From e8d191ab324b577eba47aa07bdee95509c5dc013 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?S=C3=A9bastien=20Miquel?= Date: Tue, 30 Jun 2026 09:07:40 +0200 Subject: [PATCH] =?UTF-8?q?Plus=20que=2010=E2=80=A6?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Exercices 2023.org | 21 +- Exercices 2024.org | 20 +- Exercices 2025.org | 1865 +++++++++++++++++++++++++----------- Exercices XENS MP 2023.pdf | Bin 584814 -> 827007 bytes Exercices XENS MP 2024.pdf | Bin 637225 -> 636833 bytes Exercices XENS MP 2025.pdf | Bin 664977 -> 664981 bytes 6 files changed, 1340 insertions(+), 566 deletions(-) diff --git a/Exercices 2023.org b/Exercices 2023.org index ffa338f..c9053f6 100644 --- a/Exercices 2023.org +++ b/Exercices 2023.org @@ -1,8 +1,8 @@ -# -*- org-export-switch: "centrale"; -*- +# -*- org-export-switch: "xens mp"; -*- #+title: Exercices 2023 #+author: Sébastien Miquel #+date: 02-12-2023 -# Time-stamp: <30-05-25 18:34> +# Time-stamp: <09-06-26 20:33> #+OPTIONS: @@ -162,14 +162,14 @@ python3 -c "import torch; print(f'device name [0]:', torch.cuda.get_device_name( *** XENS MP -# #+select_tags: xens -# #+exclude_tags: autre -# #+export_file_name: Exercices XENS MP 2023 - +#+select_tags: xens +#+exclude_tags: autre +#+export_file_name: Exercices XENS MP 2023 +#+options: toc:t *** Centrale -#+select_tags: cent -#+export_file_name: Exercices Centrale 2023 +# #+select_tags: cent +# #+export_file_name: Exercices Centrale 2023 *** Mines @@ -198,11 +198,6 @@ python3 -c "import torch; print(f'device name [0]:', torch.cuda.get_device_name( # #+relocate_tags: todo - -* 2024 - -#+INCLUDE: "./2024/Exercices 2024.org::*Christophe" - * ENS MP-MPI :xens: ** Algèbre diff --git a/Exercices 2024.org b/Exercices 2024.org index f15e24c..4c42326 100644 --- a/Exercices 2024.org +++ b/Exercices 2024.org @@ -1,8 +1,8 @@ -# -*- org-export-switch: "mines centrale"; -*- +# -*- org-export-switch: "xens mp"; -*- #+title: Exercices 2024 #+author: Sébastien Miquel #+date: 20-11-2024 -# Time-stamp: <26-02-26 13:42> +# Time-stamp: <14-06-26 15:13> * Meta :noexport: ** Statistiques @@ -34,10 +34,10 @@ *** XENS MP -# #+select_tags: xens -# #+exclude_tags: autre -# #+exclude_types: proof -# #+export_file_name: Exercices XENS MP 2024 +#+select_tags: xens +#+exclude_tags: autre +#+exclude_types: proof +#+export_file_name: Exercices XENS MP 2024 *** Centrale @@ -51,10 +51,10 @@ *** Mines Centrale -#+select_tags: mines cent -#+exclude_tags: autre -#+options: toc:2 -#+export_file_name: Exercices Mines Centrale 2024 +# #+select_tags: mines cent +# #+exclude_tags: autre +# #+options: toc:2 +# #+export_file_name: Exercices Mines Centrale 2024 *** todoes diff --git a/Exercices 2025.org b/Exercices 2025.org index e323737..7756ccf 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: <03-05-26 19:14> +# Time-stamp: <30-06-26 09:07> #+OPTIONS: toc:t * Meta :noexport: @@ -14,7 +14,7 @@ #+RESULTS: | ? | ! | todo | unexed | unexed xens | -| 2 | 6 | 10 | 1433 | 100 | +| 2 | 3 | 14 | 1333 | 15 | ** 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 @@ -114,7 +114,7 @@ Un sous-ensemble non vide $S$ de $\mathbb Z$ est dit direct si, pour $x,y,s,t\in 4. s Existe-t-il un ensemble $S$ direct inclus dans $\mathbb{N}$ tel que $S + S = \mathbb{N}$ ? 5. s Existe-t-il un ensemble $S$ direct inclus dans $\mathbb{Z}$ tel - que $\mathbb{N}$ soit inclus dans S+S? + que $\mathbb{N}$ soit inclus dans $S+S$ ? 6. s Existe-t-il un ensemble $S$ direct inclus dans $\mathbb{Z}$ tel que $S + S = \mathbb{Z}$ ? #+end_exercice @@ -253,7 +253,7 @@ On note $\mathbb{Z}[i\sqrt{2}] = \left\{a + ib\sqrt{2} ; (a,b) \in \mathbb{Z}^2\ # ID:8428 #+begin_exercice [ENS PLSR 2025 # 12] -Soit $(A,+)$ un groupe abélien. On dit qu'il est sans torsion lorsque $n\cdot x \neq 0$ pour tout $n \in \mathbb{N}^*$ et tout $x \in $A$ \setminus \{0\}$. Un ordre de groupe sur $(A,+)$ est une relation d'ordre totale $\leq$ sur l'ensemble $A$ telle que $\forall (x, y, z) \in A^3, \ x \leq y \Rightarrow x + z \leq y + z$. +Soit $(A,+)$ un groupe abélien. On dit qu'il est sans torsion lorsque $n\cdot x \neq 0$ pour tout $n \in \mathbb{N}^*$ et tout $x \in A \setminus \{0\}$. Un ordre de groupe sur $(A,+)$ est une relation d'ordre totale $\leq$ sur l'ensemble $A$ telle que $\forall (x, y, z) \in A^3, \ x \leq y \Rightarrow x + z \leq y + z$. 1. Montrer que si $(A, +)$ possède un ordre de groupe alors il est sans torsion. 2. Montrer que $(\mathbb{Z}^n, +)$ possède un ordre de groupe pour tout $n \in \mathbb{N}^*$. @@ -604,7 +604,8 @@ Soient $G$ un groupe admettant une partie génératrice finie et $H$ un groupe f On a une relation comme $S T^{-1} S = - U$. - On a évidemment des morphismes dans $SL_2(Z/p/Z)$. + On a évidemment des morphismes dans $SL_2(Z/p/Z)$, qui + permettent de conclure avec les questions précédentes. #+END_proof @@ -1535,7 +1536,7 @@ c'est-à-dire que $\forall x \in [a,b], f(x) \neq x \Rightarrow (f #+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 $|qx p| \leqslant \frac{1}{Q}$. - En déduire qu'il existe une infinité de couples (p,q) de $\mathbb{Z} \times \N^*$ tels que $\left|x \frac{p}{q}\right| \leqslant \frac{1}{q^2}$. + En déduire qu'il existe une infinité de couples (p,q) de $\mathbb{Z} \times \N^*$ tels que $\left|x - \frac{p}{q}\right| \leqslant \frac{1}{q^2}$. 1. On admet que $\pi$ est irrationnel. Déterminer la nature de la série $\sum \frac{1}{n \sin(n)}$. #+end_exercice @@ -1720,11 +1721,22 @@ qui est indépendante de $\tau$. #+END_proof -#+begin_exercice [ENS L 2025 # 97] :todo: +# ID:8894 +#+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. #+end_exercice #+BEGIN_proof +On va ajouter à $f$ une fonction $g$ arbitrairement petite de sorte la dérivée de $f+g$ soit positive. +Pour cela, il suffit de prendre une fonction dont la dérivée est +l'infini en chaque point où $f$ a des soucis. + +Pour cela, on fixe $\eps$. Pour tout $k$ on peut recouvrir les points +problématiques avec des intervalles de longueur totale $\leq +\frac{\eps}{2^k}$. On ajoute alors à $g$ une fonction affine par +morceaux, de dérivée $1$ sur ces intervalles, et $0$ ailleurs. + +Alors $f+g$ est finalement de dérivée positive ou $+\i$ en chaque point. #+END_proof @@ -1763,11 +1775,24 @@ 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 $\int_0^\pi \max\limits_{1\leq k\leq n}\left(a_k\frac{|\sin(kx)|}{x}\right)\,dx=\sum_{k=1}^n\frac{a_k}{k}+O(1)$. +# ID:8933 +#+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 $\int_0^\pi \max\limits_{1\leq k\leq n}\left(a_k\frac{|\sin(kx)|}{x}\right)\dx=\sum_{k=1}^n\frac{a_k}{k}+O(1)$. #+end_exercice #+BEGIN_proof +On cherche quand est-ce que $\sin (kx)$ est maximal, et $x\simeq \frac{\pi}{2}$. +On trouve $x\in \interval[{\frac{\pi}{2(k+1)}, \frac{\pi}{2k}}]$. + +On peut minorer l'intégrale, en ne prenant que le sinus concerné dans +cet intervalle. Si on remplace le sinus par $1$, on trouve +$a_k\int_{J_k} \frac{1}{x} = a_k \ln \big(1+ \frac{1}{k}\big)$, d'où +le résultat. + +Rmq : la piste consistant à étudier précisément quand est-ce que le +sinus est maximal est fortuite, on ne s'en sort pas, l'étude de quand +$kx \simeq \frac{3\pi}{2}$ est de sinus maximal dépend de +l'arithmétique de $k$. #+END_proof @@ -2018,14 +2043,20 @@ On pose : $\forall x \gt 0, \eta(x) = \sum_{n=0}^{+\i} \frac{(-1)^{n-1}}{n^x}$. 1. Montrer que $\eta(z)$ est bien définie pour tout $z \in \C$ vérifiant $\Re z \gt 0$. #+end_exercice -#+begin_exercice [ENS P 2025 # 115] :todo: +# ID:8895 +#+begin_exercice [ENS P 2025 # 115] 1. Montrer que, pour tout $n \in \N$, il existe un unique $L_n \in \R[X]$ tel que $L_n(1) = 1$ et $(1 - X^2)L_n'' - 2XL_n' + n(n+1)L_n = 0$. 1. Montrer que $\forall x \in [-1,1], \ \forall z \in \interval]{-1, 1}[, \ \frac{1}{\sqrt{1-2xz+z^2}} = \sum_{n=1}^{+\i} L_n(x)z^n$. #+end_exercice -#+BEGIN_proof :todo: +#+BEGIN_proof 1. On considère l'application $\Phi\colon P\mapsto \dots$. Elle envoie $\R_n[X]$ dans $\R_{n-1}[X]$, et de plus, si $\deg P\lt n$, alors $\deg \Phi(P) = \deg P$. Cela donne le rang de la matrice. + 2. Si on veut faire le lien avec l'équation différentielle, c'est + qu'on veut dériver par rapport à $x$. + + On obtient $\frac{z}{(1 - 2x z + z^2) \sqrt{1-2xz+z^2}}$. Donc $(1-2xz + z^2) \sum_{n=1}^{+\i}L_n'(x) z^n = z \sum_{n=1}^{+\i} L_n(x)z^n$. + Ce qui est très sympa, et doit permettre de montrer que $L_n$ vérifie ce qu'on veut. #+END_proof # ID:8869 @@ -2078,22 +2109,33 @@ de $\R_+^{*}$ dans $\C$. -# Relier à 4304, 4305 -#+begin_exercice [ENS P 2025 # 119] :todo: -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 $\forall x \in [0,1], \ g(x) - \int_{\R}^1 K(x,t)g(t) \ dt = f(x)$. + +# ID:8919 +#+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'existence et l'unicité de $g \in \mc +C^0([0,1],\R)$ telle que $\forall x \in [0,1], \ g(x) - \int_{0}^1 +K(x,t)g(t) \ dt = f(x)$. #+end_exercice #+BEGIN_proof +Unicité : simple, considérer le maximum de $g\in \Ker$. +Existence : On a $\Phi = \op{Id} - \phi$, d'inverse $\sum \phi^k$. #+END_proof +# ID:nil # No thanks #+begin_exercice [ENS L 2025 # 120] -Soient $\alpha, \theta \in ]0,1[$. Pour $f:[1,+\i[ \ra [0,1]$ continue, on pose $\|f\|_{\alpha}=\sup_{s \ra \i} s^{\alpha}|f(s)|$ et $F_{\alpha}=\{f \in \mc C^0([1,+\i[,[0,1]),\,\|f\|_{\alpha}\lt +\i\}$. - 1. Pour $f \in F_{\alpha}$, on pose $T(f): s \geq 1 \mapsto 1 - \left(1 - \frac{1}{s}\right)^{\theta} + \theta(s-1)^{\theta} \int_{-\i}^{+\i} (s+t-1)^{-\theta-1} f(t) dt$. +Soient $\alpha, \theta \in ]0,1[$. Pour $f:[1,+\i[ \ra [0,1]$ continue, on pose $\|f\|_{\alpha}=\sup_{s\geq 1} s^{\alpha}|f(s)|$ et $F_{\alpha}=\{f \in \mc C^0([1,+\i[,[0,1]),\,\|f\|_{\alpha}\lt +\i\}$. + 1. Pour $f \in F_{\alpha}$, on pose $T(f): s \geq 1 \mapsto 1 - \left(1 - \frac{1}{s}\right)^{\theta} + \theta(s-1)^{\theta} \int_{-\i}^{+\i} (s+t-1)^{-\theta-1} f(t) \dt$. Montrer que $T$ est une application lipschitzienne de $F_{\alpha}$ dans $F_{\alpha}$ (pour $\| \cdot \|_{\alpha}$). 2. On admet que, pour tout $\alpha \in ]0, 1-\theta[$, $T$ possède un unique point fixe $f_{\alpha} \in F_{\alpha}$. Montrer que $f_{\alpha}$ ne dépend pas de $\alpha$ ; on le note $f_0$. Montrer que $\int_t^{+\i} t^{-\theta} f_0(t) dt = +\i$. #+end_exercice +#+BEGIN_proof + 1. Il s'agit de montrer que $\lN T(f)\rN_{\a}\leq K \lN f\rN_{\a}$. +#+END_proof + # ID:8846 #+begin_exercice [ENS PLSR 2025 # 121] @@ -2103,9 +2145,14 @@ Montrer que $T$ est une application lipschitzienne de $F_{\alpha}$ dans $F_{\alp 1. Résoudre $x(x-1)y'' + 3xy' + y = 0$ sur $]-1,1[$. #+end_exercice +# ID:8896 #+begin_exercice [ENS PLSR 2025 # 122] Résoudre $x^2y'' + xy' + (x^2 - 1/4)y = 0$ sur $]0, 1[$. #+end_exercice +#+BEGIN_proof +Poser $y(x) = \frac{z(x)}{\sqrt{x}}$, de sorte à supprimer le terme en $y'$. On trouve finalement $\frac{A\cos x + B\sin x}{\sqrt{x}}$. +#+END_proof + # ID: nil #+begin_exercice [ENS P 2025 # 123] @@ -2116,15 +2163,24 @@ C'est faux : pour $\psi(x) = 1$, $y$ est la fonction cosinus, et entre deux zér #+END_proof +# ID:8897 #+begin_exercice [ENS PLSR 2025 # 124] Soit $f \in \mc C^2(\R, \C)$. 1. On suppose que $f'' + f' + f \xrightarrow[+\i]{} 0$. Montrer que $f \xrightarrow[+\i]{} 0$. 1. Soit $P = a_0 + a_1 X + a_2 X^2 \in \C[X]$ unitaire de degré 1 ou 2 et à racines simples dans $\C$. -On pose $\partial_P f = \sum_{k=0}^{\i} a_k f^{(k)}$. Donner une condition nécessaire et suffisante sur $P$ pour que, quelle que soit $f \in \mc C^2(\R, \C)$, $\partial_P f \xrightarrow[+\i]{} 0$ implique $f \xrightarrow[+\i]{} 0$. + + On pose $\partial_P f = \sum_{k=0}^{\i} a_k f^{(k)}$. Donner une condition nécessaire et suffisante sur $P$ pour que, quelle que soit $f \in \mc C^2(\R, \C)$, $\partial_P f \xrightarrow[+\i]{} 0$ implique $f \xrightarrow[+\i]{} 0$. 1. Soit $a, b, c \in \R$. Trouver une condition nécessaire et suffisante pour que $$\forall (x,y,z) \in \mc C^1(\R,\C)^3, \quad \left\{ \begin{array}{l} x' + ax + by + cz \xrightarrow{+\i} 0 \\ y' + bx + cy + az \xrightarrow{+\i} 0 \\ z' + cx + ay + bz \xrightarrow{+\i} 0 \end{array} \right. \Longrightarrow \left\{ \begin{array}{l} x \xrightarrow{+\i} 0 \\ y \xrightarrow{+\i} 0 \\ z \xrightarrow{+\i} 0 \end{array}\right.$$ #+end_exercice +#+BEGIN_proof + 1. $X^2 + X + 1 = (X - a)(X-b)$, il suffit de le montrer pour $f' + + \a f$, avec une condition sur la partie réelle de $f$. + 2. Cf question précédente. + 3. $X' + A X \ra 0$, avec $A$ une matrice dont $a+b+c$ est valeur propre, +#+END_proof + # ID:nil # Simple #+begin_exercice [ENS SR 2025 # 125] @@ -2141,24 +2197,40 @@ Soit $n \ge 2$. Soit $A : \R^+ \ra \M_n(\R)$ continue. On considère les solutio 1. On suppose qu'il existe $P: \R^+ \ra \mathrm{GL}_n(\R)$ continue et $D \in \M_n(\R)$ diagonale à coefficients dans $]-\i,-1]$ telles que, pour tout $t$, $A(t)=P(t)DP^{-1}(t)$. Les solutions de () ont-elles toutes pour limite 0 en $+\i$ ? #+end_exercice +# ID:8898 #+begin_exercice [ENS PLSR 2025 # 127] -On fixe un intervalle non trivial $I$.