{n-k} u^{k-1}\du = \frac{1}{n}$. -#+begin_exercice [ENS P 2025 # 100] -Soit $(a_k)_{k\geq 1}$ une suite décroissante de réels positifs telle que, pour tout $k\in\N^*$, $ka_k\leq (k+1)a_{k+1}$. Montrer que $\int_0^\pi \max_{1\leq k\leq n}\left(a_k\frac{|\sin(kx)|}{x}\right)\,dx=\sum_{k=1}^n\frac{a_k}{k}+O(1)$. +Donc $\int_0^1 f(u)^{n-1} \left(\frac{u}{f(u)}\right)^{k-1}\du = \frac{1}{n}$. Ensuite, cas d'égalité de Cauchy-Schwarz, donc les fonctions $f(u)^{n-1}$ et $f(u)^{n-1} (\frac{u}{f(u)})^2$ sont colinéaires. + +En fait, sans diviser : c'est Cauchy-Schwarz avec $f(u)^{\frac{n-1}{2}}$ et $u f(u)^{\frac{n-3}{2}}$, et ces deux fonctions sont colinéaires… + +Par ailleurs, si $f(x) = x$, ça marche. +#+END_proof + + +#+begin_exercice [ENS P 2025 # 100] :todo: +Soit $(a_k)_{k\geq 1}$ une suite décroissante de réels positifs telle que, pour tout $k\in\N^*$, $ka_k\leq (k+1)a_{k+1}$. Montrer que $\int_0^\pi \max\limits_{1\leq k\leq n}\left(a_k\frac{|\sin(kx)|}{x}\right)\,dx=\sum_{k=1}^n\frac{a_k}{k}+O(1)$. #+end_exercice +#+BEGIN_proof +#+END_proof + + +# ID:8749 #+begin_exercice [ENS PLSR 2025 # 101] Soit $f\colon \R \ra \R$ de classe $C^1$. On pose, pour $n \in \N^*$, $S_n = \frac{1}{n} \sum_{i=0}^{n-1} f\left(\frac{k}{n}\right)$. - a. Quelle est la limite de $(S_n)_{n\in\N^*}$ ? Déterminer la vitesse de convergence. - - $b$. On suppose désormais $f$ 1-périodique et de classe $C^2$. Montrer qu'il existe $C \in \R$ tel que : $\forall n \geq 1, \left| S_n - \int_0^1 f(t) \, dt \right| \leq \frac{C}{n^2}$. - - $c$. On suppose désormais $f$ 1-périodique et de classe $C^3$. Montrer qu'il existe $C \in \R$ tel que : $\forall n \geq 1, \left| S_n - \int_0^1 f(t) \, dt \right| \leq \frac{C}{n^3}$. - - $d$. Que dire si $f$ est 1-périodique et de classe $C^{\i}$ ? + 1. Quelle est la limite de $(S_n)_{n\in\N^*}$ ? Déterminer la vitesse de convergence. + 2. On suppose désormais $f$ 1-périodique et de classe $C^2$. Montrer qu'il existe $C \in \R$ tel que : $\forall n \geq 1, \left| S_n - \int_0^1 f(t) \, dt \right| \leq \frac{C}{n^2}$. + 3. On suppose désormais $f$ 1-périodique et de classe $C^3$. Montrer qu'il existe $C \in \R$ tel que : $\forall n \geq 1, \left| S_n - \int_0^1 f(t) \, dt \right| \leq \frac{C}{n^3}$. + 4. Que dire si $f$ est 1-périodique et de classe $C^{\i}$ ? #+end_exercice +#+BEGIN_proof + 1. +#+END_proof + + +# ID:8767 #+begin_exercice [ENS P 2025 # 102] -Soient $(a,b) \in \R^2$ tel que a \lt $b$, $f$ une fonction continue de $[a,b] \times [-1,1]$ dans $\R$. Pour $\lambda \in \R$, soit $I(\lambda) = \int_a^b f(t,\sin(\lambda t)) \,dt$. Montrer que $I(\lambda)$ admet une limite que l'on déterminera lorsque $\lambda$ tend vers $+\i$. +Soient $(a,b) \in \R^2$ tel que $a \lt b$, $f$ une fonction continue de $[a,b] \times [-1,1]$ dans $\R$. Pour $\lambda \in \R$, soit $I(\lambda) = \int_a^b f(t,\sin(\lambda t)) \,dt$. Montrer que $I(\lambda)$ admet une limite que l'on déterminera lorsque $\lambda$ tend vers $+\i$. #+end_exercice +#+BEGIN_proof +Posons $G(t) = \frac{1}{2}\int_{-1}^1 f(t, u)\du$. Alors $G$ est une fonction continue, et on montre que $I(\la)\ra \int_a^b G(t)\dt$. -#+begin_exercice [ENS SR 2025 # 103] +Il suffit de le montrer pour des fonctions indicatrices de rectangles, +pour lesquelles c'est clair. + +Puis on peut encadrer $f$ par de telles fonctions. +#+END_proof + + +#+begin_exercice [ENS SR 2025 # 103] :todo: Soient $N \in \N^*$ et $(x_1, \dots, x_N) \in \C^N$. Pour $y \in \R$, on note $e(y) = e^{2i\pi y}$. Soit $f\colon t \in \R \mapsto \sum_{n=1}^N x_n e(nt)$. Soient $R \in \N^*$ et $(t_1, \dots, t_R) \in \R^R$. 1. - a) Montrer que $\sum_{r=1}^R |f(t_r)|^2 \leq NR \sum_{k=1}^N |x_k|^2$. - b) Étudier le cas d'égalité dans l'inégalité précédente. - 1. Pour $t \in \R$, on pose $\Delta(t) = \inf_{n \in \Z} |n - t|$. On suppose les $t_i$ distincts. Soit $\delta \gt 0$ tel que + 1. Montrer que $\sum_{r=1}^R |f(t_r)|^2 \leq NR \sum_{k=1}^N |x_k|^2$. + 2. Étudier le cas d'égalité dans l'inégalité précédente. + 2. Pour $t \in \R$, on pose $\Delta(t) = \inf_{n \in \Z} |n - t|$. On suppose les $t_i$ distincts. Soit $\delta \gt 0$ tel que $\delta \leq \min\limits_{1\leq i\neq j\leq R} \Delta(t_i-t_j)$. Montrer que $\sum_{r=1}^R |f(t_r)|^2 \leq (2N\pi+\delta^{-1}) \sum_{r=1}^N |x_k|^2$. - $\delta \leq \min_{1\leq i\neq j\leq R} \Delta(t_i-t_j). \text{ Montrer que } \sum_{r=1}^R |f(t_r)|^2 \leq (2N\pi+\delta^{-1}) \sum_{r=1}^N |x_k|^2$.Ind. On pourra montrer que, pour une fonction $g$ de classe $\mc C^1$ sur $\R$, pour $a \in \R$ et $h \gt 0$, - - $$|g(a)| \le \frac{1}{2h} \int_{a-h}^{a+h} |g(t)| dt + \frac{1}{2} \int_{a-h}^{a+h} |g'(t)| dt$$. + Ind. On pourra montrer que, pour une fonction $g$ de classe $\mc C^1$ sur $\R$, pour $a \in \R$ et $h \gt 0$, + $$|g(a)| \le \frac{1}{2h} \int_{a-h}^{a+h} |g(t)| dt + \frac{1}{2} \int_{a-h}^{a+h} |g'(t)| dt$$. #+end_exercice +#+BEGIN_proof + 1. + 1. +#+END_proof + +# ID:8768 #+begin_exercice [ENS PLSR 2025 # 104] -On note $E$ l'ensemble des fonctions 1-périodiques et de classe $\mc C^{\i}$ de $\R$ dans - -$\C$. Soit $f \in E$. Pour $n \in \Z$, on pose $c_n(f) = \int_0^1 e^{-2in\pi t} f(t) dt$. +On note $E$ l'ensemble des fonctions 1-périodiques et de classe $\mc C^{\i}$ de $\R$ dans $\C$. Soit $f \in E$. Pour $n \in \Z$, on pose $c_n(f) = \int_0^1 e^{-2in\pi t} f(t) \dt$. 1. Montrer que $(c_n(f))_{n\in\Z}$ est sommable. - 1. On suppose que f(0)=0. Montrer qu'il existe $g\in E$ telle que $\forall t\in\R,\ f(t)=g(t)\,(e^{2i\pi t}-1)$. + 1. On suppose que $f(0)=0$. Montrer qu'il existe $g\in E$ telle que $\forall t\in\R,\ f(t)=g(t)\,(e^{2i\pi t}-1)$. #+end_exercice +#+BEGIN_proof + 1. Par IPP, on a $c_n(f)$ qui est en $O(\frac{1}{n^2})$. + 2. On a $f(t) = \sum_{n\in\Z} c_n(f) e^{-2i\pi n t}$, car la différence est + orthogonale aux polynômes trigonométriques. -#+begin_exercice [ENS P 2025 # 105] -Soient a,b\gt 0 et $m\in\Z$. Calculer $I_m(a,b)=\int_a^{+\i}e^{-ax-\frac{b}{x}}x^{m-\frac{1}{2}}dx$. + On trouve les $c_n(g)$ : il faut $c_n(f) = c_{n-1}(g) - c_n(g)$ et la somme des $c_n(f)$ fait $0$. +#+END_proof + + +#+begin_exercice [ENS P 2025 # 105] :todo: +Soient $a,b\gt 0$ et $m\in\Z$. Calculer $I_m(a,b)=\int_a^{+\i}e^{-ax-\frac{b}{x}}x^{m-\frac{1}{2}}\dx$. #+end_exercice +#+BEGIN_proof +#+END_proof + + +# ID:8769 #+begin_exercice [ENS L 2025 # 106] -Soit $n \ge 2$. Déterminer l'ensemble des matrices $A \in \M_n(\C)$ telles que l'intégrale $\int_{-\i}^{+\i} e^{t^2 A} dt$ converge. +Soit $n \ge 2$. Déterminer l'ensemble des matrices $A \in \M_n(\C)$ telles que l'intégrale $\int_{0}^{+\i} e^{t^2 A} \dt$ converge. #+end_exercice +#+BEGIN_proof +Il faut $\lN e^{t^2 A}\rN\ra 0$, donc les valeurs propres de $A$ ont des parties réelles $\lt 0$. -#+begin_exercice [ENS PLSR 2025 # 107] -Soit $f\colon \R \ra \R$ lipschitzienne. On suppose qu'il existe $R$ \gt 0 tel que, pour tout $x \in \R \setminus [-R, R]$, f(x) = 0. - 1. Montrer que $\eps \mapsto \int_{-\eps}^{-\eps} \frac{f(x)}{x} dx + \int_{-\eps}^{+\i} \frac{f(x)}{x} dx$ admet une limite en $0^+$. -On note vp $\left(\int_{-\i}^{+\i} \frac{f(x)}{x} dx\right)$ cette limite. - 1. On note $T_f\colon x \mapsto \int_{-\i}^x f(y) \ln |y x| dy + \int_x^{+\i} f(y) \ln |y x| dy$. Justifier que $T_f$ est bien définie sur $\R$. +Réciproquement, dans ce cas c'est bon. car $e^A$ est tri sup avec des coefficients diagonaux $\lt 1$, donc ses puissances tendent vers $0$ de manière exponentielle. +#+END_proof + + +#+begin_exercice [ENS PLSR 2025 # 107] :todo: +Soit $f\colon \R \ra \R$ lipschitzienne. On suppose qu'il existe $R \gt 0$ tel que, pour tout $x \in \R \setminus [-R, R]$, $f(x) = 0$. + 1. Montrer que $\eps \mapsto \int_{-\i}^{-\eps} \frac{f(x)}{x} dx + \int_{\eps}^{+\i} \frac{f(x)}{x} dx$ admet une limite en $0^+$. +On note $\op{vp}\left(\int_{-\i}^{+\i} \frac{f(x)}{x} dx\right)$ cette limite. + 1. On note $T_f\colon x \mapsto \int_{-\i}^x f(y) \ln |y- x| dy + \int_x^{+\i} f(y) \ln |y- x| dy$. Justifier que $T_f$ est bien définie sur $\R$. 1. On suppose $f$ de classe $C^1$. Montrer que $T_f$ est dérivable sur $\mathbb R$ et que : -$$\forall x \in \R, \ (T_f)'(x) = \op{vp}\left(\int_{-y}^{+\i} \frac{f(y+x)}{y} dy\right)$$. + $$\forall x \in \R, \ (T_f)'(x) = \op{vp}\left(\int_{-y}^{+\i} \frac{f(y+x)}{y} dy\right)$$. #+end_exercice +#+BEGIN_proof :todo: + 1. Si $f(0) = 0$ c'est bon, et si $f(0)\neq 0$, on peut le lui retirer, c'est bon aussi. + 2. +#+END_proof -#+begin_exercice [ENS SR 2025 # 108] + +#+begin_exercice [ENS SR 2025 # 108] :todo: 1. Pour $(p,k) \in \N^2$, montrer la convergence de $I_{p,k} = \int_0^1 \int_0^1 \frac{y^k x^p}{1 xy} \, dx \, dy$ et l'exprimer sous forme de la somme d'une série numérique. 1. On note $d_n = \op{ppcm}(1, \dots, n)$ pour $n \in \N^*$. Montrer que $I_{p,k} \in \frac{1}{d_p^2} \Z$ si $p$ \gt $k$, et $I_{p,p} \in \zeta(2) + \frac{1}{d^2} \Z$. 1. On pose $P_n = \frac{1}{n!} D^n (X^n (1 X)^n)$. Montrer que $P_n$ est à coefficients entiers. - 1. Montrer que $I_n = \int_0^1 \int_0^1 \frac{(1-y)^n P_n(x)}{1-xy} dx dy$ converge, et en donner une expression simplifiée.- e) Montrer que $I_n \in \frac{1}{d^2} (\Z + \zeta(2)\Z)$. + 1. Montrer que $I_n = \int_0^1 \int_0^1 \frac{(1-y)^n P_n(x)}{1-xy} dx dy$ converge, et en donner une expression simplifiée. + 1. Montrer que $I_n \in \frac{1}{d^2} (\Z + \zeta(2)\Z)$. #+end_exercice +# ID:8773 #+begin_exercice [ENS L 2025 # 109] Déterminer les segments $S$ de $\R$ non réduits à un point tels que l'ensemble des fonctions polynomiales à coefficients dans $\Z$ de $S$ dans $\R$ soit dense dans $(\mc C^0(S,\R), \| \|_{\i})$. #+end_exercice +#+BEGIN_proof +Mieux vaut ne pas contenir un entier, et sur tout segment $[a,b]$ +inclus dans $\interval]{0, 1}[$ c'est bon. +#+END_proof + +# ID:8770 #+begin_exercice [ENS L 2025 # 110] -On note $E$ l'ensemble des fonctions croissantes de $\mathbb R$ dans $\mathbb R$ ayant pour limites respectives 0 et 1 en $-\i$ et $+\i$. Soient $F,G,H\in E$, avec $G$ et $H$ continues. +On note $E$ l'ensemble des fonctions croissantes de $\mathbb R$ dans +$\mathbb R$ ayant pour limites respectives $0$ et $1$ en $-\i$ et +$+\i$. Soient $F,G,H\in E$, avec $G$ et $H$ continues. -On suppose qu'il existe quatre suites réelles a,b,c,d telles que $(x \mapsto F(a_nx+b_n))_n$ et $(x \mapsto F(c_nx+d_n))_n$ convergent simplement sur $\R$, respectivement vers $G$ et $H$. Montrer qu'il existe deux réels $\lambda \gt 0$ et $\mu$ tels que $\forall x \in \R, \ H(x) = G(\lambda x + \mu)$. +On suppose qu'il existe quatre suites réelles $a,b,c,d$ telles que $(x +\mapsto F(a_nx+b_n))_n$ et $(x \mapsto F(c_nx+d_n))_n$ convergent +simplement sur $\R$, respectivement vers $G$ et $H$. Montrer qu'il +existe deux réels $\lambda \gt 0$ et $\mu$ tels que $\forall x \in \R, +\ H(x) = G(\lambda x + \mu)$. #+end_exercice +#+BEGIN_proof +Si $(a_n)$ tend vers l'infini alors $F(a_n x + b)$ ne peut tendre que +vers une fonction constante (avec une cassure en un point), qui ne +sera pas continue. Donc $(a_n)$ est bornée, donc on peut supposer que +$(a_n)$ converge. De même, si $(b_n)$ tend vers l'infini, on converge +vers une fonction constante, qui ne vérifie pas les conditions de +limites. Sinon, on converge vers $F(ax + b)$, avec $a\neq 0$. +#+END_proof + #+begin_exercice [ENS L 2025 # 111] -Soit $(f_n)_{n\in\N}$ une suite de fonctions de [0,1] dans ]0,1], convergeant simplement vers une fonction $f$. - 1. Pour $n \ge 2$, on pose $t_n = \frac{1}{\ln n} \sum_{i=1}^n \frac{f_i}{i}$. Montrer que la suite $(t_n)$ converge simplement -vers $f$. - 1. On suppose que $f_0$ est à valeurs strictement positives et que, pour tout $n \ge 1$, la fonction -$f_n$ est dérivable, croissante et que $f_n' \geq \frac{nf_n}{\sigma_n}$, où $\sigma_n = \sum_{i=0}^{n-1} f_i$. On suppose également que $\sup \sigma_n(1/2) \lt +\i$. Montrer que, pour tout $x \in [0, 1/2[$, il existe $C_x \gt 0$ tel que, pour tout -$n \geq 1$ $n \text{ assez grand, } f_n(x) \leq e^{-C_x n}$. - 1. On enlève l'hypothèse sur $\sigma_n(1/2)$. Montrer qu'il existe $x_0 \in [0,1]$ tel que : -(i) $\forall x \lt x_0, \ \exists C_x \gt 0, \ \exists n_0 \in \N^*, \ \forall n \ge n_0, \ f_n(x) \le e^{-C_x n};$ (ii) $\forall x \gt x_0, \ f(x) \ge x - x_0$. +Soit $(f_n)_{n\in\N}$ une suite de fonctions de $[0,1]$ dans $]0,1]$, +convergeant simplement vers une fonction $f$. + 1. Pour $n \ge 2$, on pose $t_n = \frac{1}{\ln n} \sum_{i=1}^n + \frac{f_i}{i}$. Montrer que la suite $(t_n)$ converge simplement + vers $f$. + 2. On suppose que $f_0$ est à valeurs strictement positives et que, + pour tout $n \ge 1$, la fonction $f_n$ est dérivable, croissante + et que $f_n' \geq \frac{nf_n}{\sigma_n}$, où $\sigma_n = + \sum_{i=0}^{n-1} f_i$. On suppose également que $\sup + \sigma_n(1/2) \lt +\i$. + + Montrer que, pour tout $x \in [0, 1/2[$, + il existe $C_x \gt 0$ tel que, pour tout $n$ + assez grand, $f_n(x) \leq e^{-C_x n}$. + 3. On enlève l'hypothèse sur $\sigma_n(1/2)$. Montrer qu'il existe $x_0 \in [0,1]$ tel que : + + $\forall x \lt x_0, \ \exists C_x \gt 0, \ \exists n_0 \in \N^*, \ \forall n \ge n_0, \ f_n(x) \le e^{-C_x n};$ + + $\forall x \gt x_0, \ f(x) \ge x - x_0$. #+end_exercice +#+BEGIN_proof + 1. Sans difficultés. + 2. +#+END_proof + #+begin_exercice [ENS P 2025 # 112] Soit $f\colon x \mapsto \sum_{n=1}^{+\i} \frac{1}{n} \sin\left(\frac{x}{4^n}\right)$. - 1. Montrer que $\lim_{x \ra +\i} (\inf \{ f(t), t \ge x \}) = 0$. - 1. Montrer que $0 \lt \lim_{x \ra +\i} \left( \sup \left\{ \frac{|f(t)|}{\ln(\ln t)}, \ t \geq x \right\} \right) \lt +\i$. + 1. Montrer que $\lim\limits_{x \ra +\i} (\inf \{ f(t), t \ge x \}) = 0$. + 1. Montrer que $0 \lt \lim\limits_{x \ra +\i} \left( \sup \left\{ \frac{|f(t)|}{\ln(\ln t)}, \ t \geq x \right\} \right) \lt +\i$. #+end_exercice +#+BEGIN_proof + 1. Quand on somme, à partir de $n = \log_4(x)$, on est positif, pour + une contribution en $\sum_{\log_4(x)}^{+\i} \frac{1}{n 4^{n-\log_4(x)}}$, c'est-à-dire $\Theta(\frac{1}{\log_4(x)})$. + + La première partie est au plus en $\ln \ln (x)$. + + Si on prend un $x = 4^n \pi$, on est très proche de $0$. + + Pourquoi est-on forcément positif ?? La fonction est impaire… + 2. Simple, cf ce qui précède. +#+END_proof + #+begin_exercice [ENS L 2025 # 113] -Soit $(\lambda_n)$ une suite de réels \gt 0 telle que $\forall n \in \N, \ 2\lambda_n \leq \lambda_{n+1} \leq 3\lambda_n$. Montrer que : +Soit $(\lambda_n)$ une suite de réels $\gt 0$ telle que $\forall n \in +\N, \ 2\lambda_n \leq \lambda_{n+1} \leq 3\lambda_n$. Montrer que : -$$\forall \alpha \gt 0, \ \exists (c_1, c_2) \in (\R^{+*})^2, \ \forall t \in [1/2, 1[, \frac{c_1}{(1-t)^{\alpha}} \leq \sum_{n=1}^{+\i} \lambda_n^{\alpha} t^{\lambda_n} \leq \frac{c_2}{(1-t)^{\alpha}}$$. +$$\forall \alpha \gt 0, \ \exists (c_1, c_2) \in (\R^{+*})^2, \ +\forall t \in [1/2, 1[, \frac{c_1}{(1-t)^{\alpha}} \leq +\sum_{n=1}^{+\i} \lambda_n^{\alpha} t^{\lambda_n} \leq +\frac{c_2}{(1-t)^{\alpha}}$$. #+end_exercice +#+BEGIN_proof +Calculons $\sum_{n\geq 1} 2^{n\a} t^{2^n}$ +#+END_proof + #+begin_exercice [ENS SR 2025 # 114] On pose : $\forall x \gt 0, \eta(x) = \sum_{n=0}^{+\i} \frac{(-1)^{n-1}}{n^x}$. @@ -1199,8 +1894,7 @@ $$\sqrt{\int_0^{+\i}|f|^2}=\sup\left\{\int_0^{+\i}|fg|\;;\;g\in E\;\mathrm{tel}\ #+begin_exercice [ENS P 2025 # 119] -Soient $K \in \mc C^0([0,1]^2,\R)$ telle que $||K||_{\i} \lt 1$ et $f \in \mc C^0([0,1],\R)$. Étudier l'exis- -tence et l'unicité de $g \in \mc C^0([0,1],\R)$ telle que $\forall x \in [0,1], \ g(x) \int_{\R}^1 K(x,t)g(t) \ dt = f(x)$. +Soient $K \in \mc C^0([0,1]^2,\R)$ telle que $||K||_{\i} \lt 1$ et $f \in \mc C^0([0,1],\R)$. Étudier l'existence et l'unicité de $g \in \mc C^0([0,1],\R)$ telle que $\forall x \in [0,1], \ g(x) \int_{\R}^1 K(x,t)g(t) \ dt = f(x)$. #+end_exercice #+begin_exercice [ENS L 2025 # 120] @@ -1211,13 +1905,14 @@ Montrer que $T$ est une application lipschitzienne de $F_{\alpha}$ dans $F_{\alp #+end_exercice #+begin_exercice [ENS PLSR 2025 # 121] - 1. Expliciter le terme général d'une suite $(a_n)_{n\geq 0}$ vérifiant la relation de ré- -currence $na_{n+1} = (n+1)a_n$ pour tout $n$. - 1. Résoudre x(x-1)y'' + 3xy' + y = 0 sur ]-1,1[. + 1. Expliciter le terme général d'une suite $(a_n)_{n\geq 0}$ + vérifiant la relation de récurrence $na_{n+1} = (n+1)a_n$ pour + tout $n$. + 1. Résoudre $x(x-1)y'' + 3xy' + y = 0$ sur $]-1,1[$. #+end_exercice #+begin_exercice [ENS PLSR 2025 # 122] -Résoudre $x^2y'' + xy' + (x^2 1/4)y = 0$ sur ]0, 1[. +Résoudre $x^2y'' + xy' + (x^2 1/4)y = 0$ sur $]0, 1[$. #+end_exercice #+begin_exercice [ENS P 2025 # 123] @@ -1288,38 +1983,72 @@ Soit $G$ un sous-groupe fermé de $\mathrm{GL}_n(\R)$. On pose $L = \{A \in \M_n(\R) ; \forall t \in \R, e^{tA} \in G\}$. 1. Montrer que $L$ est un sous-espace vectoriel de $\M_n(\R)$. Ind. Considérer $\left(e^{tA/k}e^{tB/k}\right)^k$. - 1. Montrer que $\forall (A, B) \in L^2$, $AB BA \in L$. + 1. Montrer que $\forall (A, B) \in L^2$, $AB - BA \in L$. 1. Que peut-on dire de $L$ pour $G = \mathrm{SL}_n(\R)$ ? #+end_exercice #+begin_exercice [ENS PLSR 2025 # 134] Soit $n \ge 2$ un entier. Une application $f$ de classe $C^2$ définie sur un ouvert $O$ de $\R^n$, à valeurs dans $\R^n$ vérifie la propriété $\mc{P}$ si, pour tout $x \in O$, $df_x$ est composée d'une homothétie et d'une isométrie vectorielle. - 1. On suppose que n=2 et que $f$ vérifie $\mc{P}$. On note $f=(f_1,f_2)$. Montrer que $f_1$ et $f_2$ sont harmoniques, c'est-à-dire que $\Delta f_1=0$ et $\Delta f_2=0$. + 1. On suppose que $n=2$ et que $f$ vérifie $\mc{P}$. On note $f=(f_1,f_2)$. Montrer que $f_1$ et $f_2$ sont harmoniques, c'est-à-dire que $\Delta f_1=0$ et $\Delta f_2=0$. 1. Montrer que le résultat de la question a) est faux si $n \geq 3$. On pourra considérer l'application $f\colon x \in \R^n \setminus \{0\} \mapsto \frac{x}{\|x\|^2}$. #+end_exercice +#+BEGIN_proof + 1. + 2. +#+END_proof + +# ID:8776 #+begin_exercice [ENS P 2025 # 135] Soit $f\colon \R^2 \ra \R$ de classe $\mc C^2$. On dit que $f$ est harmonique si $\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0$. On dit que $f$ est homogène de degré $\lambda \geq 0$ si, pour tous $x,y \in \R$ et tout $t \in \R^+$, $f(tx,ty) = t^{\lambda}f(x,y)$. Soit $\lambda \geq 0$. Déterminer les fonctions harmoniques et homogènes de degré $\lambda$. #+end_exercice +#+BEGIN_proof +Pour $\la = 0$, $f$ est constante sur les droites, donc ne dépend que de $\theta$. + +En général, on va faire un changement de coordonnées polaires : + $r^2 \frac{\partial^2 f}{\partial r^2} + r \frac{\partial f}{\partial r} + \frac{\partial^2 f}{\partial \theta^2} = 0$. Si elle ne dépend que de $\theta$, c'est un polynôme affine en $\theta$, étant continue, elle est constante. + +En général, la condition d'homogénéité donne $r\frac{\partial f}{\partial r} = \la f$ , donc avec le laplacien, on obtient $\la^2 f + \frac{\partial^2 f}{\partial \theta^2} = 0$. Donc à $r$ fixé, $f$ dépend de $\theta$ comme $\a \cos (\sqrt{\la}\theta + c)$, ce qui impose que $\la$ soit le carré d'un entier. +#+END_proof + ** Géométrie +# ID:8728 #+begin_exercice [ENS L 2025 # 136] Montrer qu'il n'existe aucun triangle rectangle dont les longueurs des côtés sont dans $\N^*$ et dont l'aire est un carré parfait non nul. #+end_exercice +#+BEGIN_proof +Descente infinie : + +On sait paramétriser les triplets pythagoricien : $(a,b) = (m^2 - n^2, 2mn)$, dont l'aire est $mn (m-n)(m+n)$. Mais si $m$ et $n$ sont premiers entre eux, tout le monde est un carré. On peut en déduire : $x^2 - y^2 = u^2$ et $x^2 + y^2 = v^2$. On considère alors le nouveau triangle de côtés $\frac{u+v}{2}$ et $\frac{u-v}{2}$… +#+END_proof + #+begin_exercice [ENS P 2025 # 137] -Soient a, $b$, $c$, $d$ dans $\R^{+*}$. Quelle est l'aire maximale d'un quadrilatère dont les côtés successifs ont pour longueurs a, $b$, $c$, d? +Soient $a$, $b$, $c$, $d$ dans $\R_+^*$. Quelle est l'aire maximale d'un quadrilatère dont les côtés successifs ont pour longueurs $a$, $b$, $c$, $d$ ? #+end_exercice +#+BEGIN_proof +Paramétrer par la distance entre le premier sommet et le troisième, +$x$. Il faut que $|b-a|\leq x\leq a+b$. Puis pour trouver l'aire d'un +triangle $a,b,x$, on pose la hauteur $h$, qui coupe $x$ en $x = u+v$, +on écrit deux Pythagore, on fait la différence, permet de trouver $u$, +donc $h$. En posant $y = x^2$, l'aire est + + $\sqrt{y/2(a^2+b^2 - y/2) - \frac{(a^2-b^2)^2}{4}} \simeq \sqrt{y (2a^2+2b^2 - y) - (a^2 - b^2)^2}$ +La même avec $c,d$ . + +Si on dérive par rapport à $y$, on trouve $\frac{a^2+b^2 - y}{\sqrt{y (2a^2+2b^2 - y) - (a^2 - b^2)^2}}$, beurk. + +Toujours plus agréable via de la trigo… +#+END_proof #+begin_exercice [ENS PLSR 2025 # 138] 1. Quelle est l'aire maximale possible pour un rectangle de périmètre 1? 1. On considère un entier $n \geq 3$ et une liste strictement croissante $(\theta_1, \dots, \theta_n)$ à termes dans $[0, 2\pi]$. Déterminer la valeur maximale possible pour le périmètre du polygone de sommets $e^{i\theta_1}, \dots, e^{i\theta_n}$ (dans cet ordre). 1. Soit $z_1,\ldots,z_n$ des nombres complexes. On convient que $z_0=z_n$. On définit l'aire algé- brique du polygone $z_1\cdots z_n$ comme $\frac{1}{2}\sum_{k=0}^{n-1}(\op{Re}(z_k)\op{Im}(z_{k+1})-\op{Im}(z_k)\op{Re}(z_{k+1}))$. On fixe un réel p\gt 0. Parmi les listes $(z_1,\ldots,z_n)\in\C^n$ telles que le périmètre de $z_1\cdots z_n$ soit égal à $p$, déterminer celles qui maximisent l'aire algébrique du polygone associé. - - #+end_exercice ** Probabilités @@ -1382,6 +2111,7 @@ Soit $\mc{P}$ l'ensemble des nombres premiers et $\beta \gt 1$. Soit $(Y_p)_{p #+END_proof +# ID:8772 #+begin_exercice [ENS L 2025 # 144] Montrer qu'il existe $C \gt0$ tel que pour tout $n \ge 1$ et tout $(a_{i,j})_{1 \le i,j \le n} \in \{\pm 1\}^{n^2}$, il existe $(x_i)_{1 \le i \le n}$ et $(y_i)_{1 \le i \le n}$ dans $\{\pm 1\}^n$ tels que $\sum_{1 \le i \le n} a_{i,j} x_i y_j \ge C n^{3/2}$. #+end_exercice @@ -1390,9 +2120,36 @@ Montrer qu'il existe $C \gt0$ tel que pour tout $n \ge 1$ et tout $(a_{i,j})_{1 Soient $\theta \in ]0,1[$ et $X$ une variable aléatoire à valeurs dans $\R^+$ telle que $\mathbf{P}(X\gt 0)\gt 0$. Montrer que $\mathbf{P}(X\geq\theta\,\mathbf{E}(X))\geq \frac{(1-\theta)^2\mathbf{E}(X)^2}{\mathbf{E}(X^2)}$. #+end_exercice +# ID:8722 #+begin_exercice [ENS P 2025 # 146] -Soit $n \in \N$ avec $n \ge 2$. Soit $E_n = \{e_1, \dots, e_n\}$ un ensemble de cardinal $n$. Soient $\sigma_1, \dots, \sigma_n$ des variables aléatoires indépendantes suivant la loi uniforme sur $S_n$. Si $i, j \in [1, n]$, on pose $e_i e_j = e_{\sigma_i(j)}$. Montrer que la probabilité que $(E, )$ soit un groupe, sachant que admet un neutre, tend vers 0 quand $n$ tend vers l'infini. +Soit $n \in \N$ avec $n \ge 2$. Soit $E_n = \{e_1, \dots, e_n\}$ un +ensemble de cardinal $n$. Soient $\sigma_1, \dots, \sigma_n$ des +variables aléatoires indépendantes suivant la loi uniforme sur $\mc +S_n$. Si $i, j \in \db{1,n}$, on pose $e_i\star e_j = +e_{\sigma_i(j)}$. Montrer que la probabilité que $(E, \star)$ soit un +groupe, sachant que $E$ admet un neutre, tend vers $0$ quand $n$ tend +vers l'infini. #+end_exercice +#+BEGIN_proof +On note $N_i$ l'évènement « $e_i$ est un élément neutre », et $G$ : +«$(E, \star)$ forme un groupe». On s'intéresse pour commencer à +$\P(G\mid N_1)$. + +L'évènement $N_1$ signifie que $\forall i,\, \sigma_i(1) = i$ et +$\forall j,\, \sigma_1(j) = j$. + +Notons que $e_i$ admet un inverse à droite, qui est $e_{\sigma_i^{-1}(1)}$. Pour que cet élément soit un inverse à gauche, il faut que $\sigma_{\sigma_i^{-1}(1)}(i) = 1$. + +En particulier, les $\sigma_i^{-1}(1)$ sont tous distincts (sans quoi +$1$ aurait plusieurs antécédents par un même $\sigma_k$). + +Naturellement, sous la condition $N_1$, la probabilité que ces éléments soient tous distincts est $\frac{(n-1)!}{(n-1)^{n-1}}$ qui tend vers $0$. + +Formellement : !! + +Enfin, l'énoncé demande $\P(G \mid \bigcup_{i=1}^n N_i) = \frac{\P\big(G \cap \bigcup_{i=1}^n N_i\big)}{\P\big(\bigcup_{i=1}^n N_i\big)} = \frac{\P\big(\bigcup (N_i \cap G)\big)}{\P\big(\bigcup_{i=1}^n N_i\big)}$. Comme les évènements $N_i$ sont disjoints, et que les $\P(N_i)$ et les $\P(G\cap N_i)$ sont égaux, on obtient que $\P(G\mid N) = \P(G\mid N_1)$. +#+END_proof + # ID:8486 #+BEGIN_exercice [ENS L 2025 # 147] @@ -1484,11 +2241,10 @@ Soient $(X_k)_{k \in \N^*}$ une suite de variables de Rademacher indépendantes #+end_exercice #+begin_exercice [ENS P 2025 # 154] -On considère $n$ variables aléatoires de Rademacher indépendantes $(\eps_i)_{1\leq i\leq n}$. Montrer que, pour tout réel $p$ \gt 0, il existe $(c_p, C_p) \in (\R^{+*})^2$ indépendant de $n \in \N^*$ tel que, +On considère $n$ variables aléatoires de Rademacher indépendantes $(\eps_i)_{1\leq i\leq n}$. Montrer que, pour tout réel $p \gt 0$, il existe $(c_p, C_p) \in (\R^{+*})^2$ indépendant de $n \in \N^*$ tel que, -pour tout -$$(z_1, \ldots z_n) \in \C^n$$ -, $c_p \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}} \leq \left(\mathbf{E} \left|\sum_{i=1}^n \eps_i z_i\right|^p\right)^{\frac{1}{p}} \leq C_p \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}}$. +pour tout $(z_1, \ldots z_n) \in \C^n$, +$$c_p \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}} \leq \left(\mathbf{E} \left|\sum_{i=1}^n \eps_i z_i\right|^p\right)^{\frac{1}{p}} \leq C_p \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}}.$$ #+end_exercice #+begin_exercice [ENS L 2025 # 155] @@ -1502,23 +2258,34 @@ Soit $(p_n)_{n\geq 1}$ une suite d'éléments de [0,1]. Pour $n\in\N^*$, on note #+end_exercice #+BEGIN_exercice [ENS L 2025 # 157] -Montrer qu'il existe un réel c\gt 0 vérifiant la condition suivante : quel que soit $n \in \N^*$, quelle que soit $S$ partie non vide de $\mathbb{U}_n$, il existe un entier naturel $p \leq \frac{cn}{|S|}$ ainsi - -qu'une $p$-liste $(z_1,\ldots,z_p)$ d'éléments de $\mathbb{U}_n$ telle que $\left|\bigcup_{k=1}^p z_k S\right|\geq \frac{n}{2}\cdot$ +Montrer qu'il existe un réel $c\gt 0$ vérifiant la condition suivante : quel que soit $n \in \N^*$, quelle que soit $S$ partie non vide de $\mathbb{U}_n$, il existe un entier naturel $p \leq \frac{cn}{|S|}$ ainsi qu'une $p$-liste $(z_1,\ldots,z_p)$ d'éléments de $\mathbb{U}_n$ telle que $\left|\bigcup_{k=1}^p z_k S\right|\geq \frac{n}{2}$. #+END_exercice +#+BEGIN_proof +Méthode probabiliste. +#+END_proof - +# ID:8723 #+begin_exercice [ENS PLSR 2025 # 158] -Soit $p \in [0,1/2]$. On fixe une suite $(X_n)_{n \ge 1}$ de variables aléatoires $i$.i.d. à valeurs dans $\{-1,0,1\}$ et telles que $P(X_1 = 1) = P(X_1 = -1) = p$ et $P(X_1 = 0) = p$ +Soit $p \in [0,1/2]$. On fixe une suite $(X_n)_{n \ge 1}$ de variables +aléatoires i.i.d. à valeurs dans $\{-1,0,1\}$ et telles que $\P(X_1 += 1) = \P(X_1 = -1) = p$ et $P(X_1 = 0) = 1-2p$. Pour $b\in\Z, +a\in\Z^{\N^*}$ et $n\in\N^*$, on pose +$\P(b,a,n)=\P\left(\sum_{k=1}^n a_k X_k=b\right)$. -valeurs dans -$$\{-1,0,1\}$$ - et telles que $\mathbf{P}(X_1=1)=\mathbf{P}(X_1=-1)=p$ et $\mathbf{P}(X_1=1)=1-2p$. Pour $b\in\Z, a\in\Z^{\N^*}$ et $n\in\N^*$, on pose $P(b,a,n)=\mathbf{P}\left(\sum_{k=1}^n a_k X_k=b\right)$. - - 1. On suppose $a= 2^{k-1}$. Calculer P(0, a, n) pour tout $n\in \N$. - 2. On suppose $p$ = 1/4 et $a = (1)_{k\in\N}$. Calculer P(0, a, n) pour tout $n\in N$. - 3. Déterminer les valeurs de $p$ pour lesquelles $b\mapsto P(b, a, n)$ est maximale en 0 pour tout $a \in \Z^{\N^*}$. + 1. On suppose $a_k= 2^{k-1}$. Calculer $\P(0, a, n)$ pour tout $n\in \N$. + 2. On suppose $p = 1/4$ et $a_k = 1$. Calculer $\P(0, a, n)$ + pour tout $n\in \N$. + 3. s Déterminer les valeurs de $p$ pour lesquelles $b\mapsto \P(b, a, + n)$ est maximale en $0$ pour tout $a \in \Z^{\N^*}$. #+end_exercice +#+BEGIN_proof + 1. Il faut que tous les coefficients soient nuls. + 2. C'est du dénombrement : il faut autant de $1$ que de $-1$. + 3. En vérifiant jusqu'à $n = 3$, on trouve qu'il faut que $p\leq \frac{1}{4}$. + + Réciproquement, si $p\leq \frac{1}{4}$, il faut utiliser la fonction caractéristique, cf le 7341, qui est positive, et par inversion de Fourier, toute probabilité est majorée par $\P (S=0)$. +#+END_proof + #+begin_exercice [ENS PLSR 2025 # 159] Soit $n \ge 3$. Une alpiniste dispose de $n$ lieux possibles pour planter sa tente, lieux numérotés de 1 à $n$. Elle peut visiter chacun de ces lieux successivement, à partir du numéro 1, et doit décider si elle y plante sa tente. Lorsqu'elle visite le lieu $k$, elle peut savoir si elle préfère ce lieu à tous les lieux précédemment visités, mais ne sait pas si elle le préfère aux lieux non encore visités. Une fois un lieu visité, si l'alpiniste a refusé d'y installer sa tente elle ne pourra plus revenir sur ce lieu. L'alpiniste a pour objectif de maximiser la probabilité d'avoir choisi celui des $n$ lieux qui a sa préférence parmi les $n$ lieux. @@ -1561,11 +2328,9 @@ Montrer que $P(X \in H) = 1$ si et seulement si $H^{\perp} \subset \text{Ker}(C( #+end_exercice #+begin_exercice [ENS SR 2025 # 164] -Soit $\alpha \gt 0$. On considère l'équation différentielle (): $(y' = -x, x' = \alpha^2 y)$ avec -$(x,y):\R\ra\R^2$. +Soit $\alpha \gt 0$. On considère l'équation différentielle (): $(y' = -x, x' = \alpha^2 y)$ avec $(x,y):\R\ra\R^2$. 1. Si $(x_0, y_0) \in \R^2$ est fixé, justifier l'existence et l'unicité d'une solution de () vérifiant $x(0) = x_0$ et $y(0) = y_0$. Pour cette solution, on pose $I(t) = y^2(t)$ et $J(t) = \alpha^2 x^2(t)$. - 1. Montrer que les applications $T \mapsto \frac{1}{T} \int_0^T I(t) dt$ et $T \mapsto \frac{1}{T} \int_0^T J(t) dt$ admettent une -limite finie en $+\i$. + 1. Montrer que les applications $T \mapsto \frac{1}{T} \int_0^T I(t) dt$ et $T \mapsto \frac{1}{T} \int_0^T J(t) dt$ admettent une limite finie en $+\i$. 1. Soit $N \in \N^*$. On considère deux variables aléatoires $x_0, y_0$ indépendantes à valeurs dans $\frac{1}{N}\Z$ telles que, pour tout $k \in \Z$, $\mathbf{P}\left(x_0 = \frac{k}{N}\right) = \mathbf{P}\left(y_0 = \frac{k}{N}\right) = \gamma_N \exp\left(-(k/N)^2\right)$. a) Justifier l'existence de $\gamma_N \in \R^+$ pour lequel ces conditions définissent la loi des deux variables aléatoires. @@ -1748,15 +2513,9 @@ On suppose qu'il existe $\omega \in I$ tel que $\forall x \in I$, $f^n(x) \ra \o #+begin_exercice [ENS PSI 2025 # 181] Pour $n \ge 1$, on note $b_n$ le nombre de partitions d'un ensemble de cardinal $n$. -On pose -$$b_0 = 1$$ - et $B: x \mapsto \sum_{n=0}^{+\i} \frac{b_n}{n!} x^n$. +On pose $b_0 = 1$ et $B: x \mapsto \sum_{n=0}^{+\i} \frac{b_n}{n!} x^n$. -réflexive $(\forall x \in E, x \sim x)$, - - 1. Montrer que $b_n$ est le nombre de relations d'équivalence sur un ensemble de cardinal $n$. Cette notion étant hors-programme, nous en donnons la définition. Une relation ~ sur l'en- -semble $E$ est dite d'équivalence lorsque c'est une relation : -symétrique $(\forall (x,y) \in E^2, x \sim y \Rightarrow y \sim x)$, - transitive $(\forall (x, y, z) \in E^3, x \sim y \text{ et } y \sim z \Rightarrow x \sim z)$. + 1. Montrer que $b_n$ est le nombre de relations d'équivalence sur un ensemble de cardinal $n$. 1. Calculer $b_0, b_1, b_2$. 1. Montrer que, pour tout $n \in \N$, $b_{n+1} = \sum_{k=1}^n \binom{n}{k} b_k$. 1. Montrer que $B$ est dérivable sur un intervalle ouvert non vide, en déduire une équation différentielle vérifiée par $B$ puis la résoudre. @@ -2249,23 +3008,16 @@ Soit $h \in \mc C^1(\R, \R)$ telle que $\forall x \in \R, |h(x)| \leq \frac{1}{1 #+end_exercice #+begin_exercice [ENS PC 2025 # 238] -Soit, pour -$$x \in \R$$ -, $f(x) = \int_0^{+\i} e^{-y} \cos(xy) \, dy$. - +Soit, pour $x \in \R$, $f(x) = \int_0^{+\i} e^{-y} \cos(xy) \, dy$. 1. Calculer explicitement $f$. - 1. Montrer que, pour tout $k \in \N$, la dérivée $k$-ième de $f$ est bornée par k!. - 1. En quels points $x$ y a-t-il égalité entre k! et $|f^{(k)}(x)|$ ? + 1. Montrer que, pour tout $k \in \N$, la dérivée $k$-ième de $f$ est bornée par $k!$. + 1. En quels points $x$ y a-t-il égalité entre $k!$ et $|f^{(k)}(x)|$ ? #+end_exercice #+begin_exercice [ENS PC 2025 # 239] - 1. Donner les solutions de l'équation différentielle : $x'' x = \cos(2t)$. - 1. Soient c\gt 0 et $f\colon\R\ra\R$ une fonction continue telle que f(t)=0 pour tout $t$ vérifiant $|t| \ge c$. -Montrer qu'il existe une unique solution de l'équation différentielle x'' $x$ = f(t) vérifiant $\lim_{t \ra \pm \i} x(t) = 0$. - - - - $\phi(x,y) = h'(x)$ sinon. + 1. Donner les solutions de l'équation différentielle : $x'' -x = \cos(2t)$. + 1. Soient $c\gt 0$ et $f\colon\R\ra\R$ une fonction continue telle que $f(t)=0$ pour tout $t$ vérifiant $|t| \ge c$. + Montrer qu'il existe une unique solution de l'équation différentielle $x''- x = f(t)$ vérifiant $\lim_{t \ra \pm \i} x(t) = 0$. #+end_exercice ** Géométrie @@ -2359,37 +3111,67 @@ Soit $(X_n)_{n\geq 1}$ une suite $i$.i.d. de variables aléatoires suivant la lo $\begin{array}{l} \textbf{\textit{c})} \ \ \text{Soit} \ \lambda \gt 0. \ \text{Montrer que} \ \mathbf{P}(|Y_N| \gt \lambda) \leq 2e^{-\frac{\lambda^2}{2(c_1^2+\cdots+c_N^2)}}. \\ \textbf{\textit{d})} \ \ \text{Montrer que} \ N^{10} \ \mathbf{P}(|X_1+\cdots+X_N| \gt N^{3/4}) \underset{N \ra +\i}{\longrightarrow} 0. \end{array}$ #+end_exercice -* $X$ MP :xens: +* X MP :xens: ** Algèbre +# ID:nil #+begin_exercice [X MP 2025 # 256] Pour quels entiers $n \in \N^*$ le nombre réel $\cos\left(\frac{2\pi}{n}\right)$ est-il rationnel ? #+end_exercice +# ID:8757 #+begin_exercice [X MP 2025 # 257] On étudie l'équation $x^2 + y^2 = N(1 + xy)$ d'inconnue $(x, y) \in \Z^2$, où $N \in \N$. - 1. Traiter les cas $x$ = y, $N$ = 0, $N$ = 1.- b) On suppose $N \ge 2$ et on se donne (x,y) solution avec $x \ne y$. Montrer qu'on peut se ramener à $x \gt y \ge 0$. Montrer qu'il existe $z \in \Z$ tel que (y, z) soit solution et tel que y \gt $z$. -En déduire que $N$ est un carré parfait. - 1. On considère maintenant l'équation $x^2 + y^2 = -N(1 + xy)$ dans $\Z^2$. En adaptant la méthode précédente, trouver tous les couples solutions. + 1. Traiter les cas $x = y$, $N = 0$, $N = 1$. + 2. On suppose $N \ge 2$ et on se donne $(x,y)$ solution avec $x \ne + y$. Montrer qu'on peut se ramener à $x \gt y \ge 0$. Montrer qu'il + existe $z \in \Z$ tel que $(y, z)$ soit solution et tel que $y \gt + z$. En déduire que $N$ est un carré parfait. + 1. On considère maintenant l'équation $x^2 + y^2 = -N(1 + xy)$ dans + $\Z^2$. En adaptant la méthode précédente, trouver tous les + couples solutions. #+end_exercice #+begin_exercice [X MP 2025 # 258] -Soient $a \in \N$ avec $a \ge 2$ et $P = X^2 + X + a$. On suppose que, pour tout $n \in [0, a-1]$, P(n) est premier. Soit $k \in [1, a-2]$. - 1. Montrer que si k+1 est un carré alors P(a+k) n'est pas premier. - 1. Montrer que si P(a+k) n'est pas premier alors k+1 est un carré. +Soient $a \in \N$ avec $a \ge 2$ et $P = X^2 + X + a$. On suppose que, pour tout $n \in \db{0, a-1}$, $P(n)$ est premier. Soit $k \in \db{1, a-2}$. + 1. Montrer que si $k+1$ est un carré alors $P(a+k)$ n'est pas premier. + 1. Montrer que si $P(a+k)$ n'est pas premier alors $k+1$ est un carré. + + Ind. Montrera que le plus petit facteur premier $p$ de P(a+k) est supérieur ou égal à a, puis que P(a+k-p)=p. #+end_exercice +#+BEGIN_proof + 1. $P(a+k) = a^2 + 2ak + k^2 + a + k + a = a^2 + 2a (k+1) + k(k+1) = (a + (k+1))^2 - (k+1)$. + 2. $P(a-k) = a^2 - 2ak + k^2 + a - k + a = a^2 - 2a (k-1) + k (k-1) = (a - (k-1))^2 + (k-1)$ + + $P(a - (k+2)) = (a - (k+1))^2 + (k+1)$. + + !! +#+END_proof + #+begin_exercice [X MP 2025 # 259] 1. Soit $f\colon \R \ra \R$ de classe $\mc C^1$ et 1-périodique. On suppose qu'il existe $a \in \R \setminus \Q$ et $y \in \R$ tels que : $\forall x \in \R, \forall n \in \N, \sum_{k=0}^n f(x+ka) \leq \sum_{k=0}^n f(y+ka)$. Montrer que $f$ est constante. - 1. Soient $p$ un nombre premier et $n \in \N^*$. Déterminer la valuation $p$-adique de n!. + 1. Soient $p$ un nombre premier et $n \in \N^*$. Déterminer la valuation $p$-adique de $n!$. 1. Soient $m, k \in \N^*$. Montrer que $\frac{\prod_{j=1}^m \binom{2jk}{jk}}{\prod_{j=1}^m \binom{2j}{j}} \in \N$. #+end_exercice +#+BEGIN_proof + 1. + 2. + 3. +#+END_proof + +# ID:8721 #+begin_exercice [X MP 2025 # 260] -Soit $n \in \N^*$. Pour une partie $I$ de [1, n], on appelle composante de $I$ tout sous-ensemble maximal de $I$ formé d'entiers consécutifs. On note c(I) le nombre de composantes de $I$. - 1. Une permutation $\sigma \in \mc{S}_n$ est dite $i$-adaptée lorsque, pour tout $i \in I$, les entiers $\sigma(i)$ et $\sigma(i+1)$ sont consécutifs. Dénombrer les permutations $I$-adaptées en fonction de |I| et c(I). - 1. Soient $c \in \N^*$ et $p \in [1, n]$. Dénombrer les parties $I$ de [1, n] telles que |I| = $p$ et c(I) = $c$. +Soit $n \in \N^*$. Pour une partie $I$ de $\db{1,n}$, on appelle composante de $I$ tout sous-ensemble maximal de $I$ formé d'entiers consécutifs. On note $c(I)$ le nombre de composantes de $I$. + 1. Une permutation $\sigma \in \mc{S}_n$ est dite $I$-adaptée lorsque, pour tout $i \in I$, les entiers $\sigma(i)$ et $\sigma(i+1)$ sont consécutifs. Dénombrer les permutations $I$-adaptées en fonction de $|I|$ et $c(I)$. + 1. Soient $c \in \N^*$ et $p \in \db{1,n}$. Dénombrer les parties $I$ de $\db{1,n}$ telles que $|I| = p$ et $c(I) = c$. #+end_exercice +#+BEGIN_proof + 1. L'image de chaque composante, plus l'élément suivant, est un intervalle d'entiers consécutifs. En ajoutant que les éléments pas dans $I + \{0, 1\}$ sont eux mêmes des blocs, on choisit l'ordre relatif des images des blocs, puis l'orientation de chaque vrai bloc. + 2. On choisit les tailles des blocs (classique), puis on répartir les blocs. +#+END_proof + #+begin_exercice [X MP 2025 # 261] Soient $(a_n)_{n\geq 0}$ et $(b_n)_{n\geq 0}$ deux suites d'entiers relatifs. On dit que les deux séries @@ -2408,15 +3190,20 @@ Soit $G$ un groupe. Un sous-groupe $H$ de $G$ est dit maximal lorsque $H \neq G$ 1. On suppose que $G$ est fini, et on se donne un sous-groupe $H$ de $G$ tel que $\frac{|G|}{|H|}$ soit un nombre premier. Montrer que $H$ est maximal. #+end_exercice +# ID:8714 #+BEGIN_exercice [X MP 2025 # 263] Soit $\phi$ un morphisme de groupes de $\Z^{\N}$ dans $\Z$ nul sur l'ensemble $\Z^{(\N)}$ des suites presque nulles. Montrer que $\phi$ est nul. #+END_exercice +#+BEGIN_proof +Si $\forall n\in\N,\, u_n \mid u_{n+1}$ et $u_n \ra +\i$, alors l'image par $\phi$ est nulle, car l'image est divisible par $u_n$ pour tout $n$. + +Si $(u_n)$ est une suite, on peut trouver deux suites $(a_n), (b_n)$ telles que $a_n - b_n = u_n$, $a_n \wedge b_n = 1$ et $a_n \mid a_{n+1}$ et $b_n \mid b_{n+1}$. +#+END_proof +# ID:nil #+begin_exercice [X MP 2025 # 264] -On pose -$$\alpha = \frac{12 + 5i}{13}$$ -. +On pose $\alpha = \frac{12 + 5i}{13}$. 1. Montrer que $\alpha$ n'est pas une racine de l'unité. 1. Le nombre $\alpha$ est-il racine d'un polynôme unitaire à coefficients dans $\Q$ ? dans $\Z$ ? @@ -2426,10 +3213,9 @@ $$\alpha = \frac{12 + 5i}{13}$$ #+begin_exercice [X MP 2025 # 265] 1. Soient $P, Q \in \C[X]$ premiers entre eux, $z \in \C$ une racine de $A = P^2 + Q^2$. Est-ce que $z$ est racine de $B = P'^2 + Q'^2$ ? Que dire si $z$ est racine multiple de A? - b) Montrer que, si $P \in \R[X]$, $P$ s'écrit $U^2 + V^2$ avec $U$ et $V$ dans $\R[X]$ si et seulement si -$\forall x \in \R, P(x) \geq 0$. - 1. Montrer que tout $P \in \C[X]$ s'écrit $U^2 + V^2$ avec $U$ et $V$ dans $\C[X]$ si et seulement $s$ c) Montrer que tout $P \in \C[X]$ s'écrit $U^2 + V^2$ avec $U$ et $V$ dans $\C[X]$. - 1. Est-ce que tout polynôme $P \in \C[X]$ peut s'écrire $U^3 + V^3$ avec $U$ et $V$ dans $\C[X]$ ? Ind. Montrera que le plus petit facteur premier $p$ de P(a+k) est supérieur ou égal à a, puis que P(a+k-p)=p. + 2. Montrer que, si $P \in \R[X]$, $P$ s'écrit $U^2 + V^2$ avec $U$ et $V$ dans $\R[X]$ si et seulement si $\forall x \in \R, P(x) \geq 0$. + 3. Montrer que tout $P \in \C[X]$ s'écrit $U^2 + V^2$ avec $U$ et $V$ dans $\C[X]$ si et seulement $s$ c) Montrer que tout $P \in \C[X]$ s'écrit $U^2 + V^2$ avec $U$ et $V$ dans $\C[X]$. + 4. Est-ce que tout polynôme $P \in \C[X]$ peut s'écrire $U^3 + V^3$ avec $U$ et $V$ dans $\C[X]$ ? #+end_exercice #+begin_exercice [X MP 2025 # 266] @@ -2457,15 +3243,60 @@ racines complexes appartiennent à $\Q^*$. On pose $H = \max(|c_0|, \dots, |c_n| Soient $A, B \in \M_n(\R)$ de rang $1$ telles que $\tr(A) = \tr(B)$. Montrer que $A$ et $B$ sont semblables. #+end_exercice +# ID:8715 #+begin_exercice [X MP 2025 # 270] Soient $A$ et $B$ appartenant à $\M_n(\R)$, on note $k = \dim \op{Ker}(AB)$. Quelles sont les valeurs possibles pour la dimension de $\op{Ker}(BA)$ ? #+end_exercice #+BEGIN_proof On a $\rg (AB)\leq \max(\rg A, \rg B)$ et $\geq \max(\rg A + \rg B - n, 0)$. Il s'agit de montrer que toutes les valeurs peuvent être prises, indépendamment. -On peut supposer $A = J_r$. +n note $k = \dim \operatorname{Ker}(AB)$ et $k' = \dim \operatorname{Ker}(BA)$. On a $\lceil k/2 \rceil \le k' \le \min(2k, n)$. + +Pour montrer que c'est atteint, On suppose que $k \leq k' \leq 2k$ et on pose $m = k' - k$. On partitionne les indices $\{1, \dots, n\}$ en quatre ensembles : + $I_1 = \{1, \dots, m\}$, $I_2 = \{m+1, \dots, 2m\}$, $I_3 = \{2m+1, \dots, k'\}$ et $I_4 = \{k'+1, \dots, n\}$. + Dans une base $(e_1, \dots, e_n)$, on définit les endomorphismes $u$ et $v$ par : + $$u(e_i) = \begin{cases} e_{i+m} & \text{si } i \in I_1 \\ e_i & \text{si } i \in I_4 \\ 0 & \text{sinon} \end{cases} \quad \text{et} \quad v(e_i) = \begin{cases} e_{i} & \text{si } i \in I_1 \cup I_4 \\ 0 & \text{sinon} \end{cases}$$ #+END_proof +# ID:8716 +#+begin_exercice :noexport: +Soient $A$ et $B$ appartenant à $\mathcal{M}_n(\R)$, on note $k = \dim \operatorname{Ker}(AB)$ et $k' = \dim \operatorname{Ker}(BA)$. +1. Montrer que $\lceil k/2 \rceil \le k' \le \min(2k, n)$. +2. On suppose que $k \leq k' \leq 2k$ et on pose $m = k' - k$. On partitionne les indices $\{1, \dots, n\}$ en quatre ensembles : + $I_1 = \{1, \dots, m\}$, $I_2 = \{m+1, \dots, 2m\}$, $I_3 = \{2m+1, \dots, k'\}$ et $I_4 = \{k'+1, \dots, n\}$. + Dans une base $(e_1, \dots, e_n)$, on définit les endomorphismes $u$ et $v$ par : + $$u(e_i) = \begin{cases} e_{i+m} & \text{si } i \in I_1 \\ e_i & \text{si } i \in I_4 \\ 0 & \text{sinon} \end{cases} \quad \text{et} \quad v(e_i) = \begin{cases} e_{i} & \text{si } i \in I_1 \cup I_4 \\ 0 & \text{sinon} \end{cases}$$ + Déterminer $\dim \operatorname{Ker}(u \circ v)$ et $\dim \operatorname{Ker}(v \circ u)$. Conclure. +#+end_exercice +#+BEGIN_proof +1. /Inégalités :/ + On a $\operatorname{rg}(BA) \ge \operatorname{rg}(A) + \operatorname{rg}(B) - n \ge \operatorname{rg}(AB) + \operatorname{rg}(AB) - n$. + En passant aux noyaux ($n-k'$ pour $BA$ et $n-k$ pour $AB$) : $n-k' \ge 2(n-k) - n = n-2k$, d'où $k' \le 2k$. + Par symétrie, $k \le 2k'$, donc $k' \ge \lceil k/2 \rceil$. De plus, $k' \le n$ est trivial. + +2. /Calcul des noyaux :/ + Les applications linéaires sont bien définies car on donne l'image d'une base. Calculons les rangs des composées. + + - /Pour $v \circ u$ :/ + - Si $i \in I_1$, $v(u(e_i)) = v(e_{i+m})$. Comme $i+m \in I_2$ et que $v$ est nul sur $I_2$, $v(u(e_i)) = 0$. + - Si $i \in I_4$, $v(u(e_i)) = v(e_i) = e_i$. + - Sinon, $u(e_i) = 0$, donc $v(u(e_i)) = 0$. + L'image est engendrée par $(e_i)_{i \in I_4}$, famille libre de cardinal $n - k'$. + Donc $\operatorname{rg}(v \circ u) = n - k'$, soit $\dim \operatorname{Ker}(v \circ u) = k'$. + + - /Pour $u \circ v$ :/ + - Si $i \in I_1$, $u(v(e_i)) = u(e_i) = e_{i+m}$ (vecteurs indexés dans $I_2$). + - Si $i \in I_4$, $u(v(e_i)) = u(e_i) = e_i$ (vecteurs indexés dans $I_4$). + - Sinon, $v(e_i) = 0$, donc $u(v(e_i)) = 0$. + L'image est engendrée par la réunion des vecteurs de base indexés par $I_2$ et $I_4$. Ces ensembles d'indices sont disjoints, la famille est libre. + $\operatorname{rg}(u \circ v) = |I_2| + |I_4| = m + (n - k') = (k' - k) + n - k' = n - k$. + Donc $\dim \operatorname{Ker}(u \circ v) = k$. + + /Conclusion :/ + Les valeurs $k$ et $k'$ sont bien atteintes simultanément pour ce choix de matrices. Par symétrie (en échangeant $u$ et $v$), on couvre aussi le cas $k' < k$. Toutes les valeurs entières de l'intervalle $\left[\lceil k/2 \rceil, \min(2k, n)\right]$ sont possibles. +#+end_proof + + #+begin_exercice [X MP 2025 # 271] Soient $n \in \N^*$ et $C_n = \{-1, 1\}^n$. On pose $H = \{f \in \mc{L}(\R^n), \ f(C_n) = C_n\}$. Montrer que $H$ est un groupe pour la loi de composition et déterminer son cardinal. @@ -2569,9 +3400,7 @@ Pour $n \in \N^*$, on pose $\quad\displaystyle\mc{IA} = \sup \{r \in \N ; \exist #+begin_exercice [X MP 2025 # 280] 1. Soit $A \in \M_n(\R)$ une matrice diagonalisable. Donner une condition nécessaire et suffisante sur $A$ pour qu'il existe $x \in \R^n$ tel que $(x, Ax, \ldots, A^{n-1}x)$ soit une base de $\R^n$. - 1. Soient -$$b_1, b_2, b_3 \in \R$$ - et $M = \begin{pmatrix} b_1 & 0 & 0 \\ 1 & b_2 & 0 \\ 0 & 1 & b_3 \end{pmatrix}$. + 1. Soient $b_1, b_2, b_3 \in \R$ et $M = \begin{pmatrix} b_1 & 0 & 0 \\ 1 & b_2 & 0 \\ 0 & 1 & b_3 \end{pmatrix}$. - À quelle condition la matrice $M$ est-elle diagonalisable? - À quelle condition existe-t-il $x \in \R^3$ tel que $(x, Mx, M^2x)$ soit une base de $\R^3$ ? @@ -2579,7 +3408,7 @@ $$b_1, b_2, b_3 \in \R$$ #+end_exercice #+begin_exercice [X MP 2025 # 281] -$ $ Soient $V$ un $\C$ -espace vectoriel de dimension finie et $G$ un sous-groupe de $\mathrm{GL}(V)$. +Soient $V$ un $\C$ -espace vectoriel de dimension finie et $G$ un sous-groupe de $\mathrm{GL}(V)$. 1. On suppose que $G$ = GL(V). Que vaut Vect(G)? La réciproque est-elle vraie? On suppose maintenant que, pour tout $g \in G$, $g$ id est nilpotent. 1. Quels sont les éléments diagonalisables de G? @@ -2588,20 +3417,26 @@ On suppose maintenant que, pour tout $g \in G$, $g$ id est nilpotent. #+end_exercice #+begin_exercice [X MP 2025 # 282] - 1. Soit $\sum a_n z^n$ une série entière de rayon de convergence $R$ \gt 0. Soit $M \in \M_d(\C)$ une matrice complexe dont les valeurs propres sont de module strictement inférieur à $R$. Montrer que $\sum a_n M^n$ converge. - 1. Existe-t-il une série entière $\sum a_n z^n$ de rayon de convergence R\gt 0 telle que, pour toute matrice $M$ à spectre inclus dans $\overline{D(0,R)}$ et admettant une valeur propre de module $R$, la série $\sum a_n M^n$ diverge? - 1. Existe-t-il une série entière $\sum a_n z^n$ de rayon de convergence R\gt 0 telle que, pour toute matrice $M$ à spectre inclus dans $\overline{D(0,R)}$ admettant une valeur propre de module $R$, la série $\sum a_n M^n$ converge?d) Soit $f\colon z \mapsto \sum_{n=0}^{+\i} a_n z^n$ la somme d'une série entière de rayon de convergence $R$ \gt 0. + 1. Soit $\sum a_n z^n$ une série entière de rayon de convergence $R + \gt 0$. Soit $M \in \M_d(\C)$ une matrice complexe dont les + valeurs propres sont de module strictement inférieur à $R$. + Montrer que $\sum a_n M^n$ converge. + 2. Existe-t-il une série entière $\sum a_n z^n$ de rayon de + convergence $R\gt 0$ telle que, pour toute matrice $M$ à spectre + inclus dans $\overline{D(0,R)}$ et admettant une valeur propre de + module $R$, la série $\sum a_n M^n$ diverge? + 3. Existe-t-il une série entière $\sum a_n z^n$ de rayon de convergence $R\gt 0$ telle que, pour toute matrice $M$ à spectre inclus dans $\overline{D(0,R)}$ admettant une valeur propre de module $R$, la série $\sum a_n M^n$ converge? + 4. Soit $f\colon z \mapsto \sum_{n=0}^{+\i} a_n z^n$ la somme d'une série entière de rayon de convergence $R \gt 0$. -On pose -$$f^{(k)}: z \mapsto \sum_{n=k}^{+\i} n(n-1)\dots(n-k+1)a_n z^n.$$ + On pose + $$f^{(k)}: z \mapsto \sum_{n=k}^{+\i} n(n-1)\dots(n-k+1)a_n z^n.$$ -Soit $M\in\M_d(\C)$ de polynôme caractéristique $\chi_M=\prod_{i=1}^r(X-\lambda_i)^{\alpha_i}$ où les $\lambda_i$ sont distincts + Soit $M\in\M_d(\C)$ de polynôme caractéristique + $\chi_M=\prod_{i=1}^r(X-\lambda_i)^{\alpha_i}$ où les $\lambda_i$ + sont distincts et de module $\lt R$ et les $\alpha_i$ dans $\N^*$. -et de module $\lt R$ et les $\alpha_i$ dans $\N^*$. - - - Montrer l'existence de $P \in \C[X]$ tel que -$\forall i \in [1, r], \forall k \in [0, \alpha_i 1], f^{(k)}(\lambda_i) = P^{(k)}(\lambda_i)$. - - On suppose que $M$ est diagonalisable. Montrer que f(M) = P(M). + - Montrer l'existence de $P \in \C[X]$ tel que $\forall i \in \db{1,r}, \forall k \in \db{0, \a_i - 1},\, f^{(k)}(\lambda_i) = P^{(k)}(\lambda_i)$. + - On suppose que $M$ est diagonalisable. Montrer que $f(M) = P(M)$. - Est-ce toujours le cas si on ne suppose plus $M$ diagonalisable? #+end_exercice @@ -2707,37 +3542,69 @@ Dans la suite de l'exercice, on suppose $A \in \mc{S}_n(\R)$ et on note $\lambda ** Analyse +# ID:8759 #+begin_exercice [X MP 2025 # 295] 1. Soient $N_1$ et $N_2$ deux normes sur un $\R$ -espace vectoriel $E$. Montrer que si $N_1$ et $N_2$ ont la même sphère unité alors $N_1 = N_2$. 1. On pose $E = C^0([0,1],\R)$. Soit $(f,g) \in E^2$. Donner une condition nécessaire et suffisante pour que $(x,y) \in \R^2 \mapsto \|x f + y g\|_{\i}$ soit une norme sur $\R^2$. - 1. Soit $(E, \langle, \rangle)$ un espace euclidien, dont on note $\| \|$ la norme. Soit $p$ une autre norme sur $E$. On note $S$ et $S_p$ les sphères unité respectives pour $\| \|$ et $p$. Montrer que $d: x \in S \mapsto$ $\sup |\langle x,y\rangle|$ est à valeurs dans $\R^{+*}$, que $k=\sup \|y\|$ est un réel strictement positif, et enfin $y \in S_p$ que $d$ est $k$-lipschitzienne pour la norme $\| \cdot \|$. + 1. Soit $(E, \langle, \rangle)$ un espace euclidien, dont on note $\| \|$ la norme. Soit $p$ une autre norme sur $E$. On note $S$ et $S_p$ les sphères unité respectives pour $\| \|$ et $p$. Montrer que $d: x \in S \mapsto$ $\sup |\langle x,y\rangle|$ est à valeurs dans $\R^{+*}$, que $k=\sup_{y\in S_p} \|y\|$ est un réel strictement positif, et enfin que $d$ est $k$-lipschitzienne pour la norme $\| \cdot \|$. 1. On note $B = \{f \in E, \ p(f) \leq 1\}$ et, pour $x \in S, D_x = \{z \in E; |\langle x, z \rangle| \leq d(x)\}$. Montrer que $B = \bigcap_{x \in S} D_x$. #+end_exercice +#+BEGIN_proof + 1. + 2. $f,g$ non colinéaires. + 3. +#+END_proof + +# ID:7293 #+begin_exercice [X MP 2025 # 296] -Soit $E$ un $\R$ -espace vectoriel de dimension finie. Montrer que tout convexe non borné contient au moins une demi-droite. On pourra commencer par le cas d'un convexe fermé. +Soit $E$ un $\R$ espace vectoriel de dimension finie. Montrer que tout convexe non borné contient au moins une demi-droite. On pourra commencer par le cas d'un convexe fermé. #+end_exercice + #+begin_exercice [X MP 2025 # 297] -Pour $k \in \N^*$, soit $R_k$ la borne inférieure de l'ensemble $E_k$ des $r \in \R^{+*}$ tels qu'il existe une boule fermée de $\R^2$ euclidien de rayon $r$ contenant au moins $k$ points de $\Z^2$. - 1. Calculer $R_k$ pour $k$ = 2, 3, 4. +Pour $k \in \N^*$, soit $R_k$ la borne inférieure de l'ensemble $E_k$ des $r \in \R_+^*$ tels qu'il existe une boule fermée de $\R^2$ euclidien de rayon $r$ contenant au moins $k$ points de $\Z^2$. + 1. Calculer $R_k$ pour $k = 2, 3, 4$. 1. Si $k \in \N^*$, montrer que $R_k$ est le minimum de $E_k$. 1. Montrer que, pour $k \in \N^*$, $4R_k^2$ est entier. 1. Donner un équivalent de $R_k$ lorsque $k$ tend vers $+\i$. #+end_exercice +#+BEGIN_proof + 1. + 2. + 3. +#+END_proof + #+BEGIN_exercice [X MP 2025 # 298] -Soit $E$ l'espace des fonctions continues de [0,1] dans $\R$. On munit $E$ de la norme $\| \|_{\i}$. Déterminer les formes linéaires continues $\phi$ sur $E$ telles que, pour tout $(f,g) \in E^2$ tel que $\phi(fg) = 0$, on ait $\phi(f) = 0$ ou $\phi(g) = 0$. +Soit $E$ l'espace des fonctions continues de $[0,1]$ dans $\R$. On munit $E$ de la norme $\| \|_{\i}$. Déterminer les formes linéaires continues $\phi$ sur $E$ telles que, pour tout $(f,g) \in E^2$ tel que $\phi(fg) = 0$, on ait $\phi(f) = 0$ ou $\phi(g) = 0$. #+END_exercice +#+BEGIN_proof +Il y a les $f\mapsto f(a)$. + +Si $f$ ne s'annule pas et $\phi(f) = 0$, $\phi(\sqrt{f}) = 0$, donc en itérant $\phi(1) = 0$. + +Alors on a $\phi(f\times \frac{1}{f}) = 0$, donc $\phi$ va trop s'annuler : Soit $f$, on veut $\phi(f) = 0$, sinon, $\phi\big(\frac{1}{f+k}\big) = 0$ pour tout $k\gt 0$, et $\frac{1}{f+k} - \frac{1}{f}= \frac{1}{f}$. Sinon $\frac{1}{f^{1/2^k}} - \frac{1}{f^{1/2^{k+1}}}$ a un équivalent en $-\frac{\ln f}{2^{k+1}}$, donc, en renormalisant, $\phi(\ln f) = 0$. On peut appliquer ça à $e^f$ ou $e^{-f}$ pour conclure. + +Maintenant si $\phi$ est nulle sur $f$ et sur $g$ et qu'ils n'ont pas +de zéros en communs. +#+END_proof + #+begin_exercice [X MP 2025 # 299] -Soit $\rho: [0,1] \mapsto \M_n(\C)$ continue telle que, pour tout $t, \rho(t)^2 = \rho(t)$. +Soit $\rho\colon [0,1] \mapsto \M_n(\C)$ continue telle que, pour tout $t, \rho(t)^2 = \rho(t)$. 1. Montrer que $t \mapsto \op{rg} \rho(t)$ est constante. 1. Montrer l'existence de $u \in \mc C^0([0,1], \mathrm{GL}_n(\C))$ telle que $\forall t, \rho(t) = u(t)\rho(0)u^{-1}(t)$. 1. On suppose de plus que $\rho(1) = \rho(0)$. Montrer que l'on peut choisir $u$ de sorte que l'on ait aussi $u(0) = u(1)$. #+end_exercice +#+BEGIN_proof + 1. Clair. + 2. Localement, puis recollement. + 3. On peut mettre une distance sur les familles libres de $k$-vecteurs unitaires ? +#+END_proof + #+begin_exercice [X MP 2025 # 300] Soit $n \ge 2$. On note $\mc{B}_n$ l'ensemble des matrices bistochastiques de $\M_n(\R)$ c'est-à-dire les $M = (m_{i,j})_{1\leq i,j\leq n} \in \M_n(\R)$ telles que : $\forall i \in \db{1,n},\, \sum_{i=1}^n m_{i,j} = 1,\, \forall j \,\in\, \db{1,n},\sum_{i=1}^n m_{i,j} = 1$ et $\forall (i,j) \in \db{1,n}^2 m_{i,j} \geq 0$. Si $\sigma \in \mc{S}_n$, on note $P_{\sigma} = (\delta_{i,\sigma(j)})_{1 \leq i,j \leq n}$ la matrice de permutation associée à $\sigma$ ; la matrice $P_{\sigma}$ est dans $\mc{B}_n$. @@ -2773,54 +3640,95 @@ Soient $K$ une fonction continue de $[0,1]^2$ dans $\R$, $E$ l'espace des foncti 1. On suppose que $K$ est à valeurs dans $\R^{+*}$, que $\lambda \in \R^{+*}$ et que l'espace propre $E_{\lambda}(T_K)$ contient une fonction non identiquement nulle à valeurs dans $\R^+$. Montrer que $E_{\lambda}(T_K)$ est de dimension 1. #+end_exercice +# ID:nil #+begin_exercice [X MP 2025 # 303] Soit $(u_n)_{n\geq 0}$ une suite réelle telle que $u_{n+1}-\frac{u_n}{2}\ra 0$. Montrer que $u_n\ra 0$. #+end_exercice +# ID:8760 #+begin_exercice [X MP 2025 # 304] -Soient a \lt $b$ réels et $(u_n)_{n \in \N}$ une suite réelle telle que, pour tout $t \in [a, b]$, il existe une suite $(k_n)_{n\in\N}$ d'entiers tels que $tu_n-k_n\longrightarrow 0$ quand $n\ra +\i$. Montrer que la suite $(u_n)$ converge vers 0. +Soient $a \lt b$ réels et $(u_n)_{n \in \N}$ une suite réelle telle que, pour tout $t \in [a, b]$, il existe une suite $(k_n)_{n\in\N}$ d'entiers tels que $tu_n-k_n\longrightarrow 0$ quand $n\ra +\i$. Montrer que la suite $(u_n)$ converge vers $0$. #+end_exercice +#+BEGIN_proof +On peut supposer $a = 0$. On peut supposer $b = 1$ (remplacer $(u_n)$ par $tu_n$). +Alors $(u_n)$ est proche d'un entier. + +Il faut créer un $t$ qui ne marche pas, par segments emboîtés. + +Si on imagine que $u_n$ n'est pas majorée, on prend un segment qui ne marche pas pour $u_0$. Dans ce segment, un sous-segment qui ne marche pas pour une valeur suivante de $(u_n)$, qui soit assez grand etc. + +Par ailleurs, si $(u_n)$ a une valeur d'adhérence différente de $+\i$, clairement un même $t$ ne marche pour aucun. +#+END_proof + + +# À relier. #+begin_exercice [X MP 2025 # 305] Soient $\alpha \in \R^{+*}$ et $\beta = 1/\alpha$. Soit $(z_n)_{n \geq 0}$ la suite définie par $z_0 = 1$ et, pour tout $n \in \N, z_{n+1} = \frac{\alpha n + 1}{\alpha(n+1)} z_n$. 1. Donner un équivalent de $z_n$ et sa valeur exacte lorsque $\beta \in \N^*$. - 1. Soit $(x_n)_{n\geq 0}$ une suite réelle. - - On pose, pour $n \in \N$, $\mu_n = \frac{1}{n+1} \sum_{k=0}^n x_k$ et $y_n = \alpha x_n + (1-\alpha)\mu_n$. On suppose que $y_n \ra x \in \R$. Montrer que $x_n \ra x$. + 1. Soit $(x_n)_{n\geq 0}$ une suite réelle. On pose, pour $n \in \N$, $\mu_n = \frac{1}{n+1} \sum_{k=0}^n x_k$ et $y_n = \alpha x_n + (1-\alpha)\mu_n$. On suppose que $y_n \ra x \in \R$. Montrer que $x_n \ra x$. #+end_exercice +#+BEGIN_proof + 1. +#+END_proof + #+BEGIN_exercice [X MP 2025 # 306] -Pour $n \in \N$, on pose $u_n = |\{(p,q) \in \N^2, p^2 + q^2 = n\}|$. +Pour $n \in \N$, on pose $u_n = |\{(p,q) \in \N^2, ,\, p^2 + q^2 = n\}|$. 1. Déterminer la limite de la suite de terme général $\frac{1}{n}\sum u_k$. 1. Étudier la nature de la suite $(u_n)$. 