- a) Soient a et $b$ deux fonctions continues de $I$ dans $\R$. Soit $f$ une solution non nulle sur $I$ de y'' + ay' + by = 0. Montrer que les zéros de $f$ sont isolés : pour tout zéro $t_0$ de $f$ il existe un $\delta \gt 0$ tel que $f$ n'ait pas de zéro dans $|t_0 \delta, t_0 + \delta| \setminus \{t_0\}$. - 1. Soient $p_1, p_2$ deux fonctions continues de $I$ dans $\R$ telles que $\forall t \in I$, $p_1(t) \geq p_2(t)$. Soient $f, g \in \mc C^2(I, \R) \setminus \{0\}$ telles que $f'' + p_1 f = 0$ et $g'' + p_2 g = 0$. Soient $t_1 \lt t_2$ deux zéros de $f$ entre lesquels $f$ n'admet aucun autre zéro. Montrer qu'il existe un zéro de $g$ dans $[t_1, t_2]$, ainsi que dans $[t_1, t_2]$. - 1. Soient $p$,q deux fonctions continues de [0,1] dans $\mathbb R$ telles que $\forall t \in [0,1], \ q(t) \gt 0$. Pour $\lambda \in \mathbb R$, on note $f_\lambda$ la solution sur [0,1] de l'équation différentielle $y'' + (p(t) + \lambda q(t))y = 0$ avec la condition initiale $f_\lambda(0) = 0$ et $f'_\lambda(0) = 1$. On note $N_\lambda$ le nombre de zéros de $f_\lambda$. Montrer que $\lambda \mapsto N_\lambda$ est croissante et déterminer ses limites en $-\i$ et $+\i$. - 1. On admet que $(x, \lambda) \in [0, 1] \times \R \mapsto f_{\lambda}(x)$ est continue. Montrer que l'ensemble -$\{\lambda \in \R, \ f_{\lambda}(1) = 0\}$ est l'ensemble des termes d'une suite réelle strictement croissante. - 1. Montrer que $(\lambda, x) \mapsto f_{\lambda}(x)$ est continue sur $\R \times [0, 1]$. +On fixe un intervalle non trivial $I$. + 1. Soient a et $b$ deux fonctions continues de $I$ dans $\R$. Soit $f$ une solution non nulle sur $I$ de $y'' + ay' + by = 0$. Montrer que les zéros de $f$ sont isolés : pour tout zéro $t_0$ de $f$ il existe un $\delta \gt 0$ tel que $f$ n'ait pas de zéro dans $[t_0- \delta, t_0 + \delta] \setminus \{t_0\}$. + 2. Soient $p_1, p_2$ deux fonctions continues de $I$ dans $\R$ telles que $\forall t \in I$, $p_1(t) \geq p_2(t)$. Soient $f, g \in \mc C^2(I, \R) \setminus \{0\}$ telles que $f'' + p_1 f = 0$ et $g'' + p_2 g = 0$. Soient $t_1 \lt t_2$ deux zéros de $f$ entre lesquels $f$ n'admet aucun autre zéro. Montrer qu'il existe un zéro de $g$ dans $[t_1, t_2[$, ainsi que dans $]t_1, t_2]$. + 3. Soient $p,q$ deux fonctions continues de $[0,1]$ dans $\mathbb R$ telles que $\forall t \in [0,1], \ q(t) \gt 0$. Pour $\lambda \in \mathbb R$, on note $f_\lambda$ la solution sur [0,1] de l'équation différentielle $y'' + (p(t) + \lambda q(t))y = 0$ avec la condition initiale $f_\lambda(0) = 0$ et $f'_\lambda(0) = 1$. On note $N_\lambda$ le nombre de zéros de $f_\lambda$. Montrer que $\lambda \mapsto N_\lambda$ est croissante et déterminer ses limites en $-\i$ et $+\i$. + 4. On admet que $(x, \lambda) \in [0, 1] \times \R \mapsto f_{\lambda}(x)$ est continue. Montrer que l'ensemble + $\{\lambda \in \R, \ f_{\lambda}(1) = 0\}$ est l'ensemble des termes d'une suite réelle strictement croissante. + 5. Montrer que $(\lambda, x) \mapsto f_{\lambda}(x)$ est continue sur $\R \times [0, 1]$. #+end_exercice +#+BEGIN_proof + 1. + 2. + 3. + 4. +#+END_proof + +# ID:nil #+begin_exercice [ENS PLSR 2025 # 128] -Soit $\mu \in \R^+$. On considère $(E_\mu): y'' \mu(1-y^2)y' + y = 0$. +Soit $\mu \in \R^+$. On considère $(E_\mu)\colon y'' - \mu(1-y^2)y' + y = 0$. 1. Résoudre $(E_0)$. 1. Soient $x_0$ et $x_1$ deux fonctions bornées et de classe $\mc C^{\i}$ de $\R^+$ dans $\R$, et $\omega_1 \in \R$. On suppose qu'il existe des fonctions $\omega : \R^+ \ra \R$ et $\eps : \R \times \R \ra \R$ deux fois dérivables par rapport à la seconde variable telles que : - + $\omega(\mu) = 1 + \omega_1 \mu + o(\mu);$ + + $\omega(\mu) = 1 + \omega_1 \mu + o_0(\mu);$ + il existe $C: \R^+ \ra \R^+$ croissante telle que $\forall k \in \{0,1,2\}, \ \forall (\tau,\mu) \in \R^+ \times \R, \ |(\partial_2)^k \eps(\tau,\mu)| \leq C(\tau)\mu^2;$ + pour $x:(\tau,\mu)\in\R^+\times\R\mapsto x_0(\tau)+\mu x_1(\tau)+\eps(\tau,\mu)$, la fonction $t\mapsto x(\omega(\mu)t,\mu)$ est solution de $(E_\mu)$ sur $\R^+$ pour $\mu$ voisin de 0. + Calculer alors $\omega_1$ et donner une expression explicite de $x_0$ et $x_1$ en fonction de quelques constantes inconnues. #+end_exercice +#+BEGIN_proof + 1. + 2. +#+END_proof + # ID:180 # Classique #+begin_exercice [ENS L 2025 # 129] @@ -2244,41 +2316,64 @@ On sait paramétriser les triplets pythagoricien : $(a,b) = (m^2 - n^2, 2mn)$, d #+END_proof -#+begin_exercice [ENS P 2025 # 137] :todo: +# ID:8931 +#+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 $a$, $b$, $c$, $d$ ? #+end_exercice -#+BEGIN_proof :todo: -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 +#+BEGIN_proof +Notons $\theta$ l'angle entre les côtés $a$ et $b$, et $\phi$ celui entre les côtés $c$ et $d$. On a $$A = \frac{1}{2}ab\sin\theta + \frac{1}{2}cd\sin\phi$$ - $\sqrt{y/2(a^2+b^2 - y/2) - \frac{(a^2-b^2)^2}{4}} \simeq \sqrt{y (2a^2+2b^2 - y) - (a^2 - b^2)^2}$ -La même avec $c,d$ . +Le carré de la longueur de la diagonale vérifie : + $a^2+b^2 - 2ab\cos\theta = c^2+d^2 - 2cd\cos\phi$ +donc + $a^2+b^2-c^2-d^2 = 2ab\cos\theta - 2cd\cos\phi$ -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. +Extrema sous contrainte : l'aire est maximale quand + $$\vv{ab \cos \theta}{cd \cos \phi} \a \vv{ab \sin \theta}{- cd \sin \phi}$$ donc $\cos \theta \sin \phi + \sin \theta \cos \phi = 0$ , c'est-à-dire $\theta + \phi = \pi$. -Toujours plus agréable via de la trigo… +Alors les points sont cocycliques ! + +Il faut trouver la valeur commune du sinus, sachant $a^2 + b^2 - c^2-d^2 = (2ab + 2 cd)\cos \theta$, on peut élever au carré etc. + +Alternative : si on paramétrise par $x$ la longueur de la diagonale. + + Utiliser la formule de Héron pour l'aire. + + C'est l'horreur. #+END_proof +# ID:8899 #+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 + 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 algébrique 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. + 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 +#+BEGIN_proof + 1. + 2. Si on prend un truc maximal, on modifie un truc un peu -> ça doit être le milieu. + 3. On peut supposer qu'ils forment un polygone convexe. On peut + ordonner le long de cette enveloppe, et on cherche plutôt à + minimiser le périmètre en préservant l'aire, ce qui revient à + mettre chaque point sur la médiatrice des deux qui l'entourent. On + obtient que c'est un polygone régulier (tous les côtés ont la même + longueur, comme un losange). + + Ensuite, il faut montrer que les angles sont les mêmes, donc + prendre quatre points consécutifs. + + En prenant la base, l'aire s'exprime facilement. +#+END_proof + ** Probabilités @@ -2345,9 +2440,18 @@ Soit $\mc{P}$ l'ensemble des nombres premiers et $\beta \gt 1$. Soit $(Y_p)_{p 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 -#+begin_exercice [ENS 2025 # 145] :todo: -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 $\mathbf{P}(X\geq\theta\,\mathbf{E}(X))\geq \frac{(1-\theta)^2\mathbf{E}(X)^2}{\mathbf{E}(X^2)}$. +# ID:8900 +#+begin_exercice [ENS 2025 # 145] +Soient $\theta \in ]0,1[$ et $X$ une variable aléatoire à valeurs dans $\R_+$ telle que $\P(X\gt 0)\gt 0$. 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 +#+BEGIN_proof +Markov : $\P(X\geq \la)\leq $. + +Cauchy-Schwarz ? On a $\E(X \m 1_{X\geq \theta \E(X)})^2 \leq \E(X^2) \P(X \geq \theta \E(X))$. + +Et $\E(X\m 1_{X\geq \theta \E(X)})\geq (1-\theta)\E(X)$ car $\E(X\m 1_{X\leq \theta \E(X)})\leq \theta \E(X)$. +#+END_proof + # ID:8722 #+begin_exercice [ENS P 2025 # 146] @@ -2404,24 +2508,40 @@ autre premier pas ne le fait plus revenir en $0$. #+END_proof -#+begin_exercice [ENS P 2025 # 148] :todo: -Soient $p \in ]0, 1/2[$ et $(X_n)_{n \ge 1}$ une suite $i$.i.d. de variables aléatoires telle que $\mathbf{P}(X_n=1)=1-\mathbf{P}(X_n=-1)=p$. Pour $n\in\N^*$, on note $S_n=X_1+\cdots+X_n$. Montrer l'existence de $c,C_1,C_2\gt 0$ tels que $\forall u\geq 0,\ C_1e^{-cu}\leq \mathbf{P}\left(\sup_{n\geq 1}S_n\geq u\right)\leq C_2e^{-cu}$. +# ID:8920 +#+begin_exercice [ENS P 2025 # 148] +Soient $p \in ]0, 1/2[$ et $(X_n)_{n \ge 1}$ une suite i.i.d. de variables aléatoires telle que $\mathbf{P}(X_n=1)=1-\mathbf{P}(X_n=-1)=p$. Pour $n\in\N^*$, on note $S_n=X_1+\cdots+X_n$. Montrer l'existence de $c,C_1,C_2\gt 0$ tels que $\forall u\geq 0,\ C_1e^{-cu}\leq \mathbf{P}\left(\sup\limits_{n\geq 1}S_n\geq u\right)\leq C_2e^{-cu}$. #+end_exercice +#+BEGIN_proof +C'est de la ruine du joueur, il faut juste savoir que cette quantité +tend vers $0$ quand $u\ra +\i$, c'est-à-dire que $S_n$ est majorée +presque sûrement. C'est du Markov exponentielle. +#+END_proof + +# ID:8922 #+begin_exercice [ENS PLSR 2025 # 149] 1. Soit $X$ une variable aléatoire réelle et $s\gt 0$ tel que $\mathbf{E}(e^{sX})$ soit finie. Démontrer que $\forall a \gt 0$, $\mathbf{P}(X \ge a) \le e^{-sa} \mathbf{E}(e^{sX})$. - 1. Soit $(X_i)_{i \ge 1}$ une suite de variable aléatoires $i$.i.d. à valeurs dans [0, 1]. + 1. Soit $(X_i)_{i \ge 1}$ une suite de variable aléatoires i.i.d. à valeurs dans $[0, 1]$. On pose $S_n = \sum_{i=1}^n X_i$. Démontrer que $\forall t \gt 0$, $\mathbf{P}(|S_n - \mathbf{E}(S_n)| \ge t) \le 2e^{-t^2/(2n)}$. #+end_exercice +#+BEGIN_proof + 1. Markov exponentielle. + 2. On suppose les variables centrées. + Utilise $\E(e^{tX})\leq \cosh t\leq e^{t^2/2}$, puis optimiser en $t$. +#+END_proof + + +# ID:9022 #+begin_exercice [ENS PLSR 2025 # 150] Soit $(E, \mc{P}(E))$ un espace probabilisable avec $E$ dénombrable. 1. Rappeler la définition d'une probabilité sur cet espace. - 2. Pour $A$ et $B$ probabilités sur cet espace, on pose $d(A,B) = \max\limits_{S \subset E} A(S) B(S)$. Montrer que $d(A,B) = \frac{1}{2} \sum_x |A(\{x\}) - B(\{x\})|$. + 2. Pour $A$ et $B$ probabilités sur cet espace, on pose $d(A,B) = \max\limits_{S \subset E} A(S) -B(S)$. Montrer que $d(A,B) = \frac{1}{2} \sum_x |A(\{x\}) - B(\{x\})|$. 3. Soient $X$ et $Y$ deux variables aléatoires discrètes à valeurs - dans $E$ de lois respectives $A$ et $B$. Montrer que $P(X \neq Y) + dans $E$ de lois respectives $A$ et $B$. Montrer que $\P(X \neq Y) \ge d(A, B)$. 4. Les deux lois $A$ et $B$ étant fixées, montrer qu'on peut construire $X$ et $Y$ de façon à assurer l'égalité dans @@ -2432,52 +2552,91 @@ Soit $(E, \mc{P}(E))$ un espace probabilisable avec $E$ dénombrable. 2. Vient du fait que $\sum\limits_{x \mid A(x)\gt B(x)} A(x) - B(x) = \sum\limits_{x \mid A(x)\lt B(x)} B(x) - A(x)$. On prend pour $S$ l'ensemble des points où $A(x)\gt B(x)$. - 3. !! + 3. Si $X,Y$ sont à valeurs finis, dans $\db{1,m}$. + + On a $2\P(X\neq Y)\geq \E(\sum_{x_i} \m 1_{X = x_i,\,Y \neq x_i} + \m 1_{Y = x_i,\, X \neq x_i})$. + 4. Si les images sont finies, c'est simple. #+END_proof +# ID:nil #+begin_exercice [ENS PLSR 2025 # 151] -Soient $X$ et $Y$ deux variables aléatoires définies sur un même espace probabilisé $(\Omega, \mc{A}, \mathbf{P})$ et à valeurs dans [0, n]. On pose $p_k = \mathbf{P}(X = k)$ et $q_k = \mathbf{P}(Y = k)$ pour tout $k \in \db{0, n}, \op{et} d(p,q) = \max_{S \subset \db{0, n }} \overline{\mathbf{P}}(X \in S) - \mathbf{P}(Y \in S)$. - 1. Montrer que $d(p,q) \ge 0$. Que dire si d(p,q) = 0? - 1. Soit $\phi : \R \ra \R$ une fonction convexe. Comparer $\mathbf{E}(\phi(X))$ et $\phi(\mathbf{E}(X))$. - 1. On suppose de plus qu'il existe au moins deux éléments $k$ de [0,n] tels que $p_k\gt 0$. On suppose de plus que $\phi$ strictement convexe, c'est-à-dire telle que $\forall (x,y) \in \R^2, \ \forall t \in \,]0,1[\,\ x\neq y\Rightarrow \phi((1-t)x+ty)\leq (1-t)\phi(x)+t\phi(y).\ \text{Montrer que }\mathbf{E}(\phi(X))\gt \phi(\mathbf{E}(X))$. - 1. On suppose que : $\forall k \in \db{0, n }, \, p_k \gt 0$ et $q_k \gt 0$. On pose $H(p,q) = \sum_{k=0}^n p_k \ln \left( \frac{p_k}{q_k} \right)$. +Soient $X$ et $Y$ deux variables aléatoires définies sur un même espace probabilisé $(\Omega, \mc{A}, \mathbf{P})$ et à valeurs dans $\db{0,n}$. On pose $p_k = \mathbf{P}(X = k)$ et $q_k = \mathbf{P}(Y = k)$ pour tout $k \in \db{0, n}, \op{et} d(p,q) = \max_{S \subset \db{0, n }} \overline{\mathbf{P}}(X \in S) - \mathbf{P}(Y \in S)$. + 1. Montrer que $d(p,q) \ge 0$. Que dire si $d(p,q) = 0$ ? + 1. Soit $\phi \colon \R \ra \R$ une fonction convexe. Comparer + $\mathbf{E}(\phi(X))$ et $\phi(\mathbf{E}(X))$. + 1. On suppose de plus qu'il existe au moins deux éléments $k$ de + $\db{0,n}$ tels que $p_k\gt 0$. On suppose de plus que $\phi$ + strictement convexe, c'est-à-dire telle que + $\forall (x,y) \in \R^2, \ \forall t \in \,]0,1[\,\ x\neq + y\Rightarrow \phi((1-t)x+ty)\leq (1-t)\phi(x)+t\phi(y)$. Montrer + que $\mathbf{E}(\phi(X))\gt \phi(\mathbf{E}(X))$ + 1. On suppose que : $\forall k \in \db{0, n }, \, p_k \gt 0$ et $q_k + \gt 0$. On pose $H(p,q) = \sum_{k=0}^n p_k \ln \left( + \frac{p_k}{q_k} \right)$. -Montrer que $H(p,q) \ge 0$. Que dire si H(p,q) = 0? + Montrer que $H(p,q) \ge 0$. Que dire si $H(p,q) = 0$ ? #+end_exercice -#+begin_exercice [ENS L 2025 # 152] +#+begin_exercice [ENS L 2025 # 152] :todo: On considère $r_0 = 0$ et $(r_i)_{i \in \N^*} \in [0, 1]^{\N^*}$. Pour $(i, j) \in \N^* \times \N$, on pose $p_{i,j} = r_i$ si $j = i + 1, -1 - r_i$ si $j = i - 1$ et $0$ sinon.On admet l'existence d'une +1 - r_i$ si $j = i - 1$ et $0$ sinon. On admet l'existence d'une famille de variables aléatoires $(X_k^i)_{(i,k)\in\N^*\times\N}$ telles que + $X_0^{i_0} = i_0$ p.s. pour tout $i_0 \in \N^*$, - + $\mathbf{P}\left(\bigcap_{i=1}^n (X_k^{i_k} = i_{k-1})\right) = \prod_{i=1}^n p_{i_{k-1}, i_k} \text{ pour tout } (i_0, \dots, i_k) \in \N^{*k+1}$. + + $\mathbf{P}\left(\bigcap_{k=1}^n (X_k^{i_k} = i_{k-1})\right) = + \prod_{k=1}^n p_{i_{k-1}, i_k} \text{ pour tout } (i_0, \dots, i_k) + \in \N^{*k+1}$. On pose, pour $i,j\in\N^*$, $\tau^i_j=\inf\{k\in\db{ 0,+\i},\ X^i_k=j\}\in\N\cup\{+\i\}$. -Soit $b \in \N$. Calculer, pour $i \in [0, b]$, $\hat{p_i} = \mathbf{P}(\tau_0^i \lt \tau_b^i)$ en fonction des $\gamma_k = \prod_{i=1}^k \frac{1 - r_i}{r_i}$. +Soit $b \in \N$. Calculer, pour $i \in \db{0,b}$, $\hat{p_i} = +\mathbf{P}(\tau_0^i \lt \tau_b^i)$ en fonction des $\gamma_k = +\prod_{i=1}^k \frac{1 - r_i}{r_i}$. #+end_exercice +# ID:8954 #+begin_exercice [ENS PLSR 2025 # 153] Soient $(X_k)_{k \in \N^*}$ une suite de variables de Rademacher indépendantes et $X_0 =k \in \Z$ (constante). On pose, pour tout $n \in \N$, $S_n = X_0 + \cdots + X_n$. 1. Déterminer l'espérance et la variance de $S_n$. - 1. Soient $m \in \N^*$ et $k_1, \ldots, k_m \in \Z$. Que dire de la loi de $(S_n)_{n \geq m}$ conditionnée par $(S_1 = k_1, \ldots, S_m = k_m)$ ? - 1. Soient $k, N \in \N^*$ avec $N \ge k$. On considère que la marche aléatoire s'arrête dès que $S_n = 0$ ou $S_n = N$. On admet que l'arrêt est presque sûr. Déterminer la probabilité $p_k$ que la marche s'arrête sur 0 en partant de $k$. - 1. Déterminer le temps moyen d'arrêt (en 0 ou $N$ cette fois) en partant de $k$. + 1. Soient $m \in \N^*$ et $k_1, \ldots, k_m \in \Z$. Que dire de la + loi de $(S_n)_{n \geq m}$ conditionnée par $(S_1 = k_1, \ldots, + S_m = k_m)$ ? + 1. Soient $k, N \in \N^*$ avec $N \ge k$. On considère que la marche + aléatoire s'arrête dès que $S_n = 0$ ou $S_n = N$. On admet que + l'arrêt est presque sûr. Déterminer la probabilité $p_k$ que la + marche s'arrête sur $0$ en partant de $k$. + 1. Déterminer le temps moyen d'arrêt (en $0$ ou $N$ cette fois) en partant de $k$. #+end_exercice -#+begin_exercice [ENS P 2025 # 154] +#+begin_exercice Inégalité de Khintchine [ENS P 2025 # 154] :todo: 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 $(z_1, \ldots z_n) \in \C^n$, +existe $(c_p, C_p) \in (\R_+^*)^2$ indépendant de $n \in \N^*$ tel +que, pour tout $(z_1, \ldots z_n) \in \C^n$, $$c_p \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}} \leq \left(\mathbf{E} \left|\sum_{i=1}^n \eps_i z_i\right|^p\right)^{\frac{1}{p}} \leq C_p \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}}.$$ #+end_exercice +#+BEGIN_proof +On prend $p = 2$. C'est vide. + +On prend $p = 1$. On sait que $(\E \big|\sum \eps_i z_i\big|)^2\leq \E\big( |z_i|^2\big)$ par Cauchy-Schwarz. + +On a, pour un nombre complexe, $|z+u| + |z-u|\geq 2\max (|z|, |u|)$ par IT. On en déduit que $\E(|X_1 + X_2|)\geq \max \E(|X_1|), \E(|X_2|)$, ce qui est insuffisant. + +Assume not. Pour que $\E(|U|)\lt\lt \sqrt{\E(|U|^2)}$, il faut que $|U|$ prennent des valeurs très grandes, avec une petite probabilité : $\E(|U|) = G\eps$, et $\sqrt{\E(|U|^2)} = \sqrt{\eps} G$. + +Si on prend $n = 1$, c'est trivial. Si on prend $n = 2$. En préservant la norme deux, on peut mettre un $\sin \theta,\cos \theta$. + + $\E|\eps_1 z_1 h_1(t) + \eps_2 z_2 h_2(t)|$ on veut minimiser l'espérance. + $= |z_1h_1(t) + z_2 h_2(t)| + |z_1 h_1(t) - z_2 h_2(t)|$ + +Relier à 7489, et une version avec du $p$ entier, où on développe : le 7348. + +À réfléchir. +#+END_proof + # ID:8866 #+begin_exercice [ENS L 2025 # 155] @@ -2490,7 +2649,8 @@ Soit $c\in \interval]{0, 1}[$, $\P(|c^n X_n|\geq 1)\simeq e^{-1/c^n}$, donc $\P\ #+END_proof -#+begin_exercice [ENS P 2025 # 156] :todo: +# ID:8901 +#+begin_exercice [ENS P 2025 # 156] Soit $(p_n)_{n\geq 1}$ une suite d'éléments de $[0,1]$. Pour $n\in\N^*$, on note $G_n$ le graphe aléatoire $G_{n,p_n}$ d'Erdös-Renyi, c'est-à-dire un graphe aléatoire de sommets $\db{1,n}$ @@ -2505,6 +2665,11 @@ $j$. On note $I_n$ le nombre de sommets isolés de $G_n$. $p_n \leq (1-\eps)\frac{\ln(n)}{n}$. Montrer que $\mathbf{P}(I_n \geq 1) \underset{n \ra +\i}{\longrightarrow} 1$. #+end_exercice +#+BEGIN_proof + 1. Calculer l'espérance. + 2. Calculer la variance : elle est de l'ordre de l'espérance, ce qui suffit. +#+END_proof + # ID:8867 #+BEGIN_exercice [ENS L 2025 # 157] @@ -2541,7 +2706,7 @@ $\P(b,a,n)=\P\left(\sum_{k=1}^n a_k X_k=b\right)$. 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 - +# ID:nil #+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. 1. Déterminer une stratégie optimale pour l'alpiniste lorsque n=3. @@ -2569,6 +2734,7 @@ Sinon, on écrit $X = X_f \m 1_{A_f} + Y_f 1_{\ol{A_f}}$, où $\P(A_f)$ est proc #+END_proof +# ID:8936 #+begin_exercice [ENS L 2025 # 161] Pour deux variables aléatoires réelles bornées $X$ et $Y$, sur des espaces probabilisés a priori distincts, on note $X \leq_c Y$ pour @@ -2583,6 +2749,19 @@ conditions suivantes: + $M \preccurlyeq_c N$ et $X_1 \preccurlyeq_c Y_1$. On pose $S = \sum_{k=1}^M X_k$ et $T = \sum_{k=1}^N Y_k$. Montrer que $S \preccurlyeq_c T$. #+end_exercice +#+BEGIN_proof +Les conditions impliquent $\E(X) = \E(Y)$. On peut supposer $X,Y$ centrées. + +Plutôt $\E(|X-a|)\leq \E(|Y-a|)$. + +On peut aussi séparer en deux +temps : un temps on chaque les $X_i$, l'autre on change $M$ en $N$. + + + On pose $S_n = \sum_{k=1}^n X_k$ et $T_n = \sum_{k=1}^n Y_k$. On montre que $S_n \preccurlyeq_c T_n$ par récurrence sur $n$. + + On pose $g(m) = \E(f(S_m))$ pour $m \in \N$ et on montre que $g$ est convexe sur $\N$. + + la convexité de $f$ implique que pour tous $x,y \geq 0$, $f(s+x+y) - f(s+x) \geq f(s+y) - f(s)$ +#+END_proof # ID:8864 #+begin_exercice [ENS L 2025 # 162] @@ -2616,6 +2795,7 @@ On munit $\R^n$ de sa structure euclidienne canonique. Soit $X=(X_1,\ldots,X_n)^ #+END_proof +# ID:nil #+begin_exercice [ENS SR 2025 # 164] Soit $\alpha \gt 0$. On considère l'équation différentielle (): $(y' = -x, x' = \alpha^2 y)$ avec $(x,y):\R\ra\R^2$. 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 $J(t) = \alpha^2 x^2(t)$. @@ -2909,19 +3089,19 @@ Soit $n \geq 2$. On se place dans $\N^2$ et on considère le rectangle $[0,n] \t On note $u_n$ le nombre de recouvrements de $[0,n] \times [0,2]$. On munit l'ensemble des recouvrements de $[0, n] \times [0, 2]$ de la probabilité uniforme. - 1. Calculer $u_1, u_2, u_3$. Montrer que, pour tout $n \in \N^*$, $u_{n+2} = u_{n+1} + u_n$. En déduire une expression de $u_n$. + 1. Calculer $u_1, u_2, u_3$. Montrer que, pour tout $n \in \N^*$, + $u_{n+2} = u_{n+1} + u_n$. En déduire une expression de $u_n$. 1. On note $P_{1,n}$ la probabilité qu'il y ait un rectangle vertical tout à gauche. -Calculer $P_{1,n}$ et montrer que $(P_{1,n})$ admet une limite $L$. + Calculer $P_{1,n}$ et montrer que $(P_{1,n})$ admet une limite $L$. - $c$. On note $V_n$ le nombre de rectangles verticaux. Calculer $\mathbf{E}(V_n)$.Ind. On pourra écrire $V_n = \sum_{i=1}^n U_{i,n}$ où $U_{i,n}$ est l'indicatrice de l'événement : « il y a un + 3. On note $V_n$ le nombre de rectangles verticaux. Calculer $\mathbf{E}(V_n)$. -rectangle vertical en position $i$ ». + Ind. On pourra écrire $V_n = \sum_{i=1}^n U_{i,n}$ où $U_{i,n}$ est l'indicatrice de l'événement : « il y a un rectangle vertical en position $i$ ». - $d$. Montrer que $\frac{\mathbf{E}(V_n)}{n} \xrightarrow[n \ra +\i]{} L$. - -Ind. Découper la somme entre $[1, \sqrt{n}], [\sqrt{n}, n - \sqrt{n}]$ et $[n - \sqrt{n}, n]$. + 4. Montrer que $\frac{\mathbf{E}(V_n)}{n} \xrightarrow[n \ra +\i]{} L$. + Ind. Découper la somme entre $[1, \sqrt{n}], [\sqrt{n}, n - \sqrt{n}]$ et $[n - \sqrt{n}, n]$. 1. On note $V_{i,j,n}$ l'événement : « il y a un rectangle vertical en $i$ et en $j$ ». Calculer $\mathbf{E}(V_{i,j,n})$. 1. Calculer $V(V_n)$, puis en donner un équivalent quand $n \ra +\i$. #+end_exercice @@ -3421,30 +3601,41 @@ On étudie l'équation $x^2 + y^2 = N(1 + xy)$ d'inconnue $(x, y) \in \Z^2$, où couples solutions. #+end_exercice +# ID:9021 #+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 $n \in \db{0, a-1}$, $P(n)$ 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. + 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. $P(a-k) = a^2 - 2ak + k^2 + a - k + a = a^2 - 2a (k-1) + k (k-1) = (a - (k-1))^2 + (k-1)$ - - $P(a - (k+2)) = (a - (k+1))^2 + (k+1)$. - - !! + 2. Supposons que $P(a+k)$ n'est pas premier. Soit $p$ son plus petit facteur premier. + + Montrons d'abord que $p \ge a$. Si $p < a$, soit $n \in \db{0, p-1}$ le reste de $a+k$ modulo $p$. Comme $n < a$, par hypothèse $P(n)$ est premier. Or $P(n) \equiv P(a+k) \equiv 0 \pmod p$, donc $P(n) = p$. Mais $P(n) = n^2+n+a \ge a > p$, ce qui est absurde. + + On a $p^2 \le P(a+k) = (a+k)^2+(a+k)+a < (a+k+1)^2$, donc $p < a+k+1$. + + Posons $n = a+k-p$. Comme $a \le p \le a+k$, on a $0 \le n \le k \le a-2$. + + $P(n) \equiv P(a+k) \equiv 0 \pmod p$. Comme $n \in \db{0, a-1}$, $P(n)$ est premier, donc $P(n) = p$. + + L'équation $P(n) = p$ s'écrit $n^2+n+a = a+k-n$, d'où $k = n^2+2n$. + + On en déduit $k+1 = n^2+2n+1 = (n+1)^2$, ce qui montre que $k+1$ est un carré. #+END_proof +# ID:8939 #+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. 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 $m, k \in \N^*$. Montrer que $\frac{\prod_{j=1}^m \binom{2jk}{jk}}{\prod_{j=1}^m \binom{2j}{j}} \in \N$. #+end_exercice #+BEGIN_proof - 1. + 1. Donc $f$ est maximale le long de $y + ka$, pour tout $n$. En particulier, $n = 0$ donne $f(y)$ maximale, puis $f(y+a)\geq f(y-a)$ (prendre $n = 2$ et $x = y-a$) et en fait, $\forall n,\, f(y+na)\geq f(y-a)$ donc $f(y-a)$ est minimale. + + En fait, $f$ est dérivable, donc $\sum f'(y+ ka) = 0$, donc $f'$ est nulle en tout les $\+ka$. 2. 3. #+END_proof @@ -3462,22 +3653,51 @@ Soit $n \in \N^*$. Pour une partie $I$ de $\db{1,n}$, on appelle composante de $ #+END_proof +# ID:8935 #+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 -entières $\sum_{n=0}^{+\i} \frac{a_n}{n!} z^n$ et $\sum_{n=0}^{+\i} \frac{b_n}{n!} z^n$ sont congrues modulo $m$ si $a_n \equiv b_n \mod m$ pour tout $n \geq 0$. +Soient $(a_n)_{n\geq 0}$ et $(b_n)_{n\geq 0}$ deux suites d'entiers relatifs. On dit que les deux séries entières $\sum_{n=0}^{+\i} \frac{a_n}{n!} z^n$ et $\sum_{n=0}^{+\i} \frac{b_n}{n!} z^n$ sont congrues modulo $m$ si $a_n \equiv b_n \mod m$ pour tout $n \geq 0$. On note alors $\sum_{n=0}^{+\i} \frac{a_n}{n!} z^n \equiv \sum_{n=0}^{+\i} \frac{b_n}{n!} z^n \mod m$. - 1. Soit $p$ un nombre premier. Montrer que $(e^z-1)^{p-1}\equiv\sum_{n=0}^{+\i}-\frac{z^{n(p-1)}}{(n(p-1))!}\,\mathrm{mod}\,p$. - 1. Soit $m$ \gt 4 un entier non premier. - -Montrer que $m$ divise (m-1)!, et que $(e^z-1)^{m-1} \equiv 0 \mod m$. + 1. Soit $p$ un nombre premier. Montrer que $(e^z-1)^{p-1}\equiv\sum_{n=1}^{+\i}-\frac{z^{n(p-1)}}{(n(p-1))!}[p]$. + 1. Soit $m \gt 4$ un entier non premier. Montrer que $m$ divise + $(m-1)!$, et que $(e^z-1)^{m-1} \equiv 0 \mod m$. #+end_exercice +#+BEGIN_proof + 1. On a $(e^z - 1)^{p-1} = \sum_{k\in\N}$, après multiplication, on + obtient $\sum_{i_1 + \dots + i_{p-1} = k} {k\choose i_1} {k-i_1 \choose i_2}\dots {i_{p-1}\choose i_{p-1}}$ qui doit valoir $-1$ + ou $0$ modulo $p$ si $p-1 \mid k$. Si $k = p-1$, on obtient $(p-1)!$. + On peut peut-être faire du dénombrement, mais le cas $k = p-1$ est + déjà très dur : il faut faire agir par conjugaison un $p$-cycle + sur les $(p)$-cycles (au nombre de $(p-1)!$). + + En fait, c'est le nombre de surjections. + + En fait, on binôme, on a des expressions explicites des coefficients, c'est plié. + 2. On a un $a_n = \sum_{k=0}^{m-1} {m-1 \choose k} (-1)^{m-1-k} k^n$. + + Si on a l'interprétation combinatoire de $a_n$ (nombre de + surjections), on a $a_n$ qui est divisible par $(m-1)!$, par l'action. +#+END_proof + + + +# ID:8928 #+begin_exercice [X MP 2025 # 262] Soit $G$ un groupe. Un sous-groupe $H$ de $G$ est dit maximal lorsque $H \neq G$ et aucun sous-groupe de $G$ n'est compris strictement entre $H$ et $G$. Soit $n \ge 2$. - 1. Montrer que $\{\sigma \in \mc{S}_n, \ \eps(\sigma) = 1\}$ est un sous-groupe maximal de $\mc{S}_n$. - 1. Soit $k \in [1, n]$. Montrer que $\{\sigma \in \mc{S}_n, \ \sigma(k) = k\}$ est un sous-groupe maximal de $\mc{S}_n$. - 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. + 1. Montrer que $\{\sigma \in \mc{S}_n, \ \eps(\sigma) = 1\}$ est un + sous-groupe maximal de $\mc{S}_n$. + 1. Soit $k \in [1, n]$. Montrer que $\{\sigma \in \mc{S}_n, \ + \sigma(k) = k\}$ est un sous-groupe maximal de $\mc{S}_n$. + 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 +#+BEGIN_proof + 1. + 2. + 3. +#+END_proof + # ID:8714 #+BEGIN_exercice [X MP 2025 # 263] @@ -3499,33 +3719,90 @@ On pose $\alpha = \frac{12 + 5i}{13}$. 1. Soit $\alpha \in \C$ tel que $\alpha$ soit racine d'un polynôme unitaire à coefficients entiers dont toutes les racines complexes sont de module 1. Montrer que $\alpha$ est racine de l'unité. #+end_exercice +# ID:nil #+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? + 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 $A$ ? - 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 $\C[X]$ ? + 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 $\C[X]$. + 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_proof + 1. + 2. + 3. + 4. Si $X = U^3 + V^3=(U+V)(U^2 - UV + V^2)$. -#+begin_exercice [X MP 2025 # 266] -On admet le résultat suivant. Soient $c \in \C$, $U$ un voisinage de $c$ dans $\C$, $f\colon U \ra \C$ développable en série entière au voisinage de $c$ et telle que $f(z) = O((z-c)^k)$. Alors il existe $r$ \gt 0 et $z_1, \ldots, z_{2k} \in U$ distincts tels que : $\forall i \in [1, 2k], f(z_i) \in \R$ et $|c-z_i| = r$. - 1. Soient $A, B \in \R[X] \setminus \{0\}$. On suppose que les polynômes non nuls de $\mathrm{Vect}(A, B)$ sont scindés dans $\R[X]$. Montrer qu'entre deux racines de $A$ (au sens large) se trouve au moins une racine de $B$. + Soit $U + V = c$, soit $U+V = cX$, on traite les deux cas… +#+END_proof + + +#+begin_exercice [X MP 2025 # 266] :todo: +On admet le résultat suivant. Soient $c \in \C$, $U$ un voisinage de $c$ +dans $\C$, $f\colon U \ra \C$ développable en série entière au +voisinage de $c$ et telle que $f(z) = O((z-c)^k)$. Alors il existe $r +\gt 0$ et $z_1, \ldots, z_{2k} \in U$ distincts tels que : $\forall i +\in [1, 2k], f(z_i) \in \R$ et $|c-z_i| = r$. + 1. Soient $A, B \in \R[X] \setminus \{0\}$. On suppose que les polynômes non nuls + de $\mathrm{Vect}(A, B)$ sont scindés dans $\R[X]$. Montrer qu'entre deux + racines de $A$ (au sens large) se trouve au moins une racine de $B$. 1. Démontrer le résultat admis. #+end_exercice +#+BEGIN_proof + 1. + 2. On peut supposer que le coefficient en $(z-c)^k$ est non nul. +#+END_proof + +# ID:8956 #+begin_exercice [X MP 2025 # 267] Soient $F \in \R(X)$, $A = \{x \in \Q, F(x) \in \Q\}$ et $A' = \{x \in \Z, F(x) \in \Z\}$. 