1. Montrer que $(u_n)$ n'est pas bornée. #+END_exercice +#+BEGIN_proof + 1. Nombres de points entiers dans une boule. + 2. Elle diverge, car elle vaut $0$ souvent. + 3. Si on sait que tout nombre premier $4k+1$ l'est, en prenant $p_1^2\dots p_k^2$ on a plein de façons de l'écrire. + + Si on sait que si on est divisible par un nombre premier $4k+3$ et pas son carré, et que $\sum \frac{1}{p}$ diverge, pour les $4k+3$, on obtient que la proportion d'entiers pour lesquels c'est $0$ est $1$, je pense, ce qui suffit. + + Si on ne sait pas ça. Il nous reste à regarder des $2^n \cdots 5^m$ : On a $2 = 1 + 1$ et $5 = 4+1$, $25 = 25 + 0 = 16 + 9$ a deux façons de l'écrire, $125 = 100 + 25 = 121 + 4$, n'a que deux façons… + + $(a^2 + b^2) (2^2 + 1^2) = (2a - b)^2 + (2b + a)^2$ $=(2b-a)^2 + (2a+b)^2$ + + !! +#+END_proof + +# ID:8761 #+begin_exercice [X MP 2025 # 307] Soit $(a_n)_{n\in\N}$ une suite réelle vérifiant, pour tout $n\in\N$, $a_{n+1}=a_n(1-a_n)$. - 1. On suppose que $a_0 = 1/2$. Montrer que $\frac{1}{a_n} n \sim \ln n$ quand $n \ra +\i$. + 1. On suppose que $a_0 = 1/2$. Montrer que $\frac{1}{a_n}- n \sim \ln n$ quand $n \ra +\i$. 1. On suppose $a_0 \gt 1$. Déterminer la limite de $(a_n)$ puis un équivalent de $a_n$. 1. Donner un développement asymptotique à deux termes de $a_n$. #+end_exercice +# ID:8762 #+begin_exercice [X MP 2025 # 308] 1. Pour $n \ge 3$, justifier l'existence de $x_n, y_n \in \R$ avec $0 \lt x_n \lt y_n$ solutions de $x - n \ln x = 0$. 1. Donner un développement asymptotique à deux termes de $x_n$ et $y_n$. #+end_exercice #+begin_exercice [X MP 2025 # 309] -Construire une suite strictement croissante $(p_n)_{n\geq 2}$ d'entiers avec $p_2=2$ telle qu'il - -existe C\gt 0 vérifiant, pour tout $n\geq 2$, $\sum_{k=n}^{p_{n+1}-1}\frac{1}{\ln k}\geq C$, et telle que la série de terme général $2^{-(p_{n+1}-p_n)}$ diverge. +Construire une suite strictement croissante $(p_n)_{n\geq 2}$ d'entiers avec $p_2=2$ telle qu'il existe $C\gt 0$ vérifiant, pour tout $n\geq 2$, $\sum_{k=p_n}^{p_{n+1}-1}\frac{1}{\ln k}\geq C$, et telle que la série de terme général $2^{-(p_{n+1}-p_n)}$ diverge. #+end_exercice +#+BEGIN_proof +#+END_proof + + +# ID:8758 #+BEGIN_exercice [X MP 2025 # 310] On pose $\alpha = 4 \sum_{k=0}^{499999} \frac{(-1)^k}{2k+1}$. Montrer qu'exactement une des 16 premières décimales de $\alpha$ diffère de la décimale de $\pi$ correspondante. #+END_exercice +#+BEGIN_proof +Minoration et majoration d'une série alternée, en regroupant les termes deux à deux et comparaison avec une intégrale. +#+END_proof + #+begin_exercice [X MP 2025 # 311] -Soient p\gt 0 et q\gt 0 tels que $\frac{1}{p}+\frac{1}{q}=1$ et $n\in\N^*$. Montrer que, pour tout +Soient $p\gt 0$ et $q\gt 0$ tels que $\frac{1}{p}+\frac{1}{q}=1$ et $n\in\N^*$. Montrer que, pour tout $$(a_1, \dots, a_n, b_1, \dots, b_n) \in (\R^+)^{2n}, \sum_{i=1}^n a_i b_i \leq \left(\sum_{i=1}^n a_i^p\right)^{\frac{1}{p}} \left(\sum_{i=1}^n b_i^q\right)^{\frac{1}{q}}$$. #+end_exercice @@ -2871,36 +3779,73 @@ $$\forall (x,y) \in \R^2, \ \alpha F(x) F(y) \leq F(x+y) \leq \eta F(x) F(y)$$. #+end_exercice #+begin_exercice [X MP 2025 # 318] -Soient $M, m \in \R$ avec 0 \lt $m$ \lt $M$, $f \in \mc C^0(\R, [m, M])$, $q \in \R \setminus \{-1, 0, 1\}$. Soit () l'équation fonctionnelle $\forall t \in \R, g(t) = 1 + \frac{g(qt)}{f(t)}$. - 1. On suppose $m$ \gt 2 ou $M$ \lt 1/2. Montrer qu'il existe une unique solution bornée de (). - 1. Montrer que les solutions bornées de () ne s'annulent pas. +Soient $M, m \in \R$ avec $0\lt m\lt M$, $f \in \mc C^0(\R, [m, M])$, $q \in \R \setminus \{-1, 0, 1\}$. Soit $(E)$ l'équation fonctionnelle $\forall t \in \R, g(t) = 1 + \frac{g(qt)}{f(t)}$. + 1. On suppose $m\gt 2$ ou $M\lt 1/2$. Montrer qu'il existe une unique + solution bornée de $(E)$. + 2. Montrer que les solutions bornées de $(E)$ ne s'annulent pas. #+end_exercice +#+BEGIN_proof + 1. Unicité est claire. Pour l'existence, on peut imaginer que + l'équation est $g(3t) = 1 + g(t) u(t)$, avec $u(t)\gt 2$. La + fonction $g$ est entièrement déterminée par ses valeurs sur + $[1,3[$. + On est ramené à une suite $u_{n+1} = 1 + u_n v_n$. Pour $u_0$ + assez grand, on tend vers $+\i$, et pour $u_0$ assez petit vers + $-\i$, à la limite, on est bornée. + 2. ?? +#+END_proof + + + +# ID:8549 #+begin_exercice [X MP 2025 # 319] Soit $E = \R[X]$. Soit $\phi \in \mc{L}(E)$. 1. Montrer qu'il existe une unique suite $(G_n)_{n\geq 0}\in E^{\N}$ telle que : -$$\forall P \in E, \phi(P) = \sum_{n=0}^{+\i} G_n P^{(n)}$$. - - 1. Expliciter $(G_n)$ pour $\phi$ vérifiant : $\forall P \in E, \forall x \in \R, \phi(P)(x) = \int_0^x P(t) dt$. - 1. On suppose que, pour tout $P \in E$ et $a \in \R$, si $P$ admet un minimum local en a alors $\phi(P)(a) = 0$. Montrer qu'il existe $Q \in E$ tel que, pour tout $P \in E$, $\phi(P) = QP'$. - 1. On suppose que, pour tout $P \in E$ et $a \in \R$, si $P$ admet un minimum local en a alors $\phi(P)(a) \geq 0$. Montrer qu'il existe $Q, R \in E$ tels que, pour tout $P \in E$, $\phi(P) = QP' +$ -RP'' avec $R$ positif sur $\R$. - 1. Donner une preuve directe de l'égalité trouvée en b). + $$\forall P \in E, \phi(P) = \sum_{n=0}^{+\i} G_n P^{(n)}$$. + 2. Expliciter $(G_n)$ pour $\phi$ vérifiant : $\forall P \in E, + \forall x \in \R, \phi(P)(x) = \int_0^x P(t) \dt$. + 3. On suppose que, pour tout $P \in E$ et $a \in \R$, si $P$ admet un + minimum local en a alors $\phi(P)(a) = 0$. Montrer qu'il existe $Q + \in E$ tel que, pour tout $P \in E$, $\phi(P) = QP'$. + 4. On suppose que, pour tout $P \in E$ et $a \in \R$, si $P$ admet un + minimum local en a alors $\phi(P)(a) \geq 0$. Montrer qu'il existe + $Q, R \in E$ tels que, pour tout $P \in E$, $\phi(P) = QP' + RP''$ + avec $R$ positif sur $\R$. #+end_exercice +#+BEGIN_proof + 1. Le vérifier sur la base $X^n$. + 2. + 3. On a $\phi(1) = 0$. Puis $\phi((X-a)^2)(a) = 0$, donc $G_2(a) = 0$. Puis $\phi((X-a)^2 +(X-a)^3)(a) = 0$, donc $G_3(a) = 0$. + 4. Probablement similaire. +#+END_proof + #+begin_exercice [X MP 2025 # 320] Soient $f\colon \R \ra \R$ et $g\colon \R \ra \R$. On suppose qu'il existe quatre réels strictement positifs -$\alpha, \beta, A, B$ tels que $\forall (x, y) \in \R^2, \ |f(x) f(y)| \leq A |x y|^{\alpha}$ et $|g(x) g(y)| \leq B |x y|^{\beta}$ et $\alpha + \beta \gt 1$. On pose $\zeta : s \in ]1, +\i[ \mapsto \sum_{n=1}^{+\i} \frac{1}{n^s} \cdot \text{On fixe deux réels } a \lt b$. - 1. Pour une subdivision $\sigma=(x_0,\ldots,x_n)$ de [a,b], on pose $J(\sigma)=\sum_{k=0}^{n-1}f(x_k)(g(x_{k+1})-g(x_k))$ -$g(x_k)$). Montrer que $|J(\sigma) f(a)(g(b) g(a))| \le AB \zeta(\alpha + \beta) (2(b-a))^{\alpha+\beta}$. - 1. Montrer qu'il existe un réel $I_{a,b}(f,g)$ tel que, pour tout $\eps \gt 0$, il existe $\delta \gt 0$ tel que, pour toute subdivision $\sigma = (x_0, \dots, x_n)$ de [a,b], $\max_k |x_{k+1} x_k| \lt \delta \Rightarrow |J(\sigma) I_{a,b}(f,g)| \lt \eps$. +$\alpha, \beta, A, B$ tels que $\forall (x, y) \in \R^2, \ |f(x)- f(y)| \leq A |x- y|^{\alpha}$ et $|g(x)- g(y)| \leq B |x- y|^{\beta}$ et $\alpha + \beta \gt 1$. On pose $\zeta\colon s \in ]1, +\i[ \mapsto \sum_{n=1}^{+\i} \frac{1}{n^s}$. On fixe deux réels $a \lt b$. + 1. Pour une subdivision $\sigma=(x_0,\ldots,x_n)$ de $[a,b]$, on pose + $J(\sigma)=\sum_{k=0}^{n-1}f(x_k)(g(x_{k+1})-g(x_k))$. Montrer que + $|J(\sigma) f(a)(g(b) g(a))| \le AB \zeta(\alpha + \beta) + (2(b-a))^{\alpha+\beta}$. + 2. Montrer qu'il existe un réel $I_{a,b}(f,g)$ tel que, pour tout + $\eps \gt 0$, il existe $\delta \gt 0$ tel que, pour toute + subdivision $\sigma = (x_0, \dots, x_n)$ de $[a,b]$, + $\max_k |x_{k+1} x_k| \lt \delta \Rightarrow |J(\sigma) + I_{a,b}(f,g)| \lt \eps$. #+end_exercice +#+BEGIN_proof + 1. !! +#+END_proof + +# ID:8550 #+begin_exercice [X MP 2025 # 321] On note $S$ l'ensemble des nombres complexes de module 1. Soit $\gamma:[0,1]\ra S$ une fonction continue. Montrer qu'il existe une fonction continue $\theta:[0,1]\ra\R$ telle que $\gamma(t)=e^{2i\pi\theta(t)}$ pour tout $t\in[0,1]$. #+end_exercice +# ID:nil # Problème d'énoncé #+BEGIN_exercice [X MP 2025 # 322] Soit $f:[0,1]\ra\R$ continue. On pose $h:t\in[0,1]\mapsto \inf_{s\in[0,t]}f(s)$ et g=f-2h. @@ -2917,40 +3862,45 @@ On pose $\Psi(x) = \sum_{\substack{p \in \mc{P}, \alpha \in \N^* \\ p^{\alpha} \ 1. Montrer que $T(x) 2T\left(\frac{x}{2}\right) = \sum_{n=0}^{\i} (-1)^{n-1} \Psi\left(\frac{x}{n}\right) = x \ln 2 + O(\ln x)$. #+end_exercice +# ID:nil # Facile #+begin_exercice [X MP 2025 # 324] Soit $f$ une bijection de classe $\mc C^1$ de $\R^+$ sur $\R^+$, de réciproque notée $g$. 1. Montrer que, pour $x \ge 0$, $\int_a^x f(t)dt + \int_a^{f(x)} g(t)dt = x f(x)$. 1. Déduire que $\forall x, y \in \R^+, xy \leq \int_0^x f(t) dt + \int_0^y g(t) dt$. #+end_exercice +# ID:8552 #+begin_exercice [X MP 2025 # 325] -Soit $f:[0,1] \ra \R$ continue et strictement positive sur ]0,1[. - 1. Calculer $\lim_{p \ra +\i} \left( \int_0^1 f(x)^p dx \right)^{1/p}$. - 1. Calculer $\lim_{x\ra 0^+} \left( \int_0^1 f(x)^p dx \right)^{1/p}$. +Soit $f\colon [0,1] \ra \R$ continue et strictement positive sur $]0,1[$. + 1. Calculer $\lim\limits_{p \ra +\i} \left( \int_0^1 f(x)^p dx \right)^{1/p}$. + 1. Calculer $\lim\limits_{x\ra 0^+} \left( \int_0^1 f(x)^p dx \right)^{1/p}$. #+end_exercice +# ID:8551 #+begin_exercice [X MP 2025 # 326] Soit $f$ la fonction 1-périodique de $\mathbb R$ dans $\mathbb R$ telle que $\forall x \in [0,1[,f(x)=x-\frac{1}{2}$. Pour $i$ et $j$ dans $\N^*$, calculer $\int_0^1 f(ix)f(jx) dx$. #+end_exercice +# ID:nil #+begin_exercice [X MP 2025 # 327] -Pour a, $b$ \gt 0, on définit $J_{a,b} = 2 \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{(a\cos\theta)^2 + (b\sin\theta)^2}}$. +Pour $a,b\gt 0$, on définit $J_{a,b} = 2 \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{(a\cos\theta)^2 + (b\sin\theta)^2}}$. 1. Montrer que $J_{a,b} = \int_{-\i}^{+\i} \frac{dx}{\sqrt{(x^2 + a^2)(x^2 + b^2)}}$. 1. Montrer que $J_{a,b} = J_{\frac{a+b}{2}} \sqrt{ab}$ #+end_exercice +# ID:8553 #+begin_exercice [X MP 2025 # 328] -Déterminer les réels $\alpha$ et $\beta$ tels que $\int_0^{+\i} |\sin t|^{\alpha} t^{\beta} dt \lt +\i$.329. [nil] a) Pour $f \in \mc C^0(\R, \R)$, on note $I_f = \left\{ p \gt 0, \int_{\R} |f|^p \lt +\i \right\}$. Montrer que $I_f$ +Déterminer les réels $\alpha$ et $\beta$ tels que $\int_0^{+\i} |\sin t|^{\alpha} t^{\beta} dt \lt +\i$. #+end_exercice +# ID:8554 #+begin_exercice [X MP 2025 # 329] - 1. Pour -$$f \in \mc C^0(\R, \R)$$ -, on note $I_f = \{p \gt 0, \int_{\R} |f|^p \lt +\i \}$. Montrer que est un intervalle et exhiber $f$ telle que $I_f = ]a, b[$, $]0, b[$ ou $]b, +\i[$ pour $0 \lt a \lt b$. + 1. Pour $f \in \mc C^0(\R, \R)$, on note $I_f = \{p \gt 0, \int_{\R} |f|^p \lt +\i \}$. Montrer que $I_f$ est un intervalle et exhiber $f$ telle que $I_f = ]a, b[$, $]0, b[$ ou $]b, +\i[$ pour $0 \lt a \lt b$. - 1. Déterminer $\lim_{p\ra+\i} \left( \int_a^1 |f|^p \right)^{1/p}$. + 1. Déterminer $\lim\limits_{p\ra+\i} \left( \int_a^1 |f|^p \right)^{1/p}$. #+end_exercice +# ID:8281 #+begin_exercice [X MP 2025 # 330] Soit $f\colon \R \ra \R$ intégrable sur $\R$. On pose $g\colon x \in \R^* \mapsto f\left(x - \frac{1}{x}\right)$. Montrer que $g$ est intégrable sur $\R^{+*}$ et sur $\R^{-*}$. Exprimer $\int_{-\i}^0 g + \int_0^{+\i} g$ en fonction de $\int_{-\i}^{+\i} f$. #+end_exercice @@ -2973,31 +3923,37 @@ Ind. On pourra s'intéresser au déterminant de la matrice $(p_{i-1}(x_j))_{1 \l #+end_exercice #+begin_exercice [X MP 2025 # 332] -Soit $(f_n)_{n\in\N}$ une suite de fonctions de carré intégrable sur $\R$ telle que $\int_{\R} f_i f_j =$ - - $\delta_{i,j}$ pour tous $i,j\in\N$. Pour $N\in\N^*$ et $x,y\in\R$, on pose $K_N(x,y)=\sum_{i=1}^N f_k(x)f_k(y)$. Pour $p \in \N$ et $x_1, \ldots, x_p \in \R$, on pose $\phi_p(x_1, \ldots, x_p) = \det((K_N(x_i, x_j))_{1 \leq i, j \leq p})$. +Soit $(f_n)_{n\in\N}$ une suite de fonctions de carré intégrable sur $\R$ telle que $\int_{\R} f_i f_j =\delta_{i,j}$ pour tous $i,j\in\N$. Pour $N\in\N^*$ et $x,y\in\R$, on pose $K_N(x,y)=\sum_{i=1}^N f_k(x)f_k(y)$. Pour $p \in \N$ et $x_1, \ldots, x_p \in \R$, on pose $\phi_p(x_1, \ldots, x_p) = \det((K_N(x_i, x_j))_{1 \leq i, j \leq p})$. Calculer $\int_{\R^n} \dots \int_{\R^n} \phi_p(x_1, \dots, x_p) dx_1 \dots dx_p$. #+end_exercice #+begin_exercice [X MP 2025 # 333] - 1. Soit $a \in \R^{+*}$. Calculer les intégrales $\int_0^1 \frac{\ln(1+t^a)}{t} dt$ et $\int_0^1 \frac{\ln(1-t)}{t} dt$. + 1. Soit $a \in \R_+^*$. Calculer les intégrales $\int_0^1 \frac{\ln(1+t^a)}{t} dt$ et $\int_0^1 \frac{\ln(1-t)}{t} dt$. 1. Soit $(a_n)_n \in (\N^*)^{\N}$ telle que $I \in \mc{P}_f(\N) \mapsto \sum_{n \in I} a_n$ soit injective, $\mc{P}_f(\N)$ désignant l'ensemble des parties finies de $\N$. Montrer que $\sum_{n=0}^{+\i} \frac{1}{a_n} \leq 2$. $c$. Soit $(a_n)_n \in (\N^*)^{\N}$ telle qu'il n'existe pas d'entier $n$ ni de partie finie $I$ de $\N \setminus \{n\}$ telle que $a_n = \sum_{k \in I} a_k$. Montrer que $\sum_{k=1}^{+\i} \frac{1}{a_n} \le 50$. #+end_exercice +#+BEGIN_proof + 1. + 2. !! + 3. +#+END_proof + +# ID:8555 #+BEGIN_exercice [X MP 2025 # 334] Soient $(a_n)_{n\in\N}$ et $(b_n)_{n\in\N}$ deux suites réelles. -On pose $f_n: x \in \R \mapsto a_n \cos(nx) + b_n \sin(nx)$. Montrer que si $(f_n)_{n \in \N}$ converge simplement sur $\R$ alors $(a_n)_{n \in \N}$ et $(b_n)_{n \in \N}$ convergent vers 0. +On pose $f_n\colon x \in \R \mapsto a_n \cos(nx) + b_n \sin(nx)$. Montrer que si $(f_n)_{n \in \N}$ converge simplement sur $\R$ alors $(a_n)_{n \in \N}$ et $(b_n)_{n \in \N}$ convergent vers $0$. #+END_exercice +#+BEGIN_proof +On prend $x = \pi$, on obtient $a_n\ra 0$. +#+END_proof #+begin_exercice [X MP 2025 # 335] -Pour -$$n \in \N$$ -, soit $f_n : x \in \R \setminus \Z \mapsto \pi \cot(\pi x) - \sum_{k=-\i}^n \frac{1}{x+k}$. +Pour $n \in \N$, soit $f_n : x \in \R \setminus \Z \mapsto \pi \cot(\pi x) - \sum_{k=-\i}^n \frac{1}{x+k}$. 1. Montrer que $(f_n)_{n\geq 0}$ converge simplement sur $\R\setminus\Z$ vers une fonction $f$, et que l'on peut prolonger $f$ par continuité à $\R$. 1. Montrer que la fonction prolongée par continuité est de classe $\mc C^1$ sur $\R$ et vérifie : @@ -3017,9 +3973,7 @@ $$g(x) = 1 - \frac{x}{2} + \sum_{n=1}^{+\i} a_n x^{2n}$$ #+begin_exercice [X MP 2025 # 336] Soit $f \in C^0([0,1], \R)$. -Si -$$t \ge 0$$ -, on pose $g_t : x \in [0, 1] \mapsto \inf \{ f(y) + t | y - x |, y \in [0, 1] \}$. +Si $t \ge 0$, on pose $g_t \colon x \in [0, 1] \mapsto \inf \{ f(y) + t | y - x |, y \in [0, 1] \}$. 