1. On suppose $A$ infini. Montrer que $F \in \Q(X)$. - 1. On suppose $A'$ infini. Que peut-on dire de $F$ ? + 2. On suppose $A'$ infini. Que peut-on dire de $F$ ? #+end_exercice +#+BEGIN_proof + 1. + 2. $F$ est à coefficients rationnels, $F$ a une partie entière, et un + truc de degré $\lt 0$, qui tend vers $0$, donc doit être nul, donc + $F$ est un polynôme, à coefficients rationnels. On ne peut pas + dire plus. +#+END_proof + +# ID:8957 #+begin_exercice [X MP 2025 # 268] -Soit $f = \sum_{k=0}^n c_k X^k$ un polynôme de degré $n$ à coefficients entiers et dont toutes les -racines complexes appartiennent à $\Q^*$. On pose $H = \max(|c_0|, \dots, |c_n|)$. - 1. Montrer que pour le complexe $i$ on a $|f(i)|^2 \le H^2\left(\frac{n^2}{2} + n + 1\right)$. - 1. Montrer que $|f(i)|\lt 2^n$. - c) En déduire que si $n\geq 10$ alors $n\leq 5 \log_2(H)$. +Soit $f = \sum_{k=0}^n c_k X^k$ un polynôme de degré $n$ à +coefficients entiers et dont toutes les racines sont réelles. On pose $H = \max(|c_0|, \dots, |c_n|)$. + 1. Montrer que pour le complexe $i$ on a $|f(i)|^2 \le + H^2\left(\frac{n^2}{2} + n + 1\right)$. + 2. Montrer que $|f(i)|^2\geq 2^n$. + 3. sA En déduire que si $n\geq 10$ alors $n\leq 5 \log_2(H)$. + 3. sEA On peut en déduire que si $n\geq 10$ alors $n\leq 5 \log_2(H)$. + 4. Montrer que le nombre $r$ de racines de $P$ appartenant à $\Q^*$ + vérifie $2^r \le H^2 \left(\frac{n^2}{2} + n + 1\right)$. #+end_exercice +#+BEGIN_proof + 1. C'est juste $|P(i)|^2 = \Re(P(i))^2 + \Im(P(i))^2$ et on somme. + 2. On veut $\a_n \prod (i-q_i)^2\geq 2^n$, sachant $\prod q_i \geq 1$. + + Il suffit d'utiliser $1 + x^2 \ge 2|x|$. + + On a $|i- a|^2 |i-b|^2 \geq |i - \sqrt{ab}|^4$, ce qui permet de conclure. + 3. $2^n \leq H^2 \big(\frac{n^2}{2} + n + 1\big)$ + 4. Soient $q_1, \dots, q_r \in \Q^*$ les $r$ racines rationnelles non nulles de $P$. + + En écrivant $q_k = a_k/b_k$ avec $a_k, b_k \in \Z^*$ premiers entre eux, le lemme de Gauss donne la factorisation $P(X) = Q(X) \prod_{k=1}^r (b_k X - a_k)$ avec $Q \in \Z[X]$. + + On évalue en $i$ pour obtenir $|P(i)|^2 = |Q(i)|^2 \prod_{k=1}^r (a_k^2 + b_k^2)$. + + Le polynôme $Q$ est à coefficients entiers, donc $Q(i) \in \Z[i]$. Comme $P \neq 0$ et $i \notin \Q$, on a $P(i) \neq 0$, donc $Q(i) \neq 0$. L'entier algébrique $|Q(i)|^2$ est donc un entier naturel non nul, d'où $|Q(i)|^2 \ge 1$. + + De plus, $a_k$ et $b_k$ étant des entiers non nuls, on a $a_k^2 + b_k^2 \ge 2$. + + On en déduit que $|P(i)|^2 \ge 2^r$. + + En appliquant la majoration de la question 1, on conclut que $2^r \le |P(i)|^2 \le H^2 \left(\frac{m^2}{2} + m + 1\right)$. +#+END_proof + # ID:8490 #+begin_exercice [X MP 2025 # 269] @@ -3587,11 +3864,20 @@ Soient $A$ et $B$ appartenant à $\mathcal{M}_n(\R)$, on note $k = \dim \operato +# ID:8923 #+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. +Soient $n \in \N^*$. On note $A_n = \{0, 1\}^n$ et $B_n = \{-1, 1\}^n$. On considère $H_A = \{f \in \mc{L}(\R^n), \ f(A_n) = A_n\}$ et $H_B$. + 1. Montrer que $H_A$ est un groupe pour la loi de composition et déterminer son cardinal. + 2. Montrer que $H_B$ est un groupe pour la loi de composition et déterminer son cardinal. #+end_exercice #+BEGIN_proof -Par le groupe orthogonal, on peut supposer que… + 1. Un élément est déterminé par les $f(e_i)$. Par ailleurs, les + vecteurs de la base canonique sont les seuls vecteurs dont la + somme est dans l'ensemble, donc on est une permutation de la base canonique. + 2. On peut supposer que $f(1,1,\dots,1) = (1,1,\dots,1)$, quitte à + agir par une matrice diagonale. Puis, il faudrait montrer que les + voisins de ce point sont envoyés sur des voisins. Ce sont les + seuls $n$ points $f_i$ tel que $\sum f_i= (n-2) e$. #+END_proof @@ -3600,22 +3886,33 @@ Par le groupe orthogonal, on peut supposer que… Soient $X, Y \in \M_2(\mathbb{K})$ où $\mathbb{K}$ est un sous-corps de $\C$. Montrer que la matrice $A = XY + YX - \op{tr}(X)Y - \op{tr}(Y)X + (\op{tr}(X)\op{tr}(Y) - \op{tr}(XY))I_2$ est nulle. #+end_exercice -#+begin_exercice [X MP 2025 # 273] +#+begin_exercice [X MP 2025 # 273] :todo: Soient $n \in \N^*$, $P$ et $Q$ dans $\C[X]$ tels que $P$ soit scindé à racines simples, $\deg P = n$ et $\deg Q \leq n$. On admet qu'il existe une matrice $B=(b_{i,j})_{0\leq i,j\leq n-1}$ telle que, pour tout $(x,y) \in \C^2$ avec $x \neq y$, on ait $$\frac{P(x) Q(y) - P(y) Q(x)}{x - y} = \sum_{0 \le i, j \le n - 1} b_{i,j} x^i y^j$$. Montrer que dim Ker $B = |\{z \in \C, \ P(z) = Q(z) = 0\}|$. #+end_exercice +#+BEGIN_proof +On a donc $\frac{P(x) Q(y) - P(y) Q(x)}{x - y} = Y^T B X$. +Si $z$ est un zéro commun. +#+END_proof + + +# ID:nil #+begin_exercice [X MP 2025 # 274] Soit $E$ un $K$-espace vectoriel de dimension $n \ge 2$. Soit $u$ et $v$ dans $\mc{L}(E)$, $c$ = $u \circ v - v \circ u$, on suppose $\op{rg} c = 1$. - 1. Montrer qu'il existe une base de $E$ dans laquelle la matrice de $c$ est égale à $E_{n-1,n}$. + 1. Montrer qu'il existe une base de $E$ dans laquelle la matrice de + $c$ est égale à $E_{n-1,n}$. 1. Montrer que pour tout $k \in \N$, $u^k(\op{Im} c) \subset \op{Ker} c$. 1. Montrer que $\chi_u$ n'est pas irréductible dans $\mathbb{K}[X]$. - 1. Soit $u \in \mc{L}(E)$, $F$ un sous-espace vectoriel de $E$ non trivial tel que $u(F) \subset F$. Montrer que $\chi_u$ n'est pas irréductible dans $\mathbb{K}[X]$. Étudier la réciproque. + 1. Soit $u \in \mc{L}(E)$, $F$ un sous-espace vectoriel de $E$ non + trivial tel que $u(F) \subset F$. Montrer que $\chi_u$ n'est pas + irréductible dans $\mathbb{K}[X]$. Étudier la réciproque. #+end_exercice +# ID:9008 #+begin_exercice [X MP 2025 # 275] On fixe un entier $n \ge 1$ et, pour $k \in [1, n]$, on note $\mc{R}_k$ l'ensemble des matrices de rang $k$ de $\M_n(\R)$. 1. Montrer que $\mc{R}_1 = \{XY^T, (X,Y) \in (\R^n \setminus \{0\})^2\}$. @@ -3671,6 +3968,7 @@ Pour $A \in \M_n(\C)$ on pose $R(A) = \{M \in \M_n(\C), M^2 = A\}$. +# ID:9015 #+begin_exercice [X MP 2025 # 279] Pour $n \in \N^*$, on pose $\quad\displaystyle\mc{IA} = \sup \{r \in \N ; \exists A_1, \dots, A_r \in \M_n(\C), \forall i, A_i^2 = I_n \text{ et } \forall i \neq j, A_i A_j = -A_j A_i \}$. 1. Si $n$ est impair, montrer que $\mc{IA}(n) = 1$. @@ -3682,10 +3980,19 @@ Pour $n \in \N^*$, on pose $\quad\displaystyle\mc{IA} = \sup \{r \in \N ; \exist Si on en a le bon nombre, et qu'on multiplie la dimension par $2$. Prendre les $\begin{pmatrix}A & 0 \\ 0 & -A\end{pmatrix}$ et les deux antidiagonales, avec des $I_n$. - Si on en avait plus : + Pourquoi pas plus : + + Les valeurs propres de chaque $A_i$ sont $\pm 1$, et les autres + $A_i$ envoient $E_1$ sur $E_{-1}$. + + En particulier dans $\M_2(\C)$, la famille $(I_n, A_1,A_2, A_3)$ est libre (multiplier par $A_i$, et prendre la trace). + + Plus généralement, fixons une matrice $A_1$, ses espaces $E_1,E_{-1}$, et $A_2$. + + On pose $B_i = i A_i A_2^{-1}$. On a $B_i^2 = A_i A_2^{-1} A_i A_{2}^2 = I_n$. + + et $B_i B_j = A_i A_2 A_j A_2 = - B_j B_i$. #+END_proof +# ID:9009 #+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 $\R^n$. @@ -3696,15 +4003,26 @@ Pour $n \in \N^*$, on pose $\quad\displaystyle\mc{IA} = \sup \{r \in \N ; \exist - On suppose que $b_1b_2b_3 = 1$. Montrer qu'il existe un unique $(a_1, a_2) \in \R^2$ tel que $M$ soit semblable à la matrice $$M' = \begin{pmatrix} a_1 & a_2 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}.$$ #+end_exercice +# ID:9010 #+begin_exercice [X MP 2025 # 281] -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? - 1. On suppose que $G$ est fini et que $Vect(G) = \mc{L}(V)$. Quelle est la dimension de V? - 1. Si $G$ n'est plus fini mais que $Vect(G) = \mc{L}(V)$, quelle est la dimension de 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 $G$ ? + 1. On suppose que $G$ est fini et que $\vect(G) = \mc{L}(V)$. Quelle + est la dimension de $V$ ? + 1. Si $G$ n'est plus fini mais que $\vect(G) = \mc{L}(V)$, quelle est + la dimension de $V$ ? #+end_exercice +#+BEGIN_proof + 1. Non, cf SL(V). + 2. Juste l'identité. + 3. Si $G$ est fini, ils sont tous diagonalisables. + 4. On a $\op{Tr}((g-I_n) M) = 0$, pour tout $M\in\vect G$. +#+END_proof + +# ID:9011 #+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 @@ -3728,14 +4046,40 @@ On suppose maintenant que, pour tout $g \in G$, $g$ id est nilpotent. - 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 +#+BEGIN_proof + 1. + 2. oui, trivialement, prendre $a_n = \frac{1}{R^n}$ + 3. oui, prendre $a_n = \frac{1}{R^n} b_n$, avec $b_n$ qui tend vers + $0$ plus vite que toute puissance de $n$ (mais moins vite que + géométrique). + Utilise juste du binôme. + 4. +#+END_proof + + +# ID:9016 #+begin_exercice [X MP 2025 # 283] -Soient $E = \mc C^0([-1,1],\C)$, $g$ une surjection continue croissante de [-1,1] sur luimême. On considère $F$ un sous-espace vectoriel de $E$ de dimension finie stable par $f\mapsto f\circ g$. +Soient $E = \mc C^0([-1,1],\C)$, $g$ une surjection continue croissante de $[-1,1]$ sur lui-même. On considère $F$ un sous-espace vectoriel de $E$ de dimension finie stable par $f\mapsto f\circ g$. On note $\phi$ l'endomorphisme de $F$ défini par $\phi: f \mapsto f \circ g$. - 1. Montrer que 1 est la seule valeur propre de $\phi$. + 1. Montrer que $1$ est la seule valeur propre de $\phi$. 1. En déduire que $\phi = \mathrm{id}_F$. 1. Que peut-on dire des valeurs propres possibles de $\phi$ si $q$ n'est plus supposée surjective? #+end_exercice +#+BEGIN_proof + 1. Si $|\la|\lt 1$, la + surjectivité de $g$ pose problème. + + Soit $x$ tel que $f(x)\neq 0$, alors $f\circ g^n (x)$ converge, car $g$ croissante et $f$ continue, donc $\la^n f(x)$ converge, donc $\la = 1$. + + Sinon, $g^n$ converge, donc $\la^n f$ converge, donc $f = \tilde{0}$, ou $\la = 1$. + + 2. Trivial, par dimension finie, tout autre matrice a des puissances qui divergent. + 3. Si $g$ n'est pas surjective, on peut imaginer des valeurs propres de module $\lt 1$. + + Par exemple, avec $g(x)=x/2$ et $f(x)=x$, on a $\la=1/2$ +#+END_proof + # ID:nil #+begin_exercice [X MP 2025 # 284] @@ -3743,13 +4087,18 @@ Soit $p$ un nombre premier, $A$ et $B$ appartenant à $\M_n(\Z)$. Démontrer que $\op{tr}((A+B)^p) \equiv \op{tr}(A^p) + \op{tr}(B^p) \pmod{p}$. #+end_exercice +# ID:9012 #+begin_exercice [X MP 2025 # 285] Soient $n \in \N^*$ et $H$ un sous-espace vectoriel de $\M_n(\C)$ stable par produit matriciel. On note $D = \{ \delta \in \mc{L}(H) : \forall (A, B) \in H^2, \ \delta(AB) = \delta(A)B + A\delta(B) \}$. - 1. Soit $C \in H$. Montrer que $\delta : A \mapsto CA AC$ est dans $D$, et exprimer simplement $e^{\delta}$. + 1. Soit $C \in H$. Montrer que $\delta : A \mapsto CA- AC$ est dans $D$, et exprimer simplement $e^{\delta}$. 1. Soit $\delta \in D$. Montrer que $\forall A, B \in H$, $e^{\delta}(AB) = e^{\delta}(A) e^{\delta}(B)$. - 1. Retrouver le résultat de la question précédente en considérant l'application $f\colon t \in \R \mapsto e^{-t\delta} \left( e^{t\delta}(A) e^{t\delta}(B) \right)$ et en calculant f'. + 1. Retrouver le résultat de la question précédente en considérant l'application $f\colon t \in \R \mapsto e^{-t\delta} \left( e^{t\delta}(A) e^{t\delta}(B) \right)$ et en calculant $f'$. 1. Soit $\delta \in D$. Pour $\lambda \in \C$, on note $H_{\lambda}$ le sous-espace caractéristique de $\delta$ associé à $\lambda$ (éventuellement $\{0\}$). Soient $\lambda$, $\mu \in \C$, $A \in H_{\lambda}$ et $B \in H_{\mu}$. Montrer que $AB \in H_{\lambda+\mu}$. #+end_exercice +#+BEGIN_proof + 1. +#+END_proof + # ID:nil #+begin_exercice [X MP 2025 # 286] @@ -3757,41 +4106,77 @@ Soient $n \in \N^*$ et $H$ un sous-espace vectoriel de $\M_n(\C)$ stable par pro 1. Montrer qu'il existe une famille $(v_1,\ldots,v_{k+1})$ de vecteurs unitaires de $\R^k$ tels que $\langle v_i, v_j \rangle = -1/k$ pour tous $i$, $j$ distincts. #+end_exercice +# ID:9026 #+begin_exercice [X MP 2025 # 287] -Soit $E$ un $\R$ -espace vectoriel de dimension finie. - 1. Soit $f \in \mc{L}(E)$. Montrer que Tr(f id) = 0 et rg(f id) = 1 si et seulement s'il existe $a \in E$ et $\ell \in E^*$ tel que $\ell(a) = 0$ et $f = id + \ell a$. On dit alors que $f$ est une transvection. +Soit $E$ un $\R$-espace vectoriel de dimension finie. + 1. Soit $f \in \mc{L}(E)$. Montrer que $\tr(f- \op{id}) = 0$ et $\rg(f-\op{id}) = 1$ si et seulement s'il existe $a \in E$ et $\ell \in E^*$ tel que $\ell(a) = 0$ et $f = \op{id} + \ell a$. On dit alors que $f$ est une transvection. -Soit $\phi: E \times E \ra \R$ une forme bilinéaire telle que : $\forall x \in E \setminus \{0\}, \exists y \in E, \phi(x,y) \neq 0$ et $\forall (x,y) \in E^2, \phi(y,x) = -\phi(x,y)$. + 2. E Soit $\phi\colon E \times E \ra \R$ une forme bilinéaire telle que : $\forall x \in E \setminus \{0\}, \exists y \in E, \phi(x,y) \neq 0$ et $\forall (x,y) \in E^2, \phi(y,x) = -\phi(x,y)$. -Soit $G = \{u \in GL(E) : \forall x, y \in E, \phi(u(x), u(y)) = \phi(x, y)\}$. - 1. Montrer que $G$ est un sous-groupe de GL(E).- c) Montrer que $G$ contient les applications de la forme id $+\lambda \phi(a,\cdot)$ a avec $\lambda \in \R$ et $a \in E$. - 1. Montrer que $G$ est engendré par les transvections de la forme indiquée en c). + Soit $G = \{u \in GL(E) : \forall x, y \in E, \phi(u(x), u(y)) = \phi(x, y)\}$. + 3. Montrer que $G$ est un sous-groupe de $GL(E)$. + 4. Montrer que $G$ contient les applications de la forme $\op{id} +\lambda \phi(a,\cdot) a$ avec $\lambda \in \R$ et $a \in E$. + 5. Montrer que $G$ est engendré par les transvections précédentes. #+end_exercice +#+BEGIN_proof + 1. + 2. + 3. Les transvections précédentes sont exactement celles qui préservent la forme : il faut $\phi(x, \l(y) a) + \phi(\l(x) a, y) = 0$ + 4. Récurrence sur la dimension. + + L'action des transvections est transitives (si $\phi(x,y)\neq 0$, une transvection envoie $x$ sur $y$, sinon, il en faut $2$). + + Il faut en général choisir deux éléments $u,f$ tel que $\phi(u, f) = 1$, et montrer que on peut être l'identité sur le plan. Alors, on stabilisera son orthogonal. +#+END_proof + +# ID:8890 #+begin_exercice [X MP 2025 # 288] Soient $n \in \N$ et $O \in \mc{O}_n(\R)$. Calculer $\alpha_O = |\det(\psi_O)|$ où $\psi_O : A \in \mc{S}_n(\R) \mapsto$ $O^TAO$. #+end_exercice +#+BEGIN_proof + $A\mapsto OA$ est de déterminant $\pm 1$, car c'est une isométrie. +#+END_proof + +# ID:9013 #+begin_exercice [X MP 2025 # 289] -Pour $M \in GL_n(\R)$ tel que $-1 \notin Sp(M)$, on pose $T(M) = (I_n M)(I_n + M)^{-1}$. On note $\mc{A}_n(\R)$ l'ensemble des matrices antisymétriques et $\mc{B}_n(\R)$ l'ensemble des matrices +Pour $M \in GL_n(\R)$ tel que $-1 \notin Sp(M)$, on pose $T(M) = (I_n- M)(I_n + M)^{-1}$. On note $\mc{A}_n(\R)$ l'ensemble des matrices antisymétriques et $\mc{B}_n(\R)$ l'ensemble des matrices $M \in \mc{O}_n(\R)$ telles que $-1 \notin \op{Sp}(M)$. 1. Montrer que $T$ est bien définie sur $\mc{A}_n(\R)$ et $\mc{B}_n(\R)$. 1. Si $A \in \mc{A}_n(\R)$, montrer que $T(A) \in \mc{B}_n(\R)$. 1. Si $B \in \mc{B}_n(\R)$, montrer que $T(B) \in \mc{A}_n(\R)$. 1. Calculer $T \circ T(A)$ si $A \in \mc{A}_n(\R)$. - 1. Soient $x \in \R$ et $A = \begin{pmatrix} 0 & x \\ -x & 0 \end{pmatrix}$. Calculer T(A). + 1. Soient $x \in \R$ et $A = \begin{pmatrix} 0 & x \\ -x & 0 + \end{pmatrix}$. Calculer $T(A)$. 1. Déduire des questions précédentes que toute matrice de $\mc{A}_{2n}(\R)$ est orthosemblable à une matrice diagonale par blocs avec des blocs diagonaux de la forme $\begin{pmatrix} 0 & x \\ -x & 0 \end{pmatrix}$. #+end_exercice +# ID:9020 #+begin_exercice [X MP 2025 # 290] On munit $\R^n$ de sa structure euclidienne canonique. - 1. Soit $M \in \mc{S}_n^{++}(\R)$. Montrer que l'application $(x,y) \in (\R^n)^2 \mapsto \langle M^{-1}x,y \rangle$ définit un -produit scalaire sur $\R^n$. - 1. Soient $M \in \mc{S}_n^{++}(\R)$ et $N \in \mc{A}_n(\R)$. Montrer que MN est diagonalisable dans $\M_n(\C)$ à spectre inclus dans $i\R$. - 1. Soit $A \in \M_n(\R)$ diagonalisable dans $\M_n(\C)$ à spectre inclus dans $i\R$. Existe-t-il $M \in$ -$\mc{S}_n^{++}(\R)$ et $N \in \mc{A}_n(\R)$ telles que $A$ = MN? + 1. Soit $M \in \mc{S}_n^{++}(\R)$. Montrer que l'application $(x,y) + \in (\R^n)^2 \mapsto \langle M^{-1}x,y \rangle$ définit un produit + scalaire sur $\R^n$. + 1. Soient $M \in \mc{S}_n^{++}(\R)$ et $N \in \mc{A}_n(\R)$. Montrer + que $MN$ est diagonalisable dans $\M_n(\C)$ à spectre inclus dans + $i\R$. + 1. Soit $A \in \M_n(\R)$ diagonalisable dans $\M_n(\C)$ à spectre + inclus dans $i\R$. Existe-t-il $M \in\mc{S}_n^{++}(\R)$ et $N \in + \mc{A}_n(\R)$ telles que $A = MN$ ? #+end_exercice +#+BEGIN_proof + 1. + 2. Soit $S \in \mc{S}_n^{++}(\R)$ l'unique racine carrée de $M$. La matrice $MN = S^2 N$ est semblable à $S^{-1}(S^2 N)S = SNS$. + Posons $H = SNS$, alors $H$ est antisymétrique, donc + diagonalisable dans $\M_n(\C)$ et son spectre est inclus dans + $i\R$. + 3. Oui, car une telle matrice est semblable, dans $\R$ à une matrice antisymétrique de blocs $2-2$. + + On a $A = P D P^{-1} = (PP^T)(P^T)^{-1} D P^{-1}$. +#+END_proof + + +# ID:nil #+begin_exercice [X MP 2025 # 291] Soit $n \in \N^*$. On pose $J = \begin{pmatrix} 0 & -I_n \\ I_n & 0 \end{pmatrix}$. 1. Soit $M \in \M_{2n}(\R)$ telle que $M^2 = -I_{2n}$. Montrer l'équivalence : $M^T J \in \mc{S}_{2n}(\R) \Leftrightarrow M^T J M = J$. @@ -3800,11 +4185,47 @@ Soit $n \in \N^*$. On pose $J = \begin{pmatrix} 0 & -I_n \\ I_n & 0 \end{pmatrix 1. Montrer que, pour pour tout $M \in C$, $S_M J + J S_M = 0$. #+end_exercice +# ID:8879 #+begin_exercice [X MP 2025 # 292] -Les espaces $\R^p$ sont munis de leurs normes euclidiennes canoniques. Soient $d$ et $D$ des entiers $\geq 1$. Étant donné $p_0, \ldots, p_n \in \R^d$, on dit que $(p_0, \ldots, p_n)$ se plonge isométriquement dans $\Q^D$ s'il existe $q_0,\ldots,q_n\in\Q^D$ vérifiant $\|p_i-p_j\|=\|q_i-q_j\|$ pour tous $i,j\in[0,n]$. - 1. On suppose que $(p_0,\ldots,p_n)$ se plonge isométriquement dans $\Q^D$. Soit $p$ une combinaison linéaire à coefficients rationnels de $p_0, \ldots, p_n$. Montrer que $(p, p_0, \ldots, p_n)$ se plonge isométriquement dans $\Q^D$. - 1. Soient $p_0, \ldots, p_n \in \R^d$ tels que $||p_i p_j||^2 \in \Q$ pour tous $i, j \in [0, n]$. Montrer que $(p_0,\ldots,p_n)$ se plonge isométriquement dans $\Q^{4d}$. On admettra que tout entier naturel est somme de quatre carrés d'entiers. +Les espaces $\R^p$ sont munis de leurs normes euclidiennes canoniques. +Soient $d$ et $D$ des entiers $\geq 1$. Étant donné $p_0, \ldots, p_n +\in \R^d$, on dit que $(p_0, \ldots, p_n)$ se plonge isométriquement +dans $\Q^D$ s'il existe $q_0,\ldots,q_n\in\Q^D$ vérifiant +$\|p_i-p_j\|=\|q_i-q_j\|$ pour tous $i,j\in[0,n]$. + 1. On suppose que $(p_0,\ldots,p_n)$ se plonge isométriquement dans + $\Q^D$. Soit $p$ un barycentre à coefficients rationnels de $p_0, + \ldots, p_n$. Montrer que $(p, p_0, \ldots, p_n)$ se plonge + isométriquement dans $\Q^D$. + 1. Soient $p_0, \ldots, p_n \in \R^d$ tels que $||p_i- p_j||^2 \in + \Q$ pour tous $i, j \in [0, n]$. Montrer que $(p_0,\ldots,p_n)$ se + plonge isométriquement dans $\Q^{4d}$. On admettra que tout entier + naturel est somme de quatre carrés d'entiers. #+end_exercice +#+BEGIN_proof + 1. Si $p$ est la moyenne de $p_0, p_1$, on l'envoie sur la moyenne + correspondante de $q_0, q_1$. Les distances avec $q_0$ et $q_1$ correspondent + toujours. + + Pour ce qui est de la distance de $p$ à $p_2$ par exemple, elle est + entièrement déterminée (car euclidien) par les distances de $p_2$ + à $p_1, p_0$ et les distances de $p$ à $p_1$ et $p_0$, donc on est + bon. + + Ensuite, quitte à rajouter des points, on peut supposer que tous + les points sont des moyennes de deux autres points. + + Cela traite les combinaisons barycentrique. Il faut aussi traiter + par exemple l'ajout du vecteur $\vec 0$, ce qui est manifestement + faux. + 2. Considérer une base orthogonale, formée par Gram-Schmidt par + exemple. Les produits scalaires sont tous rationnels (une fois que + l'on suppose $p_0 = \vec 0$). + + Pour une telle base, on peut juste appliquer l'indication sur les + sommes de quatre carrés pour l'envoyer par une isométrie dans + $\Q^{4d}$. +#+END_proof + # ID:8875 #+BEGIN_exercice [X MP 2025 # 293] @@ -3822,20 +4243,29 @@ Les espaces $\R^p$ sont munis de leurs normes euclidiennes canoniques. Soient $d #+END_proof +# ID:9025 #+begin_exercice [X MP 2025 # 294] On munit $\R^n$ de sa structure euclidienne canonique. - 1. Soient $A \in \M_n(\R)$ et $F$ un sous-espace vectoriel de $\R^n$. Soit $(f_i)_{1 \le i \le k}$ une base or- - -thonormée de $F$. On pose : $\tau_F(A) = \sum_{i=1}^k \langle f_i, Af_i \rangle$. Montrer que $\tau_F(A)$ ne dépend pas de la - -base orthonormée choisie. + 1. Soient $A \in \M_n(\R)$ et $F$ un sous-espace vectoriel de $\R^n$. Soit $(f_i)_{1 \le i \le k}$ une base orthonormée de $F$. On pose : $\tau_F(A) = \sum_{i=1}^k \langle f_i, Af_i \rangle$. Montrer que $\tau_F(A)$ ne dépend pas de la base orthonormée choisie. Dans la suite de l'exercice, on suppose $A \in \mc{S}_n(\R)$ et on note $\lambda_1 \geq \cdots \geq \lambda_n$ les valeurs propres de A, comptées avec multiplicité. - 1. Déterminer le meilleur encadrement possible de $\tau_F(A)$ en fonction de $F$ et de Sp(A). - 1. On pose, pour $t \in \R$, $A(t) = A + tE_{1,1}$. Pour $t \in \R$, on note $\lambda_1(t) \ge \cdots \ge \lambda_n(t)$ les valeurs propres de A(t). Montrer que : $\forall t \ge 0, \lambda_n(t) \ge \lambda_n$ et $\lambda_1 \ge \lambda_2(t)$. - 1. Déterminer un équivalent simple de $\lambda_1(t)$ quand $t$ tend vers $+\i$. + 1. Déterminer le meilleur encadrement possible de $\tau_F(A)$ en + fonction de $F$ et de $Sp(A)$. + 1. On pose, pour $t \in \R$, $A(t) = A + tE_{1,1}$. Pour $t \in \R$, + on note $\lambda_1(t) \ge \cdots \ge \lambda_n(t)$ les valeurs + propres de $A(t)$. Montrer que : $\forall t \ge 0, \lambda_n(t) + \ge \lambda_n$ et $\lambda_1 \ge \lambda_2(t)$. + 1. Déterminer un équivalent simple de $\lambda_1(t)$ quand $t$ tend + vers $+\i$. #+end_exercice +#+BEGIN_proof + 1. C'est la différence de la trace, et de la trace sur $E/F$. + 2. On a $\sum_{i=n-k+1}^n \lambda_i \le \tau_F(A) \le \sum_{i=1}^k + \lambda_i$, via le minimax, + 3. Minimax aussi. + 4. Équivalent à $t$, assez clairement. +#+END_proof ** Analyse @@ -3859,20 +4289,53 @@ Soit $E$ un $\R$ espace vectoriel de dimension finie. Montrer que tout convexe n #+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 $\Z^2$. +#+begin_exercice [X MP 2025 # 297] :todo: +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. sA Si $k \in \N^*$, montrer que $R_k$ est le minimum de $E_k$. + 1. sEA On admet que la borne inférieure $R_k$ est toujours atteinte + (il existe un disque de rayon $R_k$ contenant au moins $k$ points). 1. Montrer que, pour $k \in \N^*$, $4R_k^2$ est entier. - 1. Donner un équivalent de $R_k$ lorsque $k$ tend vers $+\i$. + 1. s Donner un équivalent de $R_k$ lorsque $k$ tend vers $+\i$. #+end_exercice #+BEGIN_proof - 1. + 1. On a $R_2 = 0,5$, puisque une boule contenant deux points entiers + est forcément de rayon $\geq 0.5$ (car les deux points en question + sont à une distance d'au moins $1$), et qu'il existe une boule de + rayon $0,5$ qui contient deux entiers. + + La distance entre deux points à coordonnées entières est de la + forme $\sqrt{a^2 + b^2}$. Si les points sont voisins, elle vaut + $1$ et sinon, elle est $\geq \sqrt{2}$. + + Si un disque contient au moins trois points entiers, deux d'entre + eux sont à une distance $\geq \sqrt{2}$, donc le rayon de ce + disque est $\geq \frac{\sqrt{2}}{2}$. + + Par ailleurs, le cercle circonscrit à un carré de côté $1$ est de + rayon $\frac{\sqrt{2}}{2}$. + + On en déduit que $R_3 = R_4 = \frac{\sqrt{2}}{2}$. 2. - 3. + 3. Soit $k\in\N^*$. On considère un disque de rayon $R_k$ contenant + au moins $k$ points entiers. + + C'est le centre d'un triangle sur le cercle, mais cela n'implique + pas que ses coordonnées soient des demi-entiers. + + À un rayon $r$ fixé, on considère $(x,y)\mapsto N(r)$, définie sur + $[0,1]^2$ qui compte le nombre de sommets entiers dans le disque. + + Cette fonction vérifie une propriété de symétrie. + + Super dur !! + 4. Simple. #+END_proof +# ID:8939 #+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 $\phi(g) = 0$. #+END_exercice @@ -3883,26 +4346,41 @@ Si $f$ ne s'annule pas et $\phi(f) = 0$, $\phi(\sqrt{f}) = 0$, donc en itérant 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. +On obtient certainement que si $f\gt 0$, alors $\phi(f)$ est positive, +etc. On peut alors s'intéresser à des fonction uniquement positives. + +Si deux fonctions positives n'ont pas de zéros en commun, et sont +toutes les deux nulles, alors en sommant, on a un problème. + +Donc, on note $I$ l'ensemble des zéros communs à toutes les fonctions +positives qui annulent $f$. + +Si $I$ n'est pas réduit à un singleton, on contredit la propriété. #+END_proof +# ID:8942 #+begin_exercice [X MP 2025 # 299] 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 $u(0) = u(1)$. + 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 ? + 3. Est-ce vrai ? Si on prend une droite dans $\R^2$ qui varie, + + On peut mettre une distance sur les familles libres de $k$-vecteurs unitaires ? + + Si $u(1)$ ne convient pas, ils diffèrent par le commutateur du projecteur, c'est-à-dire $GL_p(\C)\times GL_q(\C)$. Ensuite localement, il est facile de passer de $I_p\times I_q$ à $A\times B$. #+END_proof +# ID:8943 #+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 $\mc{B}_n$. @@ -3919,23 +4397,49 @@ A, $B$ éléments distincts dans $\mc{B}_n$ et $t \in ]0, 1[$. 1. Montrer que $\mc{B}_n$ est l'enveloppe convexe des $P_{\sigma}$ pour $\sigma \in \mc{S}_n$. #+end_exercice +# ID:8891 #+begin_exercice [X MP 2025 # 301] On munit $E = \mc C^0([-1,1],\R)$ de la norme $\|\cdot\|_{\i}$. 1. Soit $n \in \N$. Montrer qu'il existe un unique $T_n \in \R[X]$ de degré $n$ tel que $\forall \theta \in \R, T_n(\cos \theta) = \cos(n\theta)$. -Soit $(a_n)_{n\geq 0}\in (\R^+)^{\N}$ telle que $\sum a_n$ converge. + 2. E Soit $(a_n)_{n\geq 0}\in (\R^+)^{\N}$ telle que $\sum a_n$ converge. 1. Soit $f\colon x \mapsto \sum_{n=0}^{+\i} a_n T_{3^n}(x)$. - Montrer que $f$ est bien définie et continue sur [-1, 1]. - - Montrer que $d(f, \R_{3^n}[X]) = \inf_{P \in \R_{3^n}[X]} \|f P\|_{\i} = \sum_{k=n+1}^{+\i} a_k$. + - Montrer que $d(f, \R_{3^n}[X]) = \inf_{P \in \R_{3^n}[X]} \|f- P\|_{\i} = \sum_{k=n+1}^{+\i} a_k$. -Ind. On pourra considérer les points $x_k = \cos(\pi(1+k3^{-n-1}))$ pour $k \in [0,3^{n+1}]$. + Ind. On pourra considérer les points $x_k = \cos(\pi(1+k3^{-n-1}))$ pour $k \in [0,3^{n+1}]$. #+end_exercice +#+BEGIN_proof + 1. + 2. + 1. + 2. +#+END_proof + +# ID:8921 #+begin_exercice [X MP 2025 # 302] -Soient $K$ une fonction continue de $[0,1]^2$ dans $\R$, $E$ l'espace des fonctions continues de [0,1] dans $\R$.- a) Si $f \in E$, soit $T_K(f)$ la fonction de [0,1] dans $\R$ telle que $\forall x \in [0,1], T_K(f)(x) =$ $\int_{-\i}^{\i} K(x,y)f(y)dy$. Montrer que $T_K$ est un endomorphisme continu de l'espace normé $(E,\|\cdot\|_{\i})$. - 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. +Soient $K$ une fonction continue de $[0,1]^2$ dans $\R$, $E$ l'espace +des fonctions continues de $[0,1]$ dans $\R$. + 1. Si $f \in E$, soit $T_K(f)$ la fonction de $[0,1]$ dans $\R$ telle + que $\forall x \in [0,1], T_K(f)(x) = \int_{-\i}^{\i} + K(x,y)f(y)\dy$. Montrer que $T_K$ est un endomorphisme continu de + l'espace normé $(E,\|\cdot\|_{\i})$. + 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 +#+BEGIN_proof + 1. + 2. S'il y avait une autre fonction, non colinéaire à $g$, en ajoutant + $g$ (qui ne peut pas s'annuler) on aurait deux fonctions + positives, ce qui serait étrange, car on peut alors créer une + fonction positive non nulle qui s'annule en un seul point, par + combinaison linéaire, ce qui n'est clairement pas possible. +#+END_proof + # ID:nil #+begin_exercice [X MP 2025 # 303] @@ -3959,15 +4463,30 @@ Par ailleurs, si $(u_n)$ a une valeur d'adhérence différente de $+\i$, clairem #+END_proof -# À relier. +# ID:9017 #+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 $x_n \ra x$. #+end_exercice -#+BEGIN_proof - 1. -#+END_proof +#+begin_proof + 1. On a $z_{n+1} = \frac{n + \beta}{n+1} z_n$. Par récurrence immédiate : + $z_n = \prod_{k=0}^{n-1} \frac{k+\beta}{k+1}$ + + Si $\beta \in \N^*$, on a $z_n = \frac{\beta(\beta+1)\dots(\beta+n-1)}{n!} = {n+\beta-1 \choose \beta-1}$. Alors $z_n \sim \frac{n^{\beta-1}}{(\beta-1)!