1. Si $t \ge 0$, montrer que $g_t$ est une fonction continue. 1. Soit $x \in [0, 1]$. Montrer que la suite $(g_n(x))_{n \ge 0}$ est croissante et qu'elle converge vers @@ -3027,21 +3981,33 @@ f(x). 1. Montrer que $(g_n)_{n\geq 0}$ converge uniformément vers $f$ sur [0,1]. #+end_exercice +# ID:8764 #+begin_exercice [X MP 2025 # 337] 1. Soit $n \in \N$. Montrer qu'il existe un unique $T_n \in \Z[X]$ tel que : $\forall x \in \R, T_n(2\cos(x)) = 2\cos(nx)$. - 1. Pour $x, y \in [-2, 2[$ avec $x \neq y$, on pose $S(x, y) = \sum_{n=0}^{+\i} \frac{1}{n} T_n(x) T_n(y)$. + 1. Pour $x, y \in [-2, 2[$ avec $x \neq y$, on pose $S(x, y) = \sum_{n=1}^{+\i} \frac{1}{n} T_n(x) T_n(y)$. - Montrer que $S_n(x, y)$ est bien défini. - - Montrer que, pour $x, y \in [-2, 2[$ avec $x \neq y$, on a $S(x, y) = -2 \ln |x y|$. + - Montrer que, pour $x, y \in [-2, 2[$ avec $x \neq y$, on a $S(x, y) = -2 \ln |x- y|$. #+end_exercice +#+BEGIN_proof + 1. + 2. + 1. C'est $\sum_{n=0}^{+\i} \frac{1}{n} \cos (n u) \cos (nu)$, donc $\cos (n \frac{(u-v)}{2}) + \cos (n \frac{(u+v)}{2})$, où $u = \arccos \frac{x}{2}$ + 2. Vient de $\sum_{n=1}^{+\infty} \frac{\cos(n\theta)}{n} = -\ln|2\sin(\frac{\theta}{2})|$, du fait que le développement du $\ln$ est valable en tout $z\in \m U$, sauf $1$. +#+END_proof + +# ID:8763 #+begin_exercice [X MP 2025 # 338] Soit $\alpha \in \R$. 1. À quelle condition sur $\alpha$ la fonction $f\colon x \mapsto \sum_{n=1}^{+\i} \frac{n^{\alpha}}{n+x}$ est-elle définie sur $\R^+$ ? 1. Lorsque $f$ est définie sur $\R^+$, déterminer sa limite, puis un équivalent, en $+\i$. - 1. On fixe un polynôme $P \in \R[X]$ de degré $d$ \gt 0, sans racine dans $[1, +\i[$. Donner une condition nécessaire et suffisante sur $(\alpha, d)$ pour que $g\colon x \mapsto \sum_{i=1}^{+\i} \frac{n^{\alpha}}{P(n+x)}$ soit définie - -sur $\R^+$. Dans ce cas, donner un équivalent de $g$ en $+\i$. + 1. On fixe un polynôme $P \in \R[X]$ de degré $d \gt 0$, sans racine dans $[1, +\i[$. Donner une condition nécessaire et suffisante sur $(\alpha, d)$ pour que $g\colon x \mapsto \sum_{i=1}^{+\i} \frac{n^{\alpha}}{P(n+x)}$ soit définie sur $\R^+$. Dans ce cas, donner un équivalent de $g$ en $+\i$. #+end_exercice +#+BEGIN_proof + 1. $\a\lt 0$. + 2. Sans difficulté. +#+END_proof + #+BEGIN_exercice [X MP 2025 # 339] 1. On fixe un entier $d \ge 0$. Soit $(c_k)_{k \le d}$ une famille de nombres complexes indexée par $\Z_{\leq d} = \{k \in \Z, k \leq d\}$. On suppose qu'il existe un réel $R$ \gt 0 telle que $(c_k z^k)_k$ soit sommable pour tout $z \in \C$ tel que |z| \gt R; pour un tel $z$, on pose $g(z) = \sum_k c_k z^k$. On suppose enfin que $c_1, \ldots, c_d$ sont tous rationnels et que $g(a) \in \Z$ pour une infinité d'entiers a. Montrer que $c_0 \in \Q$ et $c_k = 0$ pour tout $k$ \lt 0. @@ -3061,9 +4027,12 @@ $1/\theta$. On note $f(z) = \sum_{n=0}^{+\i} b_n z^n$ ce développement. #+begin_exercice [X MP 2025 # 341] 1. On pose $u_0 = 1$ et $u_{n+1} = \sum_{k=0}^n u_k u_{n-k}$ pour tout $n \in \N$. Calculer $u_n$. - 1. Pour $n \in \N$, on pose $I_n = \int_{-2}^2 x^{2n} \sqrt{4-x^2} dx$. Prouver l'existence d'une constante -c\gt 0 telle que $\forall n\in\N,\,u_n=c\,I_n$ et la déterminer. + 1. Pour $n \in \N$, on pose $I_n = \int_{-2}^2 x^{2n} \sqrt{4-x^2} dx$. Prouver l'existence d'une constante $c\gt 0$ telle que $\forall n\in\N,\,u_n=c\,I_n$ et la déterminer. #+end_exercice +#+BEGIN_proof + - +#+END_proof + #+begin_exercice [X MP 2025 # 342] Soit $m \in \N^*$. On pose $u_0 = 4^m$, $u_1 = 4^m - 1$ et, pour $k \in [1, m]$, $u_k = -1 + \frac{2m - k}{2m} u_{k+1} + \frac{k}{2m} u_{k-1}$ et $v_k = m \int_0^1 \frac{(1+x)^{2m-k}}{x} \left( (1+x)^k - (1-x)^k \right) dx$. @@ -3076,26 +4045,43 @@ Déterminer un équivalent de $\int_0^{+\i} (te^{-t})^x dt$ quand $x$ tend vers #+END_exercice +# ID:329 #+begin_exercice [X MP 2025 # 344] Soit $E$ l'ensemble des fonctions y de classe $C^2$ de $\R^+$ dans $\R$ telles que, pour tout $t \in \R^+$, $y''(t) + e^t y(t) = 0$. Soit $y \in E \setminus \{0\}$. - 1. Montrer que les zéros de y sont isolés. - 1. Montrer que les zéros de y peuvent être rangés en une suite strictement croissante $(t_n)_{n\geq 0}$ tendant vers $+\i$. + 1. Montrer que les zéros de $y$ sont isolés. + 1. Montrer que les zéros de $y$ peuvent être rangés en une suite strictement croissante $(t_n)_{n\geq 0}$ tendant vers $+\i$. 1. Donner un équivalent de $t_n$. #+end_exercice +# ID:7326 #+begin_exercice [X MP 2025 # 345] Soit $E$ un espace vectoriel euclidien de dimension $n \ge 1$. - 1. Soient $p$ un projecteur de $E$ et $a \in \mc{L}(E)$ tels que ap + pa = a. Montrer que $\op{tr} a = 0$. + 1. Soient $p$ un projecteur de $E$ et $a \in \mc{L}(E)$ tels que $ap + pa = a$. Montrer que $\op{tr} a = 0$. 1. On note $\mc{P}(E)$ l'ensemble des projecteurs orthogonaux de $E$. Pour $p \in \mc{P}(E)$, décrire l'espace tangent à $\mc{P}(E)$ en $p$. Quelle est sa dimension? #+end_exercice ** Géométrie +# ID:8766 #+begin_exercice [X MP 2025 # 346] Soit $(u, v)$ une base de $\R^2$. Donner une condition nécessaire et suffisante sur $(u, v)$ pour qu'il existe un polygone régulier à $n$ côtés dont les sommets sont tous dans $\Q u + \Q v$. #+end_exercice +#+BEGIN_proof +Si $(u,v)$ est la base canonique, s'il existe un tel polynôme régulier, alors on peut supposer que son centre est dans $\Z^2$, donc $\tan \frac{2\pi}{n}\in\Q$ : en effet c'est $\tan (x-y)$, et $\tan x,\tan y\in\Q$. Il faudrait plutôt $\tan \frac{\pi}{n}\in\Q$, et c'est le cas. + +Réciproquement, si $\tan \frac{\pi}{n}\in\Q$, c'est le cas de toutes les tangentes, donc si on place le centre en $O$ et un point en $(1, 0)$, la droite passant par $(-1, 0)$ et de pente $\tan \frac{\pi}{n}$ est rationnelle, donc son autre intersection avec le cercle l'est. + +Dans le cas général, en agissant par une homothétie + isométrie, on +peut supposer que $u = (1,0)$. On peut à nouveau centrer tout polynôme +régulier en l'origine. + +En termes de distances : $\lN n (x,y) + m (0, 1)\rN^2 = nx^2 + (ny-m)^2 = n^2 (x^2 + y^2) + m^2 - 2nm y$, donc $\frac{x^2 + y^2}{y}\in\Q$, c'est-à-dire $\sin \theta \in\Q$. En fait, on obtient même $y\in\Q$, donc on peut supposer $y = 0$, puis $x^2\in\Q$. + +Donc $(\sqrt{n}, 0)$ et $(0, 1)$. De là, ça ne doit plus être très dur. +#+END_proof + ** Probabilités @@ -3122,17 +4108,28 @@ On note $a_n = \mathbf{P}(T \geq n)$ avec, pour $k \leq N$, $a_k = 1$. Détermin #+begin_exercice [X MP 2025 # 349] Soit $n \in \N^*$. Pour $\sigma \in \mc{S}_n$, on note $|\sigma|$ le nombre de cycles dans la décomposition de $\sigma$ en cycles à supports disjoints (y compris les cycles de longueur 1).a) Pour $k \in [1, n]$, on pose $C_k = |\{\sigma \in \mc{S}_n, |\sigma| = k\}|$. - a. Pour $k \in [1, n]$, on pose $C_k = |\{\sigma \in \mc{S}_n, |\sigma| = k\}|$ + 1. Pour $k \in [1, n]$, on pose $C_k = |\{\sigma \in \mc{S}_n, |\sigma| = k\}|$ -Calculer $f_n$ où $f_n: x \mapsto \sum_{k=1}^n C_k x^k$. - 1. Soit $\sigma_n$ une variable de loi uniforme sur $\mc{S}_n$. Donner un équivalent de l'espérance de $|\sigma_n|$. + Calculer $f_n$ où $f_n: x \mapsto \sum_{k=1}^n C_k x^k$. + 1. Soit $\sigma_n$ une variable de loi uniforme sur $\mc{S}_n$. + Donner un équivalent de l'espérance de $|\sigma_n|$. 1. Montrer que $\frac{|\sigma_n|}{\ln(n)}$ tend vers 1 en probabilités quand $n \ra +\i$. #+end_exercice #+begin_exercice [X MP 2025 # 350] - 1. Soient $\lambda \gt 0$ et $X$ une variable aléatoire suivant la loi de Poisson $\mc{P}(\lambda)$. Calculer $\mathbf{E}(X(X-1)\cdots(X-p+1))$ pour tout $p \in \N^*$, et calculer $\mathbf{E}(1/(X+1))$ et $\mathbf{E}(1/(X+2))$. - 1. Soient $A$ un ensemble fini de cardinal $n$ et $p \in \N^*$. Une $p$-partition de $A$ est une partition de $X$ formée de $p$ sous-ensembles (non vides) de $X$. Soit $B$ un ensemble fini de cardinal $m$. Dénombrer, pour une $p$-partition de $\mc{F}$ de A, les applications de $A$ dans $B$ dont $\mc{F}$ est l'ensemble des fibres non vides (à savoir des ensembles non vides de la forme $f^{-1}\{b\}$ où $b \in B$). - 1. En utilisant les deux questions précédentes, exprimer le nombre de partitions de $A$ comme + 1. Soient $\lambda \gt 0$ et $X$ une variable aléatoire suivant la + loi de Poisson $\mc{P}(\lambda)$. Calculer + $\mathbf{E}(X(X-1)\cdots(X-p+1))$ pour tout $p \in \N^*$, et + calculer $\mathbf{E}(1/(X+1))$ et $\mathbf{E}(1/(X+2))$. + 1. Soient $A$ un ensemble fini de cardinal $n$ et $p \in \N^*$. Une + $p$-partition de $A$ est une partition de $X$ formée de $p$ + sous-ensembles (non vides) de $X$. Soit $B$ un ensemble fini de + cardinal $m$. Dénombrer, pour une $p$-partition de $\mc{F}$ de A, + les applications de $A$ dans $B$ dont $\mc{F}$ est l'ensemble des + fibres non vides (à savoir des ensembles non vides de la forme + $f^{-1}\{b\}$ où $b \in B$). + 1. En utilisant les deux questions précédentes, exprimer le nombre de + partitions de $A$ comme la somme d'une série numérique. #+end_exercice #+begin_exercice [X MP 2025 # 351] @@ -3142,24 +4139,19 @@ $$S_n = \sum_{i=0}^n X_i$$. 1. Pour $n \in \Z$, calculer $\mathbf{P}(S_N = n)$. 1. Montrer que : - -la somme d'une série numérique. - -$$\forall (x,y) \in (\R^{+*})^2, \quad \sum_{n \in \Z} y^n \sum_{\substack{i \in \N \\ n \geq 0}} \frac{x^{n+2i}}{n!(n+i)!} = e^{xy+1/y}$$. + $$\forall (x,y) \in (\R^{+*})^2, \quad \sum_{n \in \Z} y^n \sum_{\substack{i \in \N \\ n \geq 0}} \frac{x^{n+2i}}{n!(n+i)!} = e^{xy+1/y}$$. #+end_exercice #+begin_exercice [X MP 2025 # 352] Soient $p \in [0, 1[, m \ge 2 \text{ et } \xi = e^{2i\pi/m}]$. 1. Montrer que : - -$$\forall a,b \in \C, \quad \sum_{k \in \db{0,n }} \binom{n}{k} a^k b^{n-k} = \frac{1}{m} \sum_{j=0}^{m-1} (b + \xi^j a)^n$$. - + $$\forall a,b \in \C, \quad \sum_{k \in \db{0,n }} \binom{n}{k} a^k b^{n-k} = \frac{1}{m} \sum_{j=0}^{m-1} (b + \xi^j a)^n$$. 1. Soit $(X_i)_{i\in\N^*}$ une suite de variables aléatoires $i$.i.d. suivant la loi de Bernoulli de paramètre $p$. On pose : $A_n=(m\mid X_1+\cdots+X_n)$ et $u_n=\mathbf{P}(A_n)$. Montrer que la suite $(u_n)$ est convergente et déterminer sa limite. 1. Montrer que : $\forall n \in \N^*, \left| u_n \frac{1}{m} \right| \leq e^{-8pqn/m^2}$ où $q$ = 1 $p$. #+end_exercice #+begin_exercice [X MP 2025 # 353] -Soit $X$ une variable aléatoire discrète positive ayant un moment d'ordre 2 et telle que $\mathbf{E}(X^2) \gt 0$. Montrer que, pour $t$ \gt 0, $\mathbf{P}(X \mathbf{E}(X) \leq -t) \leq \exp\left(-\frac{t^2}{\mathbf{E}(X^2)}\right)$. +Soit $X$ une variable aléatoire discrète positive ayant un moment d'ordre 2 et telle que $\mathbf{E}(X^2) \gt 0$. Montrer que, pour $t \gt 0$, $\mathbf{P}(X \mathbf{E}(X) \leq -t) \leq \exp\left(-\frac{t^2}{\mathbf{E}(X^2)}\right)$. #+end_exercice #+BEGIN_exercice [X MP 2025 # 354] @@ -3182,34 +4174,68 @@ On pose $(X)_0 = 1$ et, pour $n \in \N^*$, $(X)_n = X(X-1) \cdots (X-n+1)$. 1. Pour $0 \le n \le k$, on note $b_{k,n}$ le nombre de façons de ranger $k$ objets indifférenciés dans $n$ tiroirs non numérotés, aucun des tiroirs n'étant vide. Montrer que $b_{k,n} = a_{k,n}$. 1. Soit $k \in \N$. Déterminer le nombre de façons de partitionner un ensemble à $k$ éléments. #+end_exercice +#+BEGIN_proof + 1. + 2. + 3. + 4. +#+END_proof +# ID:8771 #+begin_exercice [X MP 2025 # 356] On cherche à prouver l'existence d'un réel $C \gt 0$ tel que, pour toutes variables aléatoires réelles $X$ et $Y$ indépendantes et de même loi, on ait l'inégalité $\mathbf{P}(|X-Y| \leq 2) \leq C \, \mathbf{P}(|X-Y| \leq 1)$. - 1. On suppose $X$ et $Y$ à valeurs dans $\Z$. Montrer l'existence de $C' \gt 0$ indépendant de $X$ tel que $\mathbf{P}(|X Y| \leq 2) \leq C' \mathbf{P}(X = Y)$. - 1. Montrer le résultat souhaité. - 1. Montrer que $C' \geq 3$. + 1. On suppose $X$ et $Y$ à valeurs dans $\Z$. Montrer l'existence de $C' \gt 0$ indépendant de $X$ tel que $\mathbf{P}(|X - Y| \leq 2) \leq C' \mathbf{P}(X = Y)$. + 2. Montrer le résultat souhaité. + 3. Montrer que $C' \geq 3$. #+end_exercice +#+BEGIN_proof + 1. $|X-Y|\leq 2$ correspond à ou bien $X = Y$, ou bien $X = Y+1$, ce + qui est $\sum x_n x_{n+1} \leq \frac{1}{2}\sum x_n^2 + x_{n+1}^2$, + etc. + 2. Pour $X,Y$ à valeurs dans $\Z$ on est bon, et sinon, en prenant + $X',Y' = \lfloor X, Y\rfloor$, on a $\P(|X-Y|\leq 2)\leq + \P(|X'-Y'|\leq 2)\leq C \P(X'=Y') \leq C \P(|X-Y|\leq 1)$. + 3. Prendre des lois uniformes sur $\db{1,n}$. +#+END_proof + +# ID:8084 #+begin_exercice [X MP 2025 # 357] - 1. Soient $n \in \N^*$ et $p \in ]0,1[$. Existe-t-il deux variables aléatoires indépendantes $Y_1$ et $Y_2$ de même loi telles que $Y_1 + Y_2 \sim \mc{B}(n,p)$ ? - 1. On dit qu'une variable aléatoire $Z$ est infiniment divisible si, pour tout $k \in \N^*$, il existe des variables aléatoires $i$.i.d. $Y_1, \ldots, Y_k$ telles que $Y_1 + \cdots + Y_k \sim Z$, avec a priori $(Y_1, \ldots, Y_k)$ défini sur un espace probabilisé différent de celui de $Z$. + 1. Soient $n \in \N^*$ et $p \in ]0,1[$. Existe-t-il deux variables + aléatoires indépendantes $Y_1$ et $Y_2$ de même loi telles que + $Y_1 + Y_2 \sim \mc{B}(n,p)$ ? + 1. On dit qu'une variable aléatoire $Z$ est infiniment divisible si, + pour tout $k \in \N^*$, il existe des variables aléatoires + $i$.i.d. $Y_1, \ldots, Y_k$ telles que $Y_1 + \cdots + Y_k \sim + Z$, avec a priori $(Y_1, \ldots, Y_k)$ défini sur un espace probabilisé différent de celui de $Z$. -Donner un exemple d'une telle variable aléatoire. + Donner un exemple d'une telle variable aléatoire. - 1. Que dire d'une variable aléatoire $Z$ infiniment divisible de support inclus dans [0,1]? - 1. Soient $(X_i)_{i\in\N}$ une suite de variables aléatoires $i$.i.d. et $N\sim\mc{P}(\lambda)$ indépendante des $X_i$ (avec $\lambda\gt 0$). Montrer que $Z=X_1+\cdots+X_N$ est une variable aléatoire infiniment divisible. + 1. Que dire d'une variable aléatoire $Z$ infiniment divisible de + support inclus dans $[0,1]$ ? + 1. Soient $(X_i)_{i\in\N}$ une suite de variables aléatoires $i$.i.d. et + $N\sim\mc{P}(\lambda)$ indépendante des $X_i$ (avec $\lambda\gt + 0$). Montrer que $Z=X_1+\cdots+X_N$ est une variable aléatoire + infiniment divisible. #+end_exercice +#+BEGIN_proof + 1. + 2. + 3. + 4. On peut toujours découper $N$ en des lois de Poissons. +#+END_proof + +# À relier #+begin_exercice [X MP 2025 # 358] Soient $a \in ]0,1[$ et $\phi_a : x \mapsto 1 (1-x)^a$. - 1. Montrer qu'il existe une variable aléatoire $X_a$ à valeurs dans $\N^*$ telle que, pour tout $x \in [0,1], \phi_a(x) = \mathbf{E}(x^{X_a})$.b) Soit $(A_n)_{n\geq 1}$ une suite d'événements de l'espace probabilisé $(\Omega, \mc{A}, \mathbf{P})$ telle que, pour tout $n \in \N^*$, $\mathbf{P}(A_n) = \frac{a}{n}$. On pose $Y = \inf\{n \in \N^*, \ \mathbf{1}_{A_n} = 1\}$. Montrer que $Y \sim X_a$. + 1. Montrer qu'il existe une variable aléatoire $X_a$ à valeurs dans $\N^*$ telle que, pour tout $x \in [0,1], \phi_a(x) = \mathbf{E}(x^{X_a})$. + 2. Soit $(A_n)_{n\geq 1}$ une suite d'événements de l'espace probabilisé $(\Omega, \mc{A}, \mathbf{P})$ telle que, pour tout $n \in \N^*$, $\mathbf{P}(A_n) = \frac{a}{n}$. On pose $Y = \inf\{n \in \N^*, \ \mathbf{1}_{A_n} = 1\}$. Montrer que $Y \sim X_a$. On considère l'équation fonctionnelle : $\forall x \in [0,1], \phi_a(x) = x \phi(\phi_a(x))$ d'inconnue $\phi$ : $[0,1] \ra \R$. - 1. Montrer que, pour $a \in [1/2, 1]$ cette équation admet une unique solution continue, qui est - -de plus la fonction génératrice d'une variable aléatoire à valeurs dans $N$. + 1. Montrer que, pour $a \in [1/2, 1]$ cette équation admet une unique solution continue, qui est de plus la fonction génératrice d'une variable aléatoire à valeurs dans $N$. 