}$. + + Dans le cas général, + $$\ln z_n = \sum_{k=1}^n \ln\left(1 + \frac{\beta-1}{k}\right) = \sum_{k=1}^n \left(\frac{\beta-1}{k} + O\left(\frac{1}{k^2}\right)\right)$$ + En utilisant $\sum_{k=1}^n \frac{1}{k} = \ln n + \gamma + o(1)$, on obtient $\ln z_n = (\beta-1)\ln n + K + o(1)$, donc $z_n \sim C n^{\beta-1}$. + + 2. On a $x_n = (n+1)\mu_n - n\mu_{n-1}$ pour $n\geq 1$ (avec $x_0 = \mu_0$). + + On obtient $(n+\beta)\mu_n = n\mu_{n-1} + \beta y_n$. + + On veut exprimer $\mu_n$ en fonction de $y_n$. + + On cherche $\mu_n = \la_n z_n'$, où $z_n' (n+\b) = n z_{n-1}'$. + + Alors $(n+\b) \la_n z_n' = n \la_{n-1} z_{n-1}' + \b y_n$, donc $\la_n = \la_{n-1} + \frac{\b y_n}{(n+\b) z_n'}$ +#+end_proof # ID:8874 @@ -3989,7 +4508,11 @@ Pour $n \in \N$, on pose $u_n = |\{(p,q) \in \N^2 \mid p^2 + q^2 = n\}|$. Si on ne sait pas ça. Il nous reste à regarder des $5^m$ ou plutôt $5^{2m}$. - On a $5 = 4 + 1$ donc $5 = (2-i) (2+i)$, donc $5^{2m}$ a plein de façon de s'écrire comme un produit. Il faut montrer qu'elles donnent des décompositions distinctes… + Considérer les $z_k = (1+2i)^k (1-2i)^{m-k}$. Leur module au carré est $5^m$. + + Donc $5^m = a_k^2 + b_k^2$. Il reste à montrer que + $\frac{1+2i}{1-2i}$ n'est pas une racine $m$-ème de l'unité. Ou + alors prendre d'autres valeurs. #+END_proof @@ -4008,12 +4531,28 @@ Soit $(a_n)_{n\in\N}$ une suite réelle vérifiant, pour tout $n\in\N$, $a_{n+1} 1. Donner un développement asymptotique à deux termes de $x_n$ et $y_n$. #+end_exercice +# ID:9014 #+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 $n\geq 2$, $\sum_{k=p_n}^{p_{n+1}-1}\frac{1}{\ln k}\geq C$, et telle que la série de terme général $2^{-(p_{n+1}-p_n)}$ diverge. #+end_exercice -#+BEGIN_proof +#+begin_proof +On définit la suite $(p_n)_{n\geq 2}$ par $p_2 = 2$ et la relation de récurrence : +$$\forall n\geq 2, \quad p_{n+1} = p_n + \lceil \ln p_n \rceil$$ +Cette suite est strictement croissante puisque $\ln p_n \geq \ln 2 > 0$, donc $\lceil \ln p_n \rceil \geq 1$. -#+END_proof + + Pour tout $n\geq 2$, la fonction $x \mapsto 1/\ln x$ est décroissante, donc : + $$\sum_{k=p_n}^{p_{n+1}-1} \frac{1}{\ln k} \geq \frac{p_{n+1}-p_n}{\ln(p_{n+1}-1)} = \frac{\lceil \ln p_n \rceil}{\ln(p_n + \lceil \ln p_n \rceil - 1)}$$ + Or, $\lceil \ln p_n \rceil \geq \ln p_n$ et $p_{n+1}-1 < p_n + \ln p_n$. Ainsi : + $$\sum_{k=p_n}^{p_{n+1}-1} \frac{1}{\ln k} \geq \frac{\ln p_n}{\ln(p_n + \ln p_n)} = \frac{\ln p_n}{\ln p_n + \ln(1 + \frac{\ln p_n}{p_n})} \xrightarrow[n\to\infty]{} 1$$ + La suite de ces sommes étant strictement positive et de limite $1$, elle est minorée par une constante $C > 0$. + + Notons $u_n = 2^{-(p_{n+1}-p_n)} = 2^{-\lceil \ln p_n \rceil}$. On a $u_n \geq 2^{-(\ln p_n + 1)} = \frac{1}{2} e^{-\ln 2 \ln p_n} = \frac{1}{2 p_n^{\ln 2}}$. + Par comparaison série-intégrale ou étude classique des suites de type $x_{n+1} = x_n + f(x_n)$, on sait que $p_n \sim n \ln n$. + En particulier, pour $n$ assez grand, $p_n \leq n^2$ (ou toute puissance $n^\alpha$ avec $\alpha > 1$). Alors : + $u_n \geq \frac{1}{2 (n \ln n)^{\ln 2}}$ + Comme $\ln 2 < 1$, la série de terme général $(n \ln n)^{-\ln 2}$ est une série de Bertrand divergente. Par comparaison, $\sum u_n$ diverge. +#+end_proof # ID:8758 @@ -4025,44 +4564,67 @@ Minoration et majoration d'une série alternée, en regroupant les termes deux #+END_proof - +# ID:8892 #+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 $$(a_1, \dots, a_n, b_1, \dots, b_n) \in (\R^+)^{2n}, \sum_{i=1}^n a_i b_i \leq \left(\sum_{i=1}^n a_i^p\right)^{\frac{1}{p}} \left(\sum_{i=1}^n b_i^q\right)^{\frac{1}{q}}$$. #+end_exercice +# ID:6647 #+begin_exercice [X MP 2025 # 312] Soit $f\colon \R^{+*} \ra \R^{+*}$ de classe $\mc C^{\i}$ telle que $f(x) \ra 0$ quand $x \ra 0^+$ et quand $x \ra +\i$. On suppose que, pour tout $n \in \N^*$, il existe un unique $x_n \in \R^{+*}$ tel que $f^{(n)}(x_n) = 0$. 1. Montrer que la suite $(x_n)_{n\geq 1}$ est croissante. 1. Soit $n \in \N^*$. Montrer que $x^n f^{(n)}(x) \underset{x \ra 0^+}{\longrightarrow} 0$. - 1. On pose $g(x) = \frac{f(x)}{x}$ pour tout $x$ \gt 0. Montrer que, pour tout $n \ge 0$, il existe $a_{n,0},\ldots,a_{n,n}\in\Z$ tels que $g^{(n)}(x)=\sum_{k=0}^n a_{n,k}\frac{f^{(n-k)}(x)}{x^{k+1}}$ pour tout x\gt 0. + 1. On pose $g(x) = \frac{f(x)}{x}$ pour tout $x \gt 0$. Montrer que, + pour tout $n \ge 0$, il existe $a_{n,0},\ldots,a_{n,n}\in\Z$ tels + que $g^{(n)}(x)=\sum_{k=0}^n a_{n,k}\frac{f^{(n-k)}(x)}{x^{k+1}}$ pour tout $x\gt 0$. 1. Montrer que, pour tout $n \ge 0$, $(-1)^n g^{(n)}(x) \gt 0$ pour tout $x \gt 0$. #+end_exercice +# ID:8944 #+begin_exercice [X MP 2025 # 313] -Soit $f \in C^2(\R,\R)$. On suppose que : $f^2 \le 1$ et $(f')^2 + (f'')^2 \le 1$. Le but est de montrer par l'absurde que $g = f^2 + (f')^2 \le 1$. On suppose donc qu'il existe $t \in \R$ tel que : $f(t)^2 + f'(t)^2 \gt 1$. +Soit $f \in C^2(\R,\R)$. On suppose que : $f^2 \le 1$ et $(f')^2 + +(f'')^2 \le 1$. Le but est de montrer par l'absurde que $g = f^2 + +(f')^2 \le 1$. On suppose donc qu'il existe $t \in \R$ tel que : +$f(t)^2 + f'(t)^2 \gt 1$. -On pose : $E = \{x \in \R ; \forall y \in [\min(t, x), \max(t, x)], f(y)^2 + f'(y)^2 \gt 1\}$. +On pose : $E = \{x \in \R ; \forall y \in [\min(t, x), \max(t, x)], +f(y)^2 + f'(y)^2 \gt 1\}$. 1. Montrer que $E$ est un intervalle ouvert. - 1. Montrer que f' ne s'annule pas sur $E$. - 1. Conclure. + 2. Montrer que $g'$ ne s'annule pas sur $E$. + 3. Conclure. #+end_exercice +#+BEGIN_proof + 1. Clair. + 2. On a clairement que ni $f'$ ni $f$ ne s'annulent sur $E$. On a $g' = 2f'(f + f'')$. Sur $E$, $f^2 \gt (f'')^2$, soit $|f| \gt |f''|$, donc $g$ est du signe de $ff'$, qui ne s'annule pas. + 3. + + Si $ff' \gt 0$, $g$ est strictement croissante sur $E$, donc $E = (a, +\infty)$. Comme $f$ est monotone et bornée ($|f| \le 1$) et que $f''$ est bornée ($|f''| \le 1$), $f'(x) \xrightarrow[x \to +\infty]{} 0$. Alors $g(x) \xrightarrow[x \to +\infty]{} \ell^2 \le 1$, ce qui contredit la croissance de $g$ sachant $g(t) \gt 1$. +#+END_proof + +# ID:nil # Chiant. #+begin_exercice [X MP 2025 # 314] Si $(\phi_k)_{1 \le k \le 4}$ est une famille de fonctions de ]-1,1[ dans $\R$, on dit que $(\phi_k)_{1 \le k \le 4}$ vérifie (C) si $\phi_1 \lt \phi_2 \lt \phi_3 \lt \phi_4$ sur ]0, 1[ et $\phi_2 \lt \phi_4 \lt \phi_1 \lt \phi_3$ sur ]-1, 0[. 1. Montrer qu'il n'existe pas de famille $(\phi_k)_{1 \le k \le 4}$ de fonctions polynomiales vérifiant (C). Ind. On pourra étudier la valuation de $\phi_i - \phi_j$ pour $i \neq j$. 1. Existe-t-il une famille $(\phi_k)_{1 \le k \le 4}$ de fonctions de classe $C^{\i}$ vérifiant (C)? #+end_exercice +# ID:8893 #+begin_exercice [X MP 2025 # 315] -Soit $s: \R \ra \R$ telle que $(*): \forall x \in \R, s(x+1) = s(x) + \frac{1}{1+x^2}$ et $s(x) \xrightarrow[x \ra -\i]{} 0$. +Soit $s\colon \R \ra \R$ telle que $(*): \forall x \in \R, s(x+1) = s(x) + \frac{1}{1+x^2}$ et $s(x) \xrightarrow[x \ra -\i]{} 0$. 1. Montrer que, pour tout $x \in \R$, $s(x) \ge 0$. - 1. A-t-on existence et unicité de $s$ vérifiant ()? Déterminer les $s$ solutions. - 1. Que se passe-t-il si on remplace la condition $s(x) \xrightarrow[x \ra -\i]{} 0$ par la condition $s(x) \xrightarrow[x \ra +\i]{} 0$ ? + 1. A-t-on existence et unicité de $s$ vérifiant $(*)$ ? Déterminer les $s$ solutions. + 1. s Que se passe-t-il si on remplace la condition $s(x) \xrightarrow[x \ra -\i]{} 0$ par la condition $s(x) \xrightarrow[x \ra +\i]{} 0$ ? #+end_exercice +#+BEGIN_proof + 1. Si $s$ prenait une valeur $\lt 0$, on contredirait la limite en $-\i$. + 2. On peut écrire $s$ comme une série. Comme elle converge, existence et unicité. + 3. La même chose. +#+END_proof + #+begin_exercice [X MP 2025 # 316] :todo: 1. Soit $f \in \mc C^0(\R, \R)$. Montrer que $f$ est affine si et seulement si, pour tout réel $x$, on a $\frac{f(x+h) + f(x-h) - 2f(x)}{h^2} \underset{h \ra 0^+}{\longrightarrow} 0$. @@ -4074,36 +4636,50 @@ Soit $s: \R \ra \R$ telle que $(*): \forall x \in \R, s(x+1) = s(x) + \frac{1}{1 On suppose $f(0) = 0$. Alors $f(h) + f(-h) = o_0(h^2)$. + Si $f$ est $\mc C^2$, cela donne bien que $f''$ s'annule sur un + ensemble dense. + 2. Il suffit de prendre $f(x) = \a$ si $x\gt 0$ et $-\a$ sinon. #+END_proof +# ID:8945 #+begin_exercice [X MP 2025 # 317] -Soit $F: \R \ra \R^{+*}$. On suppose qu'il existe $\alpha, \eta \gt 0$ tels que : +Soit $F\colon \R \ra \R^{+*}$. On suppose qu'il existe $\alpha, \eta \gt 0$ tels que : -$$\forall (x,y) \in \R^2, \ \alpha F(x) F(y) \leq F(x+y) \leq \eta F(x) F(y)$$. +$$\forall (x,y) \in \R^2, \ \alpha F(x) F(y) \leq F(x+y) \leq \eta F(x) F(y).$$ - 1. On suppose que $F$ est de classe $\mc C^1$ et que $\frac{F'}{F}$ est bornée. Montrer qu'il existe $\gamma \in \R$ et $H: \R \ra \R^{+*}$ bornée tel que : $\forall x \in \R, \ F(x) = e^{\gamma x} H(x)$. + 1. On suppose que $F$ est de classe $\mc C^1$ et que $\frac{F'}{F}$ + est bornée. Montrer qu'il existe $\gamma \in \R$ et $H\colon \R \ra + \R^{+*}$ bornée tel que : $\forall x \in \R, \ F(x) = e^{\gamma x} + H(x)$. 1. On revient au cas général. Montrer qu'il existe une unique fonction $G: \R \ra \R^{+*}$ telle que $\frac{F}{G}$ soit bornée et $\forall (x,y) \in \R^2, \ G(x+y) = G(x) G(y)$. #+end_exercice +#+BEGIN_proof + 1. On a $\gamma = \lim \frac{\ln F(x)}{x}$, ce que l'on pose. Existence : la fonction $g = \ln F$ est de dérivée bornée et $g(n+m) = g(n) + g(m) + O(1)$. Alors $\Big|\frac{g(n)}{n} - \frac{g(kn)}{kn}\Big| \leq \frac{O(1)}{n}$, donc suite de Cauchy. + 2. +#+END_proof +# ID:8946 #+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 $(E)$ 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. +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 $\forall t \in \R, g(t) = 1 + \frac{g(qt)}{f(t)}$. On suppose $m\gt 2$ ou $M\lt 1/2$. Montrer qu'il existe une unique solution bornée de $(E)$, et qu'elle ne s'annule 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 - $[1,3[$. +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. ?? +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. + +On peut aussi le voir comme la recherche d'un point fixe de +$\Phi\colon g\mapsto 1 + \frac{g(qt)}{f}$ qui est contractante. + +Si $m\gt 2$, on peut détricoter la récurrence, écrire $g(t) = +\sum_{k=0}^{n-1}\dots + \frac{g(q^n t)}{\prod f}$, ce qui permet de +conclure. #+END_proof @@ -4132,21 +4708,24 @@ Soit $E = \R[X]$. Soit $\phi \in \mc{L}(E)$. #+END_proof +# ID:9024 #+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 $\zeta\colon s \in ]1, +\i[ \mapsto \sum_{n=1}^{+\i} \frac{1}{n^s}$. 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) + $|J(\sigma) - f(a)(g(b)- g(a))| \le AB \zeta(\alpha + \beta) (2(b-a))^{\alpha+\beta}$. 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 $[a,b]$, - $\max_k |x_{k+1} x_k| \lt \delta \Rightarrow |J(\sigma) - I_{a,b}(f,g)| \lt \eps$. + $\max_k |x_{k+1}- x_k| \lt \delta \Rightarrow |J(\sigma) + -I_{a,b}(f,g)| \lt \eps$. #+end_exercice #+BEGIN_proof - 1. !! + 1. Il faut faire une récurrence, puis prendre un $x_i$ tel que + $x_{i+1}-x_{i-1}\lt \frac{2(b-a)}{n-1}$. + 2. C'est du Cauchy, en passant par une sous-subdivision. #+END_proof @@ -4157,24 +4736,40 @@ On note $S$ l'ensemble des nombres complexes de module 1. Soit $\gamma:[0,1]\ra # 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. +Soit $f:[0,1]\ra\R$ continue. On pose $h:t\in[0,1]\mapsto \inf_{s\in[0,t]}f(s)$ et $g=f-2h$. - 1. Montrer que $g$ est continue, positive et que g(0) = 0. + 1. Montrer que $g$ est continue, positive et que $g(0) = 0$. 1. Montrer que si $f$ est affine par morceaux alors $q$ l'est aussi. - 1. On suppose que $f$ atteint son minimum en 1. On pose $q:t\in[0,1]\mapsto\inf_{s\in[t,1]}g(s)$. Montrer que $f$ = $g$ - 2q. + 1. On suppose que $f$ atteint son minimum en $1$. On pose $q:t\in[0,1]\mapsto\inf_{s\in[t,1]}g(s)$. Montrer que $f$ = $g - 2q$. #+END_exercice -#+begin_exercice [X MP 2025 # 323] +# ID:9018 +#+begin_exercice [X MP 2025 # 323] Théorème des nombres premiers Soit $\mc{P}$ l'ensemble des nombres premiers. On pose $\Psi(x) = \sum\limits_{\substack{p \in \mc{P}, \alpha \in \N^* \\ p^{\alpha} \leq x}} \ln p \text{ et } T(x) = \sum_{1 \leq n \leq x} \Psi\left(\frac{x}{n}\right)$. - 1. Montrer que $T(x) = \sum\limits_{1 \le n \le x} \ln(n) = x \ln(x) x + O(\ln x)$ quand $x \ra +\i$. - 1. Montrer que $T(x) 2T\left(\frac{x}{2}\right) = \sum_{n=0}^{\i} (-1)^{n-1} \Psi\left(\frac{x}{n}\right) = x \ln 2 + O(\ln x)$. + 1. Montrer que $T(x) = \sum\limits_{1 \le n \le x} \ln(n) = x \ln(x) - x + O(\ln x)$ quand $x \ra +\i$. + 1. Montrer que $T(x)- 2T\left(\frac{x}{2}\right) = \sum_{n=0}^{\i} (-1)^{n-1} \Psi\left(\frac{x}{n}\right) = x \ln 2 + O(\ln x)$. + 3. Montrer qu'il existe des constantes $A, B > 0$ telles que $A x \le \Psi(x) \le B x$ pour $x$ assez grand. + 4. En déduire que $\pi(x) = \Theta\left(\frac{x}{\ln x}\right)$, où $\pi(x)$ est le nombre de premiers inférieurs ou égaux à $x$. #+end_exercice #+BEGIN_proof - 1. On a $\ln (n) = \sum_{p\in\mc P,\,\a ; p^{\a}\mid n} \ln p$, donc $\sum_{1\leq n\leq x}\ln n = \sum_{p, \a} \ln p \times \lfloor \frac{x}{p^{\a}}\rfloor$ + 1. On peu prendre $x$ entier, puis faire une récurrence : - et $\sum_{1\leq n\leq x \Psi(\frac{x}{n})} =$. + On $x$ prend une nouvelle valeur $m$, on ajoute d'une part $\lm m$, d'autre part, à $T(x)$, seuls sont changés les $\Psi(\frac{x}{n})$, pour lesquels $n\mid m$, et chacun d'eux contribue $\ln p$ si $\frac{n}{m}$ est exactement une puissance de $p$. + + Pour le $= x\ln (x) - x + O(\ln x)$, c'est la comparaison intégrale. + 2. En partant de la définition de $T(x)$, on obtient $\sum_{n} \Psi(\frac{x}{n}) - 2 \sum_{n} \Psi(\frac{x}{2n})$. + + Le $x\ln 2$ vient du DL précédent. + 3. On a, comme $\Psi$ est croissante, $\Psi(x) - \Psi\left(\frac{x}{2}\right) \le \sum_{n=1}^{\infty} (-1)^{n-1} \Psi\left(\frac{x}{n}\right) \le \Psi(x)$ + + La majoration donne $\Psi(x)$ grand, et la minoration, en sommant + sur les $2^k$, donne $\Psi(x)$ petit. + 4. On a $\pi(x)\geq \frac{\Psi(x)}{\ln x}\geq A \frac{x}{\ln x}$ d'une part. + + Pour majorer $\pi(x)$, restreindre la somme au premiers entre + $\sqrt{x}$ et $x$. #+END_proof @@ -4221,38 +4816,49 @@ Déterminer les réels $\alpha$ et $\beta$ tels que $\int_0^{+\i} |\sin t|^{\alp 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 +# ID:9019 #+begin_exercice [X MP 2025 # 331] On rappelle que $\int_{\R^n} e^{-x^2/2} dx = \sqrt{2\pi}$. -Pour $n \in \N$, on pose $p_n : x \in \R \mapsto (-1)^n e^{x^2/2} \frac{d^n (e^{-x^2/2})}{d x^n}$. +Pour $n \in \N$, on pose $p_n\colon x \in \R \mapsto (-1)^n e^{x^2/2} \frac{d^n (e^{-x^2/2})}{d x^n}$. - a. Montrer que $p_n$ est polynomiale, préciser son degré et son coefficient dominant, et dé- - -montrer que $p_n$ est paire ou impaire. + 1. Montrer que $p_n$ est polynomiale, préciser son degré et son + coefficient dominant, et démontrer que $p_n$ est paire ou impaire. 1. Calculer $\int_{\R} p_m(x) p_n(x) e^{-x^2/2} dx$ pour $(m, n) \in \N^2$. 1. Soit $n \in \N^*$. Calculer l'intégrale multiple -$$I = \int_{\R} \cdots \int_{\R} \left( \prod_{1 \le i \le n} (x_j - x_i)^2 \right) \exp\left(-\frac{1}{2} \sum_{i=1}^n x_k^2\right) dx_1 \cdots dx_n$$. + $$I = \int_{\R} \cdots \int_{\R} \left( \prod_{1 \le i \lt j \le n} (x_j - x_i)^2 \right) \exp\left(-\frac{1}{2} \sum_{i=1}^n x_k^2\right) \dx_1 \cdots \dx_n$$. Ind. On pourra s'intéresser au déterminant de la matrice $(p_{i-1}(x_j))_{1 \leq i,j \leq n}$. #+end_exercice +#+BEGIN_proof + 1. + 2. ipp, donne $\delta_{m,n} n! \sqrt{2\pi}$ + 3. Le facteur est le carré du déterminant de $\big(x_j^{i-1}\big) = \det (p_{i-1}(x_j))$, ce qui donne en fait le déterminant de $\int p_i p_j e^{-x^2/2}$, c'est-à-dire d'une matrice diagonale. +#+END_proof + +# ID:nil # Sans intérêt #+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 $\phi_p(x_1, \ldots, x_p) = \det((K_N(x_i, x_j))_{1 \leq i, j \leq p})$. -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 $(a_n)_n \in (\N^*)^{\N}$ 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 $\sum_{k=1}^{+\i} \frac{1}{a_n} \le 50$. +Calculer $\int_{\R^n} \phi_p(x_1, \dots, x_p) dx_1 \dots dx_p$. #+end_exercice #+BEGIN_proof - 1. - 2. !! +Pour $p = 1$, on a $\int_{\R} \sum f_k(x)^2$. + +Pour $p = 2$, on a $\begin{pmatrix}\sum f_k(x)^2 & \sum f_k(x) f_k(y) \\ \sum f_k(x) f_k(y) & \sum f_k(y)^2\end{pmatrix}$. Donc le déterminant est $\sum_{k_1,k_2,k_3,k_4} f_{k_1}^2(y) f_{k_2}^2(x) - f_{k_1}(x) f_{k_1}(y) f_{k_2}(x)f_{k_2}(y)$. +Quand tu intègres,… +#+END_proof + + +#+begin_exercice [X MP 2025 # 333] :todo: + 2. 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 $\sum_{n=0}^{+\i} \frac{1}{a_n} \leq 2$. + 3. 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 + 2. 3. #+END_proof @@ -4268,23 +4874,28 @@ On prend $x = \pi$, on obtient $a_n\ra 0$. #+END_proof +# ID:8953 #+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 $f_n\colon x \in \R \setminus \Z \mapsto \pi \cot(\pi x) - \sum_{k=-n}^n \frac{1}{x+k}$. - 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 : + 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 $\R$. + 2. Montrer que la fonction prolongée par continuité est de classe + $\mc C^1$ sur $\R$ et vérifie : -$$\forall x \in \R, 4f'(x) = f'\left(\frac{x}{2}\right) + f'\left(\frac{x+1}{2}\right)$$. - - 1. En déduire que $f$ est identiquement nulle sur $\R$. - 1. On pose $g\colon x\mapsto \frac{x}{e^x-1}$. Justifier que $g$ est développable en série entière au voisinage de 0 et que le développement en série entière de $x\mapsto g(x)-1+\frac{x}{2}$ ne contient que des termes - -pairs. On note -$$g(x) = 1 - \frac{x}{2} + \sum_{n=1}^{+\i} a_n x^{2n}$$ -. - - 1. Pour $n \in \N^*$, donner une expression de $\zeta(2n)$ en fonction de $a_n$. Ind. On pourra considérer g(ix) pour $x \in \R$. + $$\forall x \in \R,\, 4f'(x) = f'\left(\frac{x}{2}\right) + f'\left(\frac{x+1}{2}\right)$$. + 3. En déduire que $f$ est identiquement nulle sur $\R$. + 4. On pose $g\colon x\mapsto \frac{x}{e^x-1}$. Justifier que $g$ est développable en série entière au voisinage de 0 et que le développement en série entière de $x\mapsto g(x)-1+\frac{x}{2}$ ne contient que des termes pairs. On note + $$g(x) = 1 - \frac{x}{2} + \sum_{n=1}^{+\i} a_n x^{2n}.$$ + 5. Pour $n \in \N^*$, donner une expression de $\zeta(2n)$ en fonction de $a_n$. Ind. On pourra considérer $g(ix)$ pour $x \in \R$. #+end_exercice +#+BEGIN_proof + 1. Regrouper $-k$ et $k$. + 2. Dériver… + 3. $f'$ est bornée sur $[0,1]$, puis en itérant, on trouve $f'(x) = 0$ ailleurs. +#+END_proof + # ID:8873 #+begin_exercice [X MP 2025 # 336] @@ -4327,12 +4938,44 @@ Soit $\alpha \in \R$. #+END_proof +# ID:8947 #+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. - 1. Soit $s \in \N^*$ et $P \in \C[X]$. On suppose que, pour tout entier $n$ assez grand, P(n) est la puissance $s$-ième d'un entier. Soient $\tau_1,\ldots,\tau_s$ dans $\Z$. Montrer qu'il existe une fonction gvérifiant les hypothèses de la question précédente (pour un certain d) et telle que, pour tout complexe $z$ de module assez grand, $\prod P(z + \tau_k) = g(z)^s$. En déduire qu'il existe un polynôme $Q \in \C[X]$ tel que $P = Q^s$ et $\forall k \in \Z, \ Q(k) \in \Z$. + 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 $k\lt 0$. + 1. Soit $s \in \N^*$ et $P \in \C[X]$. On suppose que, pour tout entier $n$ + assez grand, $P(n)$ est la puissance $s$-ième d'un entier. Soient + $\tau_1,\ldots,\tau_s$ dans $\Z$. Montrer qu'il existe une + fonction $g$ vérifiant les hypothèses de la question précédente + (pour un certain $d$) et telle que, pour tout complexe $z$ de + module assez grand, $\prod P(z + \tau_k) = g(z)^s$. En déduire + qu'il existe un polynôme $Q \in \C[X]$ tel que $P = Q^s$ et + $\forall k \in \Z, \ Q(k) \in \Z$. #+END_exercice +#+BEGIN_proof + 1. Si $c_1,\dots, c_d$ sont nuls, prendre $a$ très grand, on obtient + $c_0\in\N$ : limite d'entiers. + + Quitte à multiplier $g$ par un entier, on peut supposer que + $c_1,\dots, c_d$ sont entiers. + 2. $g$ est la racine $s$-ième du produit. Non seulement elle + fonctionne, mais on sait qu'elle converge. + + Alors $g$ est un polynôme, à coefficients dans $\Q$ dont les + valeurs sont, en une infinité de points, des rationnels. + + Et si on prend des entiers en $n$ points d'affilée, on est entier + tout le temps +#+END_proof + +# ID:nil # Sans intérêt. #+begin_exercice [X MP 2025 # 340] Soient $\theta \gt 1$ et $P \in \Z[X]$ unitaire de degré $n \in \N^*$ dont $\theta$ est racine de multiplicité 1 et dont les autres racines complexes sont de module \lt 1 et dont $1/\theta$ n'est pas racine. Soit $Q = X^n P(1/X)$. @@ -4343,24 +4986,39 @@ $1/\theta$. On note $f(z) = \sum_{n=0}^{+\i} b_n z^n$ ce développement. 1. On suppose que P(0) \gt 0. Montrer que $(b_n)_{n \in \N}$ est croissante. #+end_exercice + + +# ID:nil # Sans intérêt #+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. On pose $u_0 = 1$ et $u_{n+1} = \sum_{k=0}^n u_k u_{n-k}$ pour + tout $n \in \N$. Calculer $u_n$. + 2. 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 - - + - C'est Catalan. Via une série génératrice. + - Il s'agit de calculer $I_0$. La relation de récurrence vient d'une IPP. #+END_proof - +# ID:nil #+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 $v_k = m \int_0^1 \frac{(1+x)^{2m-k}}{x} \left( (1+x)^k - (1-x)^k \right) dx$. - 1. Montrer que, pour tout $k \in [1, m]$, $v_k = u_k$. - 1. Donner un équivalent de $W_m = m \int_0^1 \frac{(1+x)^m}{x} ((1+x)^m (1-x)^m) dx$. +Soit $m \in \N^*$. On pose $u_0 = 4^m$, $u_1 = 4^m - 1$ et, pour $k \in \db{1,m}$, $u_k = -1 + \frac{2m - k}{2m} u_{k+1} + \frac{k}{2m} u_{k-1}$ et $v_k = m \int_0^1 \frac{(1+x)^{2m-k}}{x} \left( (1+x)^k - (1-x)^k \right) \dx$. + 1. Montrer que, pour tout $k \in \db{1,m}$, $v_k = u_k$. + 1. Donner un équivalent de $W_m = m \int_0^1 \frac{(1+x)^m}{x} ((1+x)^m- (1-x)^m) dx$. #+end_exercice +#+BEGIN_proof + 1. + 2. On est ramené à $v_k = \frac{2m-k}{2m}v_{k+1} + \frac{k}{2m}v_{k-1}$, donc $v_{k+1} = \frac{2m v_k - k v_{k-1}}{2m-k}$. + Considérons $u_k = v_{k+1} - v_k$, on obtient $u_{k+1} = \frac{k+1}{2m-k-1}u_k$, donc on trouve $u_k = \frac{k! (2m-k)!}{(2m)!}$, et on veut un équivalent de la somme des inverses des coefficients binomiaux. Ce qui est trivial. +#+END_proof + +# ID:8948 #+BEGIN_exercice [X MP 2025 # 343] Déterminer un équivalent de $\int_0^{+\i} (te^{-t})^x dt$ quand $x$ tend vers $+\i$. #+END_exercice +#+BEGIN_proof +Chercher le maximum de la fonction. +#+END_proof # ID:329 @@ -4409,30 +5067,38 @@ Un tiroir contient 2n chaussettes, constituant $n$ paires. On tire successivemen Indication : Pour simplifier le résultat, on pourra utiliser un raisonnement probabiliste pour établir que $\quad\displaystyle\sum_{k=n}^{2n} \binom{k}{n} 2^{-k} = 1$. #+end_exercice +# ID:8952 #+begin_exercice [X MP 2025 # 348] -On organise un tournoi avec une infinité $(J_n)_{n\in\N}$ de joueurs. Les modalités sont les suivantes : $J_0$ et $J_1$ s'affrontent, le gagnant affronte $J_2$ et ainsi de suite : le gagnant de chaque partie affronte le joueur suivant lors de la partie suivante. On considère tous les matchs comme indépendants et on note $p_n = \mathbf{P}(J_n \text{ remporte son premier match})$. Le tournoi s'arrête lorsqu'un joueur remporte deux matchs successifs. On note $T$ la variable aléatoire donnant le nombre de matchs joués jusqu'à l'arrêt du tournoi. Pour les deux premières questions, on fixe - -$$\alpha \in ]0,1[$$ - et on suppose que : $\forall n \geq 2, \ p_n = 1 - \frac{1}{n^{\alpha}}$. +On organise un tournoi avec une infinité $(J_n)_{n\in\N}$ de joueurs. Les modalités sont les suivantes : $J_0$ et $J_1$ s'affrontent, le gagnant affronte $J_2$ et ainsi de suite : le gagnant de chaque partie affronte le joueur suivant lors de la partie suivante. On considère tous les matchs comme indépendants et on note $p_n = \mathbf{P}(J_n \text{ remporte son premier match})$. Le tournoi s'arrête lorsqu'un joueur remporte deux matchs successifs. On note $T$ la variable aléatoire donnant le nombre de matchs joués jusqu'à l'arrêt du tournoi. Pour les deux premières questions, on fixe $$\alpha \in ]0,1[$$ et on suppose que : $\forall n \geq 2, \ p_n = 1 - \frac{1}{n^{\alpha}}$. 1. Montrer que $T$ est presque sûrement finie. 1. Montrer que $T$ est d'espérance finie. - 1. Dans cette question, on fixe $N \geq 2$ et la condition de victoire devient : un joueur remporte le tournoi quand il a gagné $N$ matchs consécutifs. Ainsi le cas précédent correspond au cas N=2. On suppose que, pour tout $n \in \N^*$, $p_n=p \in ]0,1[$. + 1. Dans cette question, on fixe $N \geq 2$ et la condition de victoire devient : un joueur remporte le tournoi quand il a gagné $N$ matchs consécutifs. Ainsi le cas précédent correspond au cas $N=2$. On suppose que, pour tout $n \in \N^*$, $p_n=p \in ]0,1[$. -On note $a_n = \mathbf{P}(T \geq n)$ avec, pour $k \leq N$, $a_k = 1$. Déterminer une relation de récurrence entre les $a_n$. + On note $a_n = \mathbf{P}(T \geq n)$ avec, pour $k \leq N$, $a_k = 1$. Déterminer une relation de récurrence entre les $a_n$. #+end_exercice +#+BEGIN_proof + 1. Simple. + 2. Simple, via $\ln (1+u) = u + O(u^2)$. + 3. Simple. +#+END_proof + +# ID:6755 #+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 $C_k = |\{\sigma \in \mc{S}_n, |\sigma| = k\}|$. - +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 $1$). 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 $|\sigma_n|$. - 1. Montrer que $\frac{|\sigma_n|}{\ln(n)}$ tend vers 1 en probabilités quand $n \ra +\i$. + 1. Montrer que $\frac{|\sigma_n|}{\ln(n)}$ tend vers $1$ en + probabilités quand $n \ra +\i$. #+end_exercice +# ID:nil #+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 @@ -4449,53 +5115,79 @@ Soit $n \in \N^*$. Pour $\sigma \in \mc{S}_n$, on note $|\sigma|$ le nombre de c partitions de $A$ comme la somme d'une série numérique. #+end_exercice +# ID:9023 #+begin_exercice [X MP 2025 # 351] -Soient $p \in ]0,1[$ et t\gt 0. Soient $(X_n)_{n\in\N}$ une suite de variables aléatoires $i$.i.d. vérifiant $\mathbf{P}(X_n=1)=p$ et $\mathbf{P}(X_n=-1)=1-p$ et $N\sim\mc{P}(t)$ indépendante des $X_n$. On pose : - -$$S_n = \sum_{i=0}^n X_i$$. +Soient $p \in ]0,1[$ et $t\gt 0$. Soient $(X_n)_{n\in\N}$ une suite de +variables aléatoires $i$.i.d. vérifiant $\mathbf{P}(X_n=1)=p$ et +$\mathbf{P}(X_n=-1)=1-p$ et $N\sim\mc{P}(t)$ indépendante des $X_n$. +On pose : $S_n = \sum_{i=0}^n X_i$. 1. Pour $n \in \Z$, calculer $\mathbf{P}(S_N = n)$. 1. Montrer que : - $$\forall (x,y) \in (\R^{+*})^2, \quad \sum_{n \in \Z} y^n \sum_{\substack{i \in \N \\ n \geq 0}} \frac{x^{n+2i}}{n!(n+i)!} = e^{xy+1/y}$$. + $$\forall (x,y) \in (\R^{+*})^2, \quad \sum_{n \in \Z} y^n \sum_{\substack{i \in \N \\ n+i \geq 0}} \frac{x^{n+2i}}{i!(n+i)!} = e^{x(y+1/y)}$$. #+end_exercice +# ID:nil # Cf année précédente. #+begin_exercice [X MP 2025 # 352] Soient $p \in [0, 1[, m \ge 2 \text{ et } \xi = e^{2i\pi/m}]$. 1. Montrer que : $$\forall a,b \in \C, \quad \sum_{k \in \db{0,n }} \binom{n}{k} a^k b^{n-k} = \frac{1}{m} \sum_{j=0}^{m-1} (b + \xi^j a)^n$$. 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 $p$. + 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 $t \gt 0$, $\mathbf{P}(X \mathbf{E}(X) \leq -t) \leq \exp\left(-\frac{t^2}{\mathbf{E}(X^2)}\right)$. +#+begin_exercice [X MP 2025 # 353] :todo: +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_proof :todo: +Pour tout $\lambda > 0$, par Markov exponentielle : +$$\P(X - \mu \leq -t) = \P(e^{-\lambda X} \geq e^{-\lambda(\mu-t)}) \leq e^{\lambda(\mu-t)} \E(e^{-\lambda X}).$$ -#+BEGIN_exercice [X MP 2025 # 354] -Soit $(X_n)_{n\geq 1}$ une suite de variables aléatoires $i$.i.d. à valeurs dans $\N^*$. On suppose +#+END_proof -de plus que $\mathbf{E}(X_1^2) \lt +\i$, et on pose $S_n = \sum_{i=1}^n X_i$ et $T_n = \sum_{i=1}^n \frac{1}{S_i}$ pour $n \geq 1$. + +#+BEGIN_exercice [X MP 2025 # 354] :todo: +Soit $(X_n)_{n\geq 1}$ une suite de variables aléatoires i.i.d. à valeurs dans +$\N^*$. On suppose de plus que $\mathbf{E}(X_1^2) \lt +\i$, et on pose +$S_n = \sum_{i=1}^n X_i$ et $T_n = \sum_{i=1}^n \frac{1}{S_i}$ pour $n +\geq 1$. 1. Montrer que, pour tout $\omega$, $(T_n(\omega))_{n\geq 1}$ a une limite dans $[0,+\i]$. 1. Montrer qu'il existe une constante $C\gt 0$ et une suite strictement croissante $(n_k)_{k \ge 1}$ d'entiers $\geq 1$ vérifiant $n_{k+1} \geq 2n_k$ et $\mathbf{P}(S_{n_k} \geq 2n_k \mathbf{E}(X_1)) \leq \frac{C}{2^k}$ pour tout $k \geq 1$. 1. En déduire que $(T_n)_{n\geq 1}$ tend presque sûrement vers $+\i$. 1. Montrer que $\mathbf{V}(T_n) \leq \sum_{i=1}^n \mathbf{E}\left(\frac{1}{S_i^2}\right)$ pour tout $n \geq 1$. #+END_exercice +#+BEGIN_proof :todo: + 1. Trivial. + 2. On a $\P(S_{n_k}\geq 2 n_k \E(X_1))\leq \frac{n_k \V(X_1)}{n_k^2}$, donc pas dur… + 3. BC, presque sûrement le précédent n'arrive qu'un nombre fini de + fois. Les autres fois, $\frac{1}{S_i}\geq \frac{1}{n_i}$, ce qui + n'est pas suffisant, mais en fait si, parce que les $S_n$ sont + croissantes. + 4. ?? +#+END_proof +# ID:8955 #+begin_exercice [X MP 2025 # 355] On pose $(X)_0 = 1$ et, pour $n \in \N^*$, $(X)_n = X(X-1) \cdots (X-n+1)$. 1. Montrer que $((X)_n)_{n\geq 0}$ est une base de $\R[X]$. - 1. Pour $k \in \N$, on décompose $X^k = \sum_{n=0}^{+\i} a_{k,n}(X)_n$. Déterminer $a_{k,0}$ et $a_{k,n}$ pour $n \geq k$. - 1. En considérant une variable aléatoire $Z$ suivant la loi de Poisson de paramètre 1, montrer que $\forall k \in \N, \sum_{i=0}^{+\i} a_{k,n} = \frac{1}{e} \sum_{i=0}^{+\i} \frac{i^k}{i!}$. - 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. + 2. Pour $k \in \N$, on décompose $X^k = \sum_{n=0}^{+\i} a_{k,n}(X)_n$. Déterminer $a_{k,0}$ et $a_{k,n}$ pour $n \geq k$. + 3. En considérant une variable aléatoire $Z$ suivant la loi de Poisson + de paramètre 1, montrer que $\forall k \in \N, \sum_{n=0}^{+\i} a_{k,n} = \frac{1}{e} \sum_{i=0}^{+\i} \frac{i^k}{i!}$. + 4. Pour $0 \le n \le k$, on note $b_{k,n}$ le nombre de façons de + ranger $k$ objets différenciés dans $n$ tiroirs indifférenciés, + aucun des tiroirs n'étant vide. Montrer que $b_{k,n} = a_{k,n}$. + 5. Soit $k \in \N$. Déterminer le nombre de façons de partitionner un ensemble à $k$ éléments. #+end_exercice #+BEGIN_proof - 1. + 1. $a_{k,0} = 0$ sauf $k = 0$. 