1. Montrer que ce n'est pas le cas pour a = 1/3. #+end_exercice @@ -3218,18 +4244,34 @@ Soit $(X_n)$ une suite de variables aléatoires indépendantes telles que $\math 1. Soit $\lambda \in \R^+$. Déterminer la limite de $\left(\mathbf{E}\left(e^{-\lambda\frac{S_n}{n}}\right)\right)_{n\geq 1}$. 1. Soit $f \in \mc C^0(\R^{+*}, \R)$ dérivable sur $]1, +\i[$ et telle que : $\forall x \gt 1, f(x-1) + xf'(x) = 0$ et $\forall x \in [0, 1], f(x) = 1$. Montrer qu'il existe une unique fonction $f$ qui respecte ces conditions, qu'elle est strictement positive sur $\R^+$ et tend vers 0 en $+\i$. - 1. On définit $\phi(\lambda) = \int_0^{+\i} e^{-\lambda t} f(t) dt$, avec $f$ la fonction de la question précédente. Mon- $\text{trer qu'il existe } k\gt 0 \text{ tel que, pour tout } \lambda \in \R^+, \lim_{n \ra +\i} \mathbf{E}\left(e^{-\lambda \frac{S_n}{n}}\right) = e^{-k}\phi(\lambda)$. + 1. On définit $\phi(\lambda) = \int_0^{+\i} e^{-\lambda t} f(t) dt$, avec $f$ la fonction de la question précédente. Montrer qu'il existe $k\gt 0$ tel que, pour tout $\lambda \in \R^+, \lim\limits_{n \ra +\i} \mathbf{E}\left(e^{-\lambda \frac{S_n}{n}}\right) = e^{-k}\phi(\lambda)$. #+end_exercice +#+BEGIN_proof + 1. +#+END_proof + #+begin_exercice [X MP 2025 # 360] -Soient $X$ une variable aléatoire à support fini à valeurs dans $\Z^2$ et telle que $-X \sim$ -$X, (X_k)_{k \geq 1}$ une suite $i$.i.d. de variables aléatoires suivant la loi de $X$. Pour $n \in \N^*$, on pose $S_n = X_1 + \cdots + X_n$. +Soient $X$ une variable aléatoire à support fini à valeurs dans $\Z^2$ et telle que $-X \sim X$, $(X_k)_{k \geq 1}$ une suite $i$.i.d. de variables aléatoires suivant la loi de $X$. Pour $n \in \N^*$, on pose $S_n = X_1 + \cdots + X_n$. 1. Montrer que, si $n \in \N^*$, $\mathbf{E}(\|S_n\|^2) = n \, \mathbf{E}(\|X\|^2)$ et $\mathbf{P}(S_{2n} = 0) = \sum_{x \in \Z^2} \mathbf{P}(S_n = x)^2$. 1. Montrer qu'il existe $c \in \R^{+*}$ tel que $\forall n \in \N^*, \mathbf{P}(S_{2n} = 0) \geq \frac{c}{n}$. 1. Démontrer que $P(\exists n \ge 1, S_n = 0) = 1$. #+end_exercice +#+BEGIN_proof + 1. Première partie triviale, seconde par la position à l'instant $n$, et la symétrie, sympa. + 2. C'est Cauchy-Schwarz. + 3. Il faut sans doute utiliser l'espérance ? -* $X$ PSI :autre: + Sinon, à partir de $0$ il a une probabilité $\a$ de ne jamais + revenir. + + La probabilité de ne jamais revenir à partir de $2$ est $\geq \a + \a \P(S_2 = 0)$, puis $\a + \a \P(S_2 =0) \a$. + + On a $\P («S_{2n} = 0») = \sum_{k=1}^{2n-2} \P(R = k) \P (S_{2n-k})$. Si la somme des $\P(R = k)$ est $\lt \frac{1}{2}$. !! +#+END_proof + + +* X PSI :autre: ** Algèbre @@ -3327,17 +4369,18 @@ $\textbf{\textit{b}}) \ \ \text{Soit} \ b \in ]0,\pi/2[. \ \text{Montrer que} \ #+end_exercice #+begin_exercice [X PSI 2025 # 378] -Soit $f \in \mc C^{\i}(\R, \R)$ telle que $f(t) \ra 0$ quand $t \ra +\i$. Soit () l'équation différentielle y' + y = f(t). +Soit $f \in \mc C^{\i}(\R, \R)$ telle que $f(t) \ra 0$ quand $t \ra +\i$. Soit () l'équation différentielle $y' + y = f(t)$. 1. Montrer que toute solution y de () tend vers 0 lorsque $t \ra +\i$. 1. On suppose de plus que $f(t) \sim \frac{1}{t \ra +\i} \frac{1}{t^{\alpha}}$ avec $\alpha \gt 0$. Soit y une solution non nulle de (). Déterminer un équivalent de y(t) quand $t \ra +\i$. #+end_exercice +# ID:8538 #+BEGIN_exercice [X PSI 2025 # 379] -Pour $f \in \mc C^1(\R^+, \R^{+*})$, on considère le problème de Cauchy $f' = -f^2$ et f(0) = 1. +Pour $f \in \mc C^1(\R^+, \R^{+*})$, on considère le problème de Cauchy $f' = -f^2$ et $f(0) = 1$. 1. Résoudre l'équation différentielle. - Soit $h \in ]0, 1/2[$. On définit la suite $(y_n)$ par $y_0 = 1$ et $y_{n+1} y_n = -hy_n^2$. +Soit $h \in ]0, 1/2[$. On définit la suite $(y_n)$ par $y_0 = 1$ et $y_{n+1} y_n = -hy_n^2$. 1. Montrer que $y_n \ra 0$. 1. Montrer que $\frac{1}{y_n} = 1 + nh + o(n)$. #+END_exercice @@ -3376,7 +4419,7 @@ mètre $p \in [0,1]$. Pour $n \in \N$, on pose $Y_n = \sum_{k=1}^n \frac{X_k}{k+ 1. Trouver $\alpha \in \R$ tel que, pour tout $\eps \gt 0$, on ait $\lim_{n \ra \i} \mathbf{P}(|Y_n \alpha| \gt \eps) = 0$. #+end_exercice -* $X$ PC :autre: +* X PC :autre: ** Algèbre @@ -4328,7 +5371,8 @@ On pose $u_0=\frac{1}{2}$ et $\forall n, u_n=u_{n-1}(1-u_{n-1})$. 3. On a $\frac{1}{u_{n+1}} - \frac{1}{u_n} = \frac{1}{u_n (1-u_n)} - \frac{1}{u_n}$ $=\frac{1}{u_n (1-u_n)}\big(1 - (1-u_n)\big) = \frac{1}{1-u_n} = 1 + u_n + O(u_n^2)$. #+END_proof -* Mines - MP + +* Mines - MP :mines: ** Algèbre @@ -4508,21 +5552,38 @@ a) Écrire la décomposition en éléments simples de $\frac{P'}{D}$. b) Montrer que $Conv(\mathcal{Z}(P')) \subset Conv(\mathcal{Z}(P))$. c) Soit $H$ un demi-plan fermé de $\C$ contenant au moins une racine de P'. Montrer que $H$ - contient au moins une racine de $P$. Démontrer ensuite que $P(H) = \C$. #+end_exercice +# ID:7528 #+begin_exercice [Mines MP 2025 # 483] -Quelle est la dimension du $\mathbb{Q}$ -sous-espace vectoriel de $\mathbb{R}$ engendré par $\mathbb{U}_5$ ? +Quelle est la dimension du $\mathbb{Q}$-sev de $\mathbb{R}$ engendré par $\mathbb{U}_5$, c'est-à-dire $\R \cap \vect_{\Q} \m U_5$ ? #+end_exercice +#+BEGIN_proof +On note $\om = e^{2i\frac{\pi}{5}}$ $\m U_5 = \{1, \om, \ol{\om}, \om^2, \ol{\om^2}\}$ et $F = \R\cap \vect \m U_5$. + +Comme la somme des racines $5$-ème de l'unité fait $0$, on a $1 + 2 \cos \frac{2\pi}{5} + 2 \cos \frac{4\pi}{5} = 0$, et $\cos \frac{4\pi}{5} = 2 \cos^2 \frac{2\pi}{5} - 1$, donc $\cos \frac{2\pi}{5}$ est racine du polynôme $P = 1 + 2X + 4 X^2 - 2 = 4X^2 + 2X - 1$. On peut par ailleurs vérifier que ce polynôme n'a pas de racine rationnelle, car si $P(\frac{p}{q}) = 0$, avec $p,q$ premiers entre eux, alors $4p^2 + 2pq - q^2 = 0$, donc $p\mid q^2$, donc $p = \pm 1$, et de même, $q\mid 4$, et on peut vérifier que $\pm 1,\pm \frac{1}{2}$ et $\pm \frac{1}{4}$ ne sont pas racines de $P$. + +On en déduit que $\cos \frac{2\pi}{5}$ est irrationnel. + + $F$ contient $1$, et $\cos \frac{2\pi}{5} = \frac{\om +\ol{om}}{2}$ qui est irrationnel donc $\dim_{\Q} F\geq 2$. Par ailleurs, l'expression $\cos \frac{2\pi}{5} = \frac{\sqrt{5}-1}{4}$ justifie que $\Q[\cos \frac{\sqrt{5}-1}{4}] = \Q[\sqrt{5}]$ est un corps, de dimension $2$ sur $\Q$. + +L'espace $G=\vect_{\Q} \m U_5$ est stable par multiplication (car $\m U_5$ l'est) donc doit être un $\Q[\sqrt{5}]$-espace vectoriel. Si dimension en tant que $\Q$-ev doit donc être un multiple de celle de $F$. + +On a clairement $\dim_{\Q} G\leq 5$ et $G\neq F$, et $\dim F\geq 2$, donc nécessairement, $\dim_{\Q} F = 2$ et $\dim_{\Q} G = 4$. +#+END_proof + #+begin_exercice [Mines MP 2025 # 484] Soit $A \in \mathcal{M}_n(\mathbb{Z})$. Pour tout $x \in \mathbb{Z}^n$, on pose $\Lambda(x) = \operatorname{pgcd}(x_1, \ldots, x_n)$. Montrer l'équivalence des énoncés suivants : i) $\Lambda(Ax) = \Lambda(x)$ pour tout $x \in \mathbb{R}^n$ ; ii) det $A = \pm 1$. #+end_exercice #+begin_exercice [Mines MP 2025 # 485] -Soit $A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix}$.- a) Calculer $A^n$ pour $n \in \mathbb{Z}$. -b) Trouver $B \in \mathcal{M}_4(\mathbb{Q})$ telle que $B^2 = A$. +Soit $A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix}$. + 1. Calculer $A^n$ pour $n \in \mathbb{Z}$. + 2. Trouver $B \in \mathcal{M}_4(\mathbb{Q})$ telle que $B^2 = A$. #+end_exercice +# ID:8707 #+begin_exercice [Mines MP 2025 # 486] -Soient $A, B \in \mathcal{M}_n(\mathbb{K})$ et $P \in \mathbb{K}[X]$ tel que $P(0) \neq 0$. On suppose que AB = P(A). Montrer que $A$ est inversible puis que $A$ et $B$ commutent. +Soient $A, B \in \mathcal{M}_n(\mathbb{K})$ et $P \in \mathbb{K}[X]$ tel que $P(0) \neq 0$. On suppose que $AB = P(A)$. Montrer que $A$ est inversible puis que $A$ et $B$ commutent. #+end_exercice #+begin_exercice [Mines MP 2025 # 487] @@ -4537,7 +5598,7 @@ b) Montrer que $M^2 = (\operatorname{tr} M)I_n$. c) Calculer $\chi_M$. À quelle #+end_exercice #+begin_exercice [Mines MP 2025 # 489] -Déterminer les matrices $A \in \mathcal{M}_n(\mathbb{R})$ telles que, pour toute matrice $B \in \mathcal{M}_n(\mathbb{R})$, AB = 0 implique BA = 0. +Déterminer les matrices $A \in \mathcal{M}_n(\mathbb{R})$ telles que, pour toute matrice $B \in \mathcal{M}_n(\mathbb{R})$, $AB = 0$ implique $BA = 0$. #+end_exercice #+begin_exercice [Mines MP 2025 # 490] @@ -4693,9 +5754,27 @@ b) i) Soient $u \in \mathcal{A}$ tel que $\operatorname{rg}(u) = r \geq 1$ et $f c) Montrer que $A = \mathcal{L}(E)$. #+end_exercice +# ID:8717 #+begin_exercice [Mines MP 2025 # 514] -Soient $E$ un $\mathbb{K}$ -espace vectoriel et $u \in \mathcal{L}(E)$ vérifiant $\operatorname{Im} u^2 = \operatorname{Ker} u^3$. Montrer que : $\operatorname{Im} u = \operatorname{Ker} u^4$. +Soient $E$ un $\mathbb{K}$-espace vectoriel et $u \in \mathcal{L}(E)$ vérifiant $\operatorname{Im} u^2 = \operatorname{Ker} u^3$. Montrer que : $\operatorname{Im} u = \operatorname{Ker} u^4$. #+end_exercice +#+begin_proof +Procedons par double inclusion. + 1. Montrons que $\operatorname{Im} u \subset \operatorname{Ker} u^4$. + Soit $y \in \operatorname{Im} u$. Il existe $x \in E$ tel que $y = u(x)$. + Alors $u^4(y) = u^5(x) = u^3(u^2(x))$. + Or $u^2(x) \in \operatorname{Im} u^2$ et par hypothèse $\operatorname{Im} u^2 = \operatorname{Ker} u^3$, donc $u^3(u^2(x)) = 0$. + Ainsi $u^4(y) = 0$, c'est-à-dire $y \in \operatorname{Ker} u^4$. + + 2. Montrons que $\operatorname{Ker} u^4 \subset \operatorname{Im} u$. + Remarquons d'abord que $\operatorname{Ker} u \subset \operatorname{Ker} u^3 = \operatorname{Im} u^2 \subset \operatorname{Im} u$. + Soit $x \in \operatorname{Ker} u^4$. Alors $u^3(u(x)) = u^4(x) = 0$, donc $u(x) \in \operatorname{Ker} u^3$. + Par hypothèse, $u(x) \in \operatorname{Im} u^2$, donc il existe $z \in E$ tel que $u(x) = u^2(z) = u(u(z))$. + On en déduit que $u(x - u(z)) = 0$, donc $x - u(z) \in \operatorname{Ker} u$. + Comme vu précédemment, $\operatorname{Ker} u \subset \operatorname{Im} u$, donc $x - u(z) \in \operatorname{Im} u$. + Finalement, $x = u(z) + (x - u(z))$ est somme de deux éléments de $\operatorname{Im} u$, donc $x \in \operatorname{Im} u$. +#+end_proof + #+begin_exercice [Mines MP 2025 # 515] Soit $A \in \mathcal{M}_n(\C)$. On note S(A) sa classe de similitude. On suppose S(A) bornée. a) Montrer que $A$ est diagonale à l'aide des matrices de dilatation $I_n + (\lambda - 1)E_{i,i}$, où $\lambda \in \C$ @@ -4722,6 +5801,10 @@ c) L'équivalence précédente reste-t-elle vraie si on ne suppose plus $M$ inve #+begin_exercice [Mines MP 2025 # 518] Déterminer les applications $f$ de $\mathbb{R}^n$ dans $\mathcal{M}_n(\mathbb{R})$ telles que, pour tout $X \in \mathbb{R}^n$ et toute $P \in GL_n(\mathbb{R})$, on ait $f(PX) = Pf(X)P^{-1}$. #+end_exercice +#+BEGIN_proof +Pour $X = E_1$, on commute avec toutes les matrices de la forme $\begin{pmatrix}1 & L \\ 0 & P\end{pmatrix}$. À relier à un plus bas. +#+END_proof + #+begin_exercice [Mines MP 2025 # 519] Soient $n \geq2$ et $f \in \mathcal{L}(\mathcal{M}_n(\C))$. On suppose que, pour toute matrice $A \in \mathrm{GL}_n(\C)$, la matrice f(A) appartient à $\mathrm{GL}_n(\C)$. @@ -4873,8 +5956,8 @@ Soit $A \in \mathcal{M}_n(\mathbb{R})$ une matrice diagonalisable. On pose : + $\mathcal{C}(A) = \{ B \in \mathcal{M}_n(\mathbb{R}), AB = BA \}$ + $\mathcal{C}'(A) = \{ B \in \mathcal{M}_n(\mathbb{R}) ; \forall C \in \mathcal{C}(A), CB = BC \}$. # -a) Montrer que $\mathcal{C}(A)$ est un sous-espace vectoriel de $\mathcal{M}_n(\mathbb{R})$ et en déterminer la dimension. À quelle condition a-t-on $C(A) = \mathbb{R}[A]$ ? -b) Montrer que $\mathcal{C}'(A)$ est un sous-espace vectoriel de $\mathcal{M}_n(\mathbb{R})$ et en déterminer la dimension. Montrer que $C'(A) = \mathbb{R}[A]$. + 1. Montrer que $\mathcal{C}(A)$ est un sous-espace vectoriel de $\mathcal{M}_n(\mathbb{R})$ et en déterminer la dimension. À quelle condition a-t-on $C(A) = \mathbb{R}[A]$ ? + 2. Montrer que $\mathcal{C}'(A)$ est un sous-espace vectoriel de $\mathcal{M}_n(\mathbb{R})$ et en déterminer la dimension. Montrer que $C'(A) = \mathbb{R}[A]$. #+end_exercice #+begin_exercice [Mines MP 2025 # 543] @@ -5144,17 +6227,33 @@ a) Montrer que $\operatorname{rg} N = n 1$. b) On suppose que $MN^T = N^TM = NM^T$. Montrer que $\mathbb{R}^n = \operatorname{Im} M \oplus \operatorname{Im} N$. c) Étudier la réciproque de b). #+end_exercice +# ID:8597 #+begin_exercice [Mines MP 2025 # 586] -a) Soit $A \in \mathcal{S}_n(\mathbb{R})$. Montrer l'équivalence des énoncés suivants : -i) $x^T A x \geq 0$ pour tout $x \in \mathbb{R}^n$, ii) $\operatorname{Sp} A \subset \mathbb{R}^+$. -b) Montrer que, pour tout $x \in \mathbb{R}^n$, u) Sp $A \subset \mathbb{R}^+$. b) Montrer que, pour tout $A \in \mathcal{M}_n(\mathbb{R})$, $A^T A \in \mathcal{S}_n^+(\mathbb{R})$. -c) Montrer que, pour tout $S \in \mathcal{S}_n^+(\mathbb{R})$, il existe $A \in \mathcal{M}_n(\mathbb{R})$ telle que $S = A^T A$, puis déterminer $\{A \in \mathcal{M}_n(\mathbb{R}), A^T A = S\}$. + 1. Soit $A \in \mathcal{S}_n(\mathbb{R})$. Montrer l'équivalence des énoncés suivants : + + pour tout $x \in \mathbb{R}^n$ + + $\operatorname{Sp} A \subset \mathbb{R}^+$. + 2. Montrer que, pour tout $A \in \mathcal{M}_n(\mathbb{R})$, $A^T A \in \mathcal{S}_n^+(\mathbb{R})$. + 3. Montrer que, pour tout $S \in \mathcal{S}_n^+(\mathbb{R})$, il existe $A \in \mathcal{M}_n(\mathbb{R})$ telle que $S = A^T A$, puis déterminer $\{A \in \mathcal{M}_n(\mathbb{R}), A^T A = S\}$. #+end_exercice +#+BEGIN_proof + 1. + 2. + 3. Si $S = P D P^{-1}$, on a $S = P \sqrt{D} \sqrt{D} P^T$. + Si on multiplie $A$ à gauche par un $O_n$, cela préserve. Réciproquement, c'est clair. +#+END_proof + + +# ID:8596 #+begin_exercice [Mines MP 2025 # 587] -a) Soit $A \in \mathcal{M}_n(\mathbb{R})$. Montrer que $A \in \mathcal{A}_n(\mathbb{R})$ si et seulement si les coefficients diagonaux de $P^{-1}AP$ sont nuls pour toute matrice $P \in \mathcal{O}_n(\mathbb{R})$. -b) Soit $A \in \mathcal{A}_n(\mathbb{R})$. Montrer que le rang de $A$ est pair. + 1. Soit $A \in \mathcal{M}_n(\mathbb{R})$. Montrer que $A \in \mathcal{A}_n(\mathbb{R})$ si et seulement si les coefficients diagonaux de $P^{-1}AP$ sont nuls pour toute matrice $P \in \mathcal{O}_n(\mathbb{R})$. + 2. Soit $A \in \mathcal{A}_n(\mathbb{R})$. Montrer que le rang de $A$ est pair. #+end_exercice +#+BEGIN_proof + 1. Se ramène au cas $n = 2$, sympa. + 2. Pas de valeur propre réelle ! +#+END_proof + #+begin_exercice [Mines MP 2025 # 588] On pose $\sigma(A) = \sum_{i=1}^n \sum_{j=1}^n a_{i,j}^2$ pour $A = (a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_n(\mathbb{R})$. @@ -5165,9 +6264,14 @@ c) Calculer $\sigma(A)$ lorsque $A$ représente une projection orthogonale dans d) Déterminer les matrices $P \in GL_n(\mathbb{R})$ telles que $\forall A \in \mathcal{M}_n(\mathbb{R}), \ \sigma(P^{-1}AP) = \sigma(A)$. #+end_exercice +# ID:8598 #+begin_exercice [Mines MP 2025 # 589] Soient $n,m\in\N^*,\,S\in\mathcal{S}_m(\mathbb{R})$. On pose $A=\left(\mathrm{Tr}(S^{i+j-2})\right)_{1\leqslant i,j\leqslant n}$. Montrer que $\operatorname{rg}(A) = \min(n, \operatorname{deg}(\pi_S))$, où l'on a noté $\pi_S$ le polynôme minimal de $\hat{S}$. #+end_exercice +#+BEGIN_proof + $S$ est diagonalisable, et le polynôme minimal est de degré le nombre de valeurs propres distinctes. +#+END_proof + #+begin_exercice [Mines MP 2025 # 590] Soit $(E, \langle , \rangle)$ un espace euclidien. @@ -5193,6 +6297,10 @@ On munit $\mathbb{R}^n$ de son produit scalaire canonique. Soit $A \in \mathcal{ #+begin_exercice [Mines MP 2025 # 594] Déterminer les applications $f$ de $\mathbb{R}^n$ dans $\mathcal{M}_n(\mathbb{R})$ telles que, pour tout $X \in \mathbb{R}^n$ et toute $P \in \mathcal{O}_n(\mathbb{R})$, on ait $f(PX) = Pf(X)P^{-1}$. #+end_exercice +#+BEGIN_proof +Il y a l'application $X\mapsto X^T X$, ainsi que les $X\mapsto X A X^T$. En fait non. +#+END_proof + #+begin_exercice [Mines MP 2025 # 595] Soit $A \in \mathcal{S}_n(\mathbb{R})$. Déterminer le nombre de matrices $B \in \mathcal{M}_n(\mathbb{R})$ telles que $A = B^2$. @@ -5624,9 +6732,12 @@ a) On suppose $f$ dérivable sur $I$. Montrer que $f \in S$. b) On suppose que $f \in S$ et que $f$ admet un maximum en $x_0 \in I$. Montrer que $f$ est dérivable en $x_0$. #+end_exercice +# ID:8539 #+begin_exercice [Mines MP 2025 # 665] -On pose $f(x) = e^{-1/x^2}$ pour $x \in \mathbb{R}^*$, et f(0) = 0. a) Montrer que $f$ est de classe $C^{\infty}$ et qu'il existe une suite $(P_n)_{n\geq 0} \in \mathbb{R}[X]^{\N}$ telle que $\forall x \in \mathbb{R}^*, \ \forall n \in \N, \ f^{(n)}(x) = P_n(1/x)e^{-1/x^2}$. -b) Montrer que $f$ n'est solution sur $\mathbb{R}$ d'aucune équation différentielle de la forme y'=a(x)y avec $a: \mathbb{R} \to \mathbb{R}$ continue.- c) Montrer que $P_n$ est scindé sur $\mathbb{R}$ quel que soit $n \in \N$. +On pose $f(x) = e^{-1/x^2}$ pour $x \in \mathbb{R}^*$, et $f(0) = 0$. + 1. Montrer que $f$ est de classe $C^{\infty}$ et qu'il existe une suite $(P_n)_{n\geq 0} \in \mathbb{R}[X]^{\N}$ telle que $\forall x \in \mathbb{R}^*, \ \forall n \in \N, \ f^{(n)}(x) = P_n(1/x)e^{-1/x^2}$. + 2. Montrer que $f$ n'est solution sur $\mathbb{R}$ d'aucune équation différentielle de la forme $y'=a(x)y$ avec $a: \mathbb{R} \to \mathbb{R}$ continue. + 3. Montrer que $P_n$ est scindé sur $\mathbb{R}$ quel que soit $n \in \N$. #+end_exercice #+begin_exercice [Mines MP 2025 # 666] @@ -6302,11 +7413,16 @@ c) Déterminer toutes les solutions de (). #+begin_exercice [Mines MP 2025 # 761] On pose $E=\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ et $\phi$ l'endomorphisme de $E$ défini par : - $\forall f \in E, \forall t \in \mathbb{R}, \phi(f)(t) = f'(t) + tf(t)$. a) Déterminer les éléments propres de $\phi$. - -b) Déterminer les éléments propres de $\phi^2$. -c) Résoudre l'équation différentielle : $y'' + 2ty + (t^2 + 3)y = 0$. + $\forall f \in E, \forall t \in \mathbb{R}, \phi(f)(t) = f'(t) + tf(t)$. + 1. Déterminer les éléments propres de $\phi$. + 2. Déterminer les éléments propres de $\phi^2$. + 3. Résoudre l'équation différentielle : $y'' + 2ty + (t^2 - 3)y = 0$. #+end_exercice +#+BEGIN_proof + 1. Cours de sup. + 2. Si $\la = 0$, noyau de $f$. +#+END_proof + #+begin_exercice [Mines MP 2025 # 762] Soit (E) l'équation différentielle sur $\mathbb{R}^{+*}$ : ty'' + ty' y = 0. @@ -6666,9 +7782,28 @@ Soit $p \in [0,1[$. On dispose d'une urne contenant des boules blanches et noire a) Déterminer la loi de $X_1$ et sa fonction génératrice. #+end_exercice +# ID:8718 #+begin_exercice [Mines MP 2025 # 812] -Alice et Bob possèdent chacun un sac avec $n$ jetons numérotés de 1 à $n$. Alice tire un jeton au hasard. Bob tire ensuite des jetons, sans remise, jusqu'à ce que le numéro tiré soit supérieur ou égal au numéro tiré par Alice. On note $Y$ le nombre de jetons tirés par Bob. +Alice et Bob possèdent chacun un sac avec $n$ jetons numérotés de 1 à $n$. Alice tire un jeton au hasard. Bob tire ensuite des jetons, sans remise, jusqu'à ce que le numéro tiré soit supérieur ou égal au numéro tiré par Alice. On note $Y$ le nombre de jetons tirés par Bob. Déterminer la loi de $Y$. #+end_exercice +#+BEGIN_proof +Soit $X$ le numéro tiré par Alice. $X$ suit une loi uniforme sur $\db{1,n}$. +Bob effectue des tirages sans remise. $Y$ prend ses valeurs dans $\db{1,n}$. +D'après la formule des probabilités totales : +$$P(Y=k) = \sum_{i=1}^n P(Y=k \mid X=i) P(X=i) = \frac{1}{n} \sum_{i=1}^n P(Y=k \mid X=i).$$ +Si $X=i$, pour que Bob s'arrête exactement au $k$-ième tirage, il doit avoir tiré $k-1$ jetons strictement inférieurs à $i$, puis un jeton supérieur ou égal à $i$. Cela nécessite que $i \ge k$. +Le nombre total d'arrangements de $k$ jetons parmi $n$ est $\frac{n!}{(n-k)!}$. +Le nombre d'arrangements favorables (les $k-1$ premiers dans $\db{1,i-1}$ et le $k$-ième dans $\db{i,n}$) est $\frac{(i-1)!}{(i-k)!} \times (n-i+1)$. +Ainsi, pour $i \ge k$ : +$$P(Y=k \mid X=i) = \frac{(i-1)!}{(i-k)!} (n-i+1) \frac{(n-k)!}{n!} = {i-1 \choose k-1}(k-1)! (n-i+1) \frac{(n-k)!}{n!}.$$ +En injectant dans la somme et en sortant les termes constants : +$$P(Y=k) = \frac{(k-1)! (n-k)!}{n \cdot n!} \sum_{i=k}^n {i-1 \choose k-1} (n-i+1).$$ +Calculons la somme $S = \sum_{i=k}^n {i-1 \choose k-1} \sum_{j=i}^n 1 = \sum_{j=k}^n \sum_{i=k}^j {i-1 \choose k-1} = \sum_{j=k}^n {j \choose k} = {n+1 \choose k+1}.$ +Finalement : +$$P(Y=k) = \frac{(k-1)! (n-k)!}{n \cdot n!} \frac{(n+1)!}{(k+1)!(n-k)!} = \frac{n+1}{n} \frac{(k-1)!}{(k+1)!} = \frac{n+1}{n k(k+1)}.$$ +La loi de $Y$ est donnée par $\forall k\in\db{1,n},\, P(Y=k) = \frac{n+1}{n k(k+1)}$. +#+end_proof + #+begin_exercice [Mines MP 2025 # 813] On pose, pour $k \in \N^*$ : $\mathbf{P}(X = k) = \frac{k-1}{2^k}$. @@ -6859,11 +7994,12 @@ e) On suppose que $n$ est premier et on admet qu'alors le polynôme $1+X+X^2+\cd #+end_exercice #+begin_exercice [Mines MP 2025 # 841] -Soient $X_1, \ldots, X_n$ $i$.i.d. de loi $\mathcal{B}(p), U = (X_1 \cdots X_n)$ et $M = U^T U$. +Soient $X_1, \ldots, X_n$ i.i.d. de loi $\mathcal{B}(p)$, $U = (X_1 \cdots X_n)$ et $M = U^T U$. -a) Déterminer la loi de rg(M) et de tr(M). -$\boldsymbol{b}$ ) Calculer la probabilité de l'événement « $M$ est un projecteur ». -c) Ici $n=2, V=\begin{pmatrix}1\\1\end{pmatrix}$ et $S=VMV^T$. Déterminer l'espérance et la variance de $S$. + 1. Déterminer la loi de $\rg(M)$ et de $\tr(M)$. + 2. Calculer la probabilité de l'événement « $M$ est un projecteur ». + c. Ici $n=2, V=\begin{pmatrix}1\\1\end{pmatrix}$ et $S=VMV^T$. + Déterminer l'espérance et la variance de $S$. #+end_exercice #+begin_exercice [Mines MP 2025 # 842] @@ -9307,7 +10443,7 @@ b) Montrer, par récurrence, que, pour tout $n \in \N^*$, $\mathbf{V}(D_n) = n!$ #+end_exercice -* Centrale - MP +* Centrale - MP :cent: ** Algèbre @@ -10961,7 +12097,7 @@ b) Déterminer l'espérance et la variance de la plus grande valeur propre de $M #+end_exercice -* Autres Écoles - MP +* Autres Écoles - MP :autres: ** Algèbre @@ -10979,11 +12115,12 @@ Soit $A \in \mathcal{M}_3(\mathbb{R})$ telle que $A^2 = 0$ et $A \neq 0$. Montrer que $A$ est semblable à $\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$. #+end_exercice +# ID:8547 #+begin_exercice [IMT # 1357] -a) Soient $n \in \N^*$, $u, v \in \mathcal{L}(\mathbb{R}^n)$ nilpotents et non nuls tels que $u \circ v = v \circ u$. Montrer que $\operatorname{rg}(u \circ v) < \operatorname{rg}(v)$. -b) Soient $u_1, \ldots, u_n \in \mathcal{L}(\mathbb{R}^n)$ nilpotents et commutant deux à deux. + 1. Soient $n \in \N^*$, $u, v \in \mathcal{L}(\mathbb{R}^n)$ nilpotents et non nuls tels que $u \circ v = v \circ u$. Montrer que $\operatorname{rg}(u \circ v) < \operatorname{rg}(v)$. + 2. Soient $u_1, \ldots, u_n \in \mathcal{L}(\mathbb{R}^n)$ nilpotents et commutant deux à deux. -Montrer que $u_1 \circ \cdots \circ u_n = 0$. + Montrer que $u_1 \circ \cdots \circ u_n = 0$. #+end_exercice #+begin_exercice [CCINP # 1358] @@ -11556,19 +12693,44 @@ On considère une pièce ayant une probabilité $p \in ]0,1[$ d'obtenir pile et #+end_exercice #+begin_exercice [IMT # 1440] -Soit $S$ la somme de $N$ dés équilibrés à 6 faces, où $N$ suit la loi uniforme sur $\db{1,52}$. Déterminer la probabilité des événements (S=1), (S=2), (S=3). Calculer l'espérance de $S$. +Soit $S$ la somme de $N$ dés équilibrés à 6 faces, où $N$ suit la loi uniforme sur $\db{1,52}$. Déterminer la probabilité des événements $(S=1)$, $(S=2)$, $(S=3)$. Calculer l'espérance de $S$. #+end_exercice +# ID:8719 #+begin_exercice [IMT # 1441] On joue des parties indépendantes d'un jeu où la probabilité de gagner est de $\frac{2}{3}$. Soit $A_n$ l'événement « les parties $n$ et n+1 sont gagnées, mais ce sont les premières à être gagnées consécutivement ». On note $p_n = \mathbf{P}(A_n)$. -a) Calculer $p_1$ et $p_2$. -b) Montrer que, pour tout $n \in \N^*$, $p_{n+2} = \frac{1}{3}p_{n+1} + \frac{2}{9}p_n$. + 1. Calculer $p_1$ et $p_2$. + 2. Montrer que, pour tout $n \in \N^*$, $p_{n+2} = \frac{1}{3}p_{n+1} + \frac{2}{9}p_n$. #+end_exercice +#+begin_proof + 1. L'événement $A_1$ correspond à gagner les parties 1 et 2. Comme les parties sont indépendantes : + $$p_1 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}.$$ + L'événement $A_2$ correspond à gagner 2 et 3, sans avoir gagné 1 et 2. Cela impose que la partie 1 soit perdue, puis 2 et 3 gagnées : + $$p_2 = \frac{1}{3} \times \left(\frac{2}{3}\right)^2 = \frac{4}{27}.$$ + 2. Pour réaliser $A_{n+2}$ (le premier doublé gagnant est en position $n+2, n+3$), on raisonne sur le résultat des premières parties (système complet d'événements) : + + Si la partie 1 est perdue (probabilité $1/3$), on se retrouve dans la même situation décalée d'un rang. Il faut donc réaliser l'événement $A_{n+1}$ sur la suite des parties. La contribution est $\frac{1}{3}p_{n+1}$. + + Si la partie 1 est gagnée (probabilité $2/3$), alors la partie 2 doit nécessairement être perdue (sinon le premier doublé serait en 1, 2). La probabilité de ce début est $\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$. On se retrouve alors dans la situation initiale décalée de deux rangs, et il faut réaliser $A_n$. La contribution est $\frac{2}{9}p_n$. + On en déduit la relation de récurrence : + $$p_{n+2} = \frac{1}{3}p_{n+1} + \frac{2}{9}p_n.$$ +#+end_proof + +# ID:8720 #+begin_exercice [IMT # 1442] -On dispose de $N$ coffres. Avec probabilité $p$, on place dans l'un des coffres un trésor (le choix du coffre est effectué sous loi uniforme). Quelle est la probabilité que le $N$-ième coffre contienne un trésor sachant que les $N$-1 autres coffres sont vides? +On dispose de $N$ coffres. Avec probabilité $p$, on place dans l'un des coffres un trésor (le choix du coffre est effectué sous loi uniforme). Quelle est la probabilité que le $N$-ième coffre contienne un trésor sachant que les $N-1$ autres coffres sont vides? #+end_exercice +#+begin_proof +Notons $T$ l'événement "un trésor est placé" et $C_i$ l'événement "le trésor est dans le coffre $i$". +L'énoncé donne $P(T)=p$ et $P(C_i | T) = \frac{1}{N}$. +Ainsi, $P(C_i) = \frac{p}{N}$. Les événements $C_i$ sont deux à deux disjoints. + +Soit $V$ l'événement "les $N-1$ premiers coffres sont vides". $V$ est le complémentaire de $\bigcup_{i=1}^{N-1} C_i$. On a $P(V) = 1 - \sum_{i=1}^{N-1} P(C_i) = 1 - (N-1)\frac{p}{N}$. + +On cherche $P(C_N | V) = \frac{P(C_N \cap V)}{P(V)}$. +Or si le trésor est dans le coffre $N$, les autres sont vides, donc $C_N \subset V$, d'où $P(C_N \cap V) = P(C_N) = \frac{p}{N}$. +Finalement : $P(C_N | V) = \frac{\frac{p}{N}}{1 - \frac{p(N-1)}{N}} = \frac{p}{N - p(N-1)}$. +#+end_proof #+begin_exercice [IMT # 1443] Soient $n, N \in \N \setminus \{0, 1\}$. On considère $N$ clients et $n$ fournisseurs. Chaque client peut choisir individuellement un fournisseur. On note $X_i$ le nombre de clients ayant choisi le fournisseur numéro $i$. @@ -12877,3 +14039,10 @@ d) Rappeler le développement en série entière de $x \mapsto \ln(1-x)$. 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