2. - 3. - 4. + 3. On a $\E(Z(Z-1)\dots (Z-n+1)) = \la^n$, donc le terme de gauche + est l'espérance, et il reste à calculer $\E(Z^k)$. + 4. En évaluant la question 2 en les $k$, on obtient une relation de + récurrence, qui est clairement la même que pour $b_{k,n}$. + 5. C'est la somme des précédents. #+END_proof # ID:8771 @@ -4543,19 +5235,28 @@ On cherche à prouver l'existence d'un réel $C \gt 0$ tel que, pour toutes var #+END_proof -# À relier +# ID:8951 #+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 $x \in [0,1], \phi_a(x) = \mathbf{E}(x^{X_a})$. - 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 ce n'est pas le cas pour a = 1/3. +Soient $a \in ]0,1[$ et $\phi_a\colon x \mapsto 1- (1-x)^a$. + 1. Montrer qu'il existe une variable aléatoire $X_a$ à valeurs dans + $\N^*$ telle que, pour tout $x \in [0,1], \phi_a(x) = + \mathbf{E}(x^{X_a})$. + 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 $Y \sim X_a$. #+end_exercice +# ID:nil # Chiant. +#+begin_exercice [X MP 2025 # 358] +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 ce n'est pas le cas pour $a = 1/3$. +#+end_exercice + +# ID:nil # Chiant, see 8950 #+begin_exercice [X MP 2025 # 359] Soit $(X_n)$ une suite de variables aléatoires indépendantes telles que $\mathbf{P}(X_n=0)=1-\frac{1}{n}$ et $\mathbf{P}(X_n=n)=\frac{1}{n}$. On pose, pour $n\in\N^*$, $S_n=X_1+\cdots+X_n$. 1. Soit $\lambda \in \R^+$. Déterminer la limite de $\left(\mathbf{E}\left(e^{-\lambda\frac{S_n}{n}}\right)\right)_{n\geq 1}$. @@ -4564,27 +5265,28 @@ Montrer qu'il existe une unique fonction $f$ qui respecte ces conditions, qu'ell 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. + 1. L'écrire, et utiliser $\ln (1+x) = x + O(x^2)$ pour se ramener à Riemann. #+END_proof + +# ID:8949 #+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. + 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 ? - - Sinon, à partir de $0$ il a une probabilité $\a$ de ne jamais + 3. 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 $\a + \a \P(S_2 =0) \a$. + La probabilité de ne jamais revenir après l'instant $2$ est $\geq \a + \a \P(S_2 = 0)$. - 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 $\lt \frac{1}{2}$. !! + La probabilité de ne jamais revenir après l'instant $4$ est $\geq \a + \a \P(S_2 = 0) + \a \P(S_4 = 0)$, etc. #+END_proof @@ -5342,6 +6044,7 @@ Montrer le caractère $C^{\i}$ sur $\R^2$ de la fonction définie par $$\forall x \neq y, f(x, y)=\frac{e^x-e^y}{x-y} \text { et } \forall x, f(x, x)=e^x$$ #+END_exercice +# ID:nil #+BEGIN_exercice [X 2025] Soit $\alpha\gt 0$. On définit $$z_0=1 \text { et } \forall n \in \N, z_{n+1}=\frac{\alpha n+1}{\alpha(n+1)} z_n$$ @@ -5356,9 +6059,10 @@ $$z_0=1 \text { et } \forall n \in \N, z_{n+1}=\frac{\alpha n+1}{\alpha(n+1)} z_ To check. #+END_proof +# ID:nil #+BEGIN_exercice [ENS 2025, MPI] -Soit $E$ un espace préhilbertien de dimension infinie. Soit $K$ une partie de $E$ non vide, bornée et dont la frontière est compacte. -Montrer que $K$ est d'intérieur vide. +Soit $E$ un espace préhilbertien de dimension infinie. Soit $K$ une partie de $E$ non vide, bornée et dont la frontière est compacte. Montrer que $K$ est d'intérieur vide. + Question supplémentaire : et si on remplace l'hypothèse "préhilbertien" par "normé" ? #+END_exercice @@ -5367,6 +6071,7 @@ Dans $\mc{S}_n^{++}(\R)$, on définit la relation d'ordre strict $\gt : A\gt B \ Montrer que l'application $A \mapsto A^{-1}$ est décroissante sur $\mc{S}_n^{++}(\R)$. #+END_exercice +# ID:nil #+BEGIN_exercice [X 2025] Soit $d \in \N^*$. On note $\quad\displaystyle f\colon z \mapsto \sum_{k \leq d, k \in \Z} c_k z^k$, et on suppose que $f$ est définie sur le complémentaire d'un disque centré en 0. On suppose également que $c_1, \ldots, c_d \in \Q$ et qu'il existe une infinité de $z \in \Z$ tels que $f(z) \in \Z$. @@ -5384,6 +6089,7 @@ Via un polynôme $P_k$ tel que $\forall x \in \R, \quad P_k(2 \cos (x))=2 \cos ( #+END_proof +# ID:nil #+BEGIN_exercice [ENS SR 2025] Soit $n \in \N^*$. On note $E=\left\{A \in S_n^+(\R), \op{rg}(A)=1\right\}$. 1. Montrer que $A \in E \Longleftrightarrow \exists U \in \R^n \setminus\{0\}, A=U U^T$. @@ -5400,12 +6106,14 @@ z(c)= \pm e_j #+END_exercice +# ID:nil #+BEGIN_exercice On note $A=\left\{\left(a_n\right) \in \R^{\N}, \forall n \in \N, n a_{n+1}=(n+1) a_n\right\}$. 1. Etudier $A$. 2. Trouver les solutions sur $\interval]{-1, 1}[$ de $\quad\displaystyle (H): x(x-1) y''+3 x y'+y=0$ #+END_exercice +# ID:nil #+BEGIN_exercice [ENS 2025] Soient $\alpha \in \N^*$, ppremier impair. On dit qu'une partie $D \subset \Z / p^\alpha \Z$ est $f$ génératrice pour $f\colon \Z / p^\alpha \Z \times$ $\Z / p^\alpha \Z \ra \Z / p^\alpha \Z$ si @@ -5417,6 +6125,7 @@ $$\forall y \in \Z / p^\alpha \Z, \exists n \geq 2, \exists d_1, \ldots, d_n \in 3. Avec $f:(x, y) \mapsto x y$, montrer qu'il existe au moins une partie $f$-génératrice de cardinal 2. On admettra que le groupe des inversibles de $\Z / p^\alpha \Z$ est cyclique. #+END_exercice +# ID:nil #+BEGIN_exercice [ENS25, SR] Pour $f, g \in E=C^0([-1,1], \R)$, on note $(f \mid g)=\int_{-1}^1 f g$. @@ -5429,23 +6138,25 @@ Pour tout entier naturel $n$, on pose $\quad\displaystyle L_n=\left(X^2-1\right) 5. Etudier la rationnalité des racines de $P_n$ #+END_exercice +# ID:nil #+BEGIN_exercice [X 2025] -Soit $\sum\left(a_n z^n\right)$ une série entière de rayon $R\gt 0$ et de somme $f$. Soeint $p \in \N^*$ et $M \in \M_p(\C)$ telle que $\forall \lambda \in \op{Sp}(M),|\lambda|\lt R$. +Soit $\sum\left(a_n z^n\right)$ une série entière de rayon $R\gt 0$ et de somme $f$. Soient $p \in \N^*$ et $M \in \M_p(\C)$ telle que $\forall \lambda \in \op{Sp}(M),|\lambda|\lt R$. 1. Montrer que $\sum\left(a_n M^n\right)$ converge. 2. Peut-on trouver une suite $\left(a_n\right)$ telle que le résultat soit vrai pour toute matrice $M \in \M_p(\C)$ telle que $\forall \lambda \in \op{Sp}(M),|\lambda| \leq R$. #+END_exercice - +# ID:nil #+BEGIN_exercice [X 2025] On note $B_n(\R)$ l'ensemble des matrices $M \in O_n(\R)$ telles que $\op{det}(M)=1$ et $-1 \notin \op{Sp}(M)$. On note -$$T: M \mapsto\left(I_n-M\right)\left(I_n+M\right)^{-1}$$ +$T\colon M \mapsto\left(I_n-M\right)\left(I_n+M\right)^{-1}$ 1. Montrer que $T$ est bien définie sur $A_n(\R)$ et que $T\left(A_n(\R)\right) \subset B_n(\R)$. 2. Montrer que $T\left(B_n(\R)\right) \subset A_n(\R)$. 3. On prend $n=2$. Soit $M=\left(\begin{array}{cc}0 & -\tan (\theta) \\ \tan (\theta) & 0\end{array}\right)$ avec $|\theta|\lt \frac{\pi}{2}$. Que dire de $T(M)$ et $T^2(M)$ ? #+END_exercice +# ID:nil #+BEGIN_exercice [ENS 2025] On dit qu'une matrice est de Bordaud si ses coefficients diagonaux sont exactement ses valeurs propres comptées avec multiplicité. 1. Montrer que $A$ est semblable à une matrice de Bordaud si et @@ -5472,23 +6183,9 @@ On dit qu'une matrice est de Bordaud si ses coefficients diagonaux sont exacteme Trouver deux dés non biaisés tels que la probabilité de la somme soit la même que pour deux dés usuels. Les valeurs des faces sont des entiers naturels, pas forcément distincts et les dés peuvent être différents. #+END_exercice -#+BEGIN_exercice -Ci-dessous, version alternative du même exercice -Soient $f, g\colon \R \ra \R$ deux fonctions telles que: - + Il existe $\alpha, \beta\gt 0$ tels que $\alpha+\beta\gt 1$; - + Il existe $A, B\gt 0$ tels que : $\quad\displaystyle\forall x, y \in \R, \quad |f(x)-f(y)| \leq A|x-y|^\alpha, \quad |g(x)-g(y)| \leq B|x-y|^\beta$. - -Soit $S=\left\{x_0, x_1, \ldots, x_n\right\}$ avec $x_0\lt x_1\lt \cdots\lt x_n, a=x_0, b=x_n$. On définit : - -$$J_S(f, g):=\sum_{i=0}^{n-1} f\left(x_i\right)\left(g\left(x_{i+1}\right)-g\left(x_i\right)\right)$$ - - 1. Montrer que : $\quad\displaystyle \left|J_S(f, g)-f(a)(g(b)-g(a))\right| \leq A B|2(b-a)|^{\alpha+\beta} \zeta(\alpha+\beta)$, où $\zeta$ désigne la fonction zêta de Riemann. - 2. Montrer qu'il existe une unique valeur $I \in \R$ telle que : - - $$\forall \eps\gt 0, \exists \delta\gt 0, \text { si } \max_{0 \leq i\lt n}\left|x_{i+1}-x_i\right|\lt \delta \Rightarrow\left|J_S(f, g)-I\right|\lt \eps$$ -#+END_exercice +# ID:nil #+BEGIN_exercice Soient $f$ et $g$ deux fonctions définies sur $\R$ vérifiant : @@ -5560,10 +6257,11 @@ c) Ici, $p \geq5$. Montrer que pour $k \in [1, p-1]$, il existe un unique $k^* \ $$k \cdot k^* \equiv 1 \ [p^2].$$ Montrer que $\sum_{k=1}^{p-1} k^* \equiv 0 \ [p^2]$. #+end_exercice +# ID:8992 #+begin_exercice [Mines MP 2025 # 462] On note $\tau(n)$ (resp. $\sigma(n)$ ) le nombre de diviseurs positifs de $n$ (resp. la somme des diviseurs positifs de n). -a) Montrer que si $n \wedge m = 1$ alors $\sigma(nm) = \sigma(n) \sigma(m)$. -b) Montrer que si $\sigma(n)$ est premier alors $\tau(n)$ l'est aussi. + 1. Montrer que si $n \wedge m = 1$ alors $\sigma(nm) = \sigma(n) \sigma(m)$. + 2. Montrer que si $\sigma(n)$ est premier alors $\tau(n)$ l'est aussi. #+end_exercice #+begin_exercice [Mines MP 2025 # 463] @@ -5592,9 +6290,9 @@ Soient $p$ un nombre premier, et $G$ un groupe fini d'ordre 2p. Montrer que $G$ #+begin_exercice [Mines MP 2025 # 467] On note $S_n$ le groupe des permutations de $\{1, \ldots, n\}$ et $M_n$ le maximum des ordres des éléments de $S_n$. -a) Montrer que $n \leq M_n \leq n!$. -b) Si $n \geq 4$, montrer que $M_n \leqslant \max_{2 \leqslant k \leqslant n-1} (k M_{n-k})$. -c) Montrer que $M_n = O(3^{n/3})$. + 1. Montrer que $n \leq M_n \leq n!$. + 2. Si $n \geq 4$, montrer que $M_n \leqslant \max_{2 \leqslant k \leqslant n-1} (k M_{n-k})$. + 3. Montrer que $M_n = O(3^{n/3})$. #+end_exercice #+begin_exercice [Mines MP 2025 # 468] @@ -5706,8 +6404,9 @@ $$\operatorname{Conv}(I) = \left\{ \sum_{k=0}^{n} \lambda_k x_k \; ; \; (\lambda Pour $P \in \C[X]$, on note $\mathcal{Z}(P)$ l'ensemble de ses racines complexes. Soit $P \in \C[X]$ non constant. -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$. + 1. Écrire la décomposition en éléments simples de $\frac{P'}{D}$. + 2. Montrer que $Conv(\mathcal{Z}(P')) \subset Conv(\mathcal{Z}(P))$. + 3. 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 @@ -5729,9 +6428,18 @@ On a clairement $\dim_{\Q} G\leq 5$ et $G\neq F$, et $\dim F\geq 2$, donc néces #+END_proof +# ID:8985 #+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$. +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 : + + $\Lambda(Ax) = \Lambda(x)$ pour tout $x \in \mathbb{R}^n$ ; + + $\det A = \pm 1$. #+end_exercice +#+BEGIN_proof +De droite à gauche : $A^{-1}$ est à coefficients entiers. + +De gauche à droite : réduire modulo $p$, pour $p$ divisant $\det A$. +#+END_proof + #+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}$. @@ -5771,8 +6479,8 @@ b) Calculer rg $M$ en fonction de $z$. #+begin_exercice [Mines MP 2025 # 492] Soit $(a_1,\ldots,a_n)\in\mathbb{K}^n$. On pose $a_{n+1}=a_1$. Soit $M=(m_{i,j})_{1\leqslant i,j\leqslant n}\in\mathcal{M}_n(\mathbb{K})$ définie par : $m_{i,j}=a_j^i+a_{j+1}^i$. -a) Calculer $\det M$. -$\boldsymbol{b}$ ) Étudier l'inversibilité de $M$. + 1. Calculer $\det M$. + 2. Étudier l'inversibilité de $M$. #+end_exercice #+begin_exercice [Mines MP 2025 # 493] @@ -5860,17 +6568,20 @@ c) Montrer que $B^{k+1}A^k = B$. #+end_exercice #+begin_exercice [Mines MP 2025 # 507] -Soient $\mathbb{K}$ un sous-corps de $\C$ et $A,B\in\mathcal{M}_n(\mathbb{K})$ deux matrices semblables. Montrer que Com $A$ et Com $B$ sont semblables. +Soient $\mathbb{K}$ un sous-corps de $\C$ et $A,B\in\mathcal{M}_n(\mathbb{K})$ deux matrices semblables. Montrer que $\op{Com}A$ et $\op{Com}B$ sont semblables. #+end_exercice +# ID:8989 #+begin_exercice [Mines MP 2025 # 508] Soient $n \geq2$ et $\mathbb{K} = \mathbb{R}$ ou $\C$. -a) Calculer $\operatorname{Com}(J_r)$ où $J_r = \begin{pmatrix} I_r & 0 \\ 0 & 0 \end{pmatrix}$ avec $0 \leqslant r \leqslant n$. + 1. Calculer $\op{Com}(J_r)$ où $J_r = \begin{pmatrix} I_r & 0 \\ 0 & 0 \end{pmatrix}$ avec $0 \leqslant r \leqslant n$. -b) Montrer que Com(AB) = Com(A) Com(B) pour tout couple $(A, B) \in \mathcal{M}_n(\mathbb{K})^2$. + 2. Montrer que $\op{Com}(AB) = \op{Com}(A) \op{Com}(B)$ pour tout couple $(A, B) \in \mathcal{M}_n(\mathbb{K})^2$. -c) Exprimer $\operatorname{rg}(\operatorname{Com}(A))$ en fonction de $\operatorname{rg} A$. d) Étudier l'injectivité de $\gamma: A \mapsto \text{Com}(A)$. Quelle est son image? + 3. Exprimer $\op{rg}(\op{Com}(A))$ en fonction de + $\op{rg} A$. + 4. Étudier l'injectivité de $\gamma\colon A \mapsto \op{Com}(A)$. Quelle est son image? #+end_exercice #+begin_exercice [Mines MP 2025 # 509] @@ -6182,12 +6893,14 @@ c) Démontrer l'existence d'une base de $E$ dans laquelle la matrice de $u$ est #+begin_exercice [Mines MP 2025 # 552] Soit $A \in \mathcal{M}_3(\C)$. On pose $G_A = \{c \in \C^*, cA \text{ est semblable à } A\}$. -a) Montrer que $G_A$ est un sous-groupe de $\C^*$.- b) Montrer que $G_A$ est infini si et seulement si $A$ est nilpotente, et qu'alors $G_A = \C^*$. + 1. Montrer que $G_A$ est un sous-groupe de $\C^*$. + 2. Montrer que $G_A$ est infini si et seulement si $A$ est + nilpotente, et qu'alors $G_A = \C^*$. #+end_exercice #+begin_exercice [Mines MP 2025 # 553] -a) Déterminer les $\lambda \in \mathbb{R}$ pour lesquels il existe $A, B \in GL_n(\C)$ telles que $AB = \lambda BA$. -b) Déterminer, parmi les $\lambda$ trouvés en a), ceux pour lesquels toutes les matrices A,B vérifiant cette condition sont diagonalisables. + 1. Déterminer les $\lambda \in \mathbb{R}$ pour lesquels il existe $A, B \in GL_n(\C)$ telles que $AB = \lambda BA$. + 2. Déterminer, parmi les $\lambda$ trouvés en a), ceux pour lesquels toutes les matrices A,B vérifiant cette condition sont diagonalisables. #+end_exercice #+begin_exercice [Mines MP 2025 # 554] @@ -6240,15 +6953,16 @@ Soient $u \in \mathcal{L}(\C^n)$ et $\alpha > 0$. Montrer qu'il existe une base #+begin_exercice [Mines MP 2025 # 562] Soient $n \geq2$ et $G$ un sous-groupe de $\mathrm{GL}_n(\C)$ tel qu'il existe $N \in \N^*$ tel que $A^N = I_n$ pour tout $A \in G$. -a) Montrer que tous les éléments de $G$ sont diagonalisables. + 1. Montrer que tous les éléments de $G$ sont diagonalisables. + 2. Soit $(M_i)_{1 \leq i \leq m}$ une base de $\vect(G)$ constituée d'éléments de $G$. -b) Soit $(M_i)_{1 \leq i \leq m}$ une base de Vect(G) constituée d'éléments de $G$. Soit $f: A \in G \mapsto$ $(\operatorname{tr}(AM_1),\ldots,\operatorname{tr}(AM_m))$. + Soit $f: A \in G \mapsto$ $(\operatorname{tr}(AM_1),\ldots,\operatorname{tr}(AM_m))$. -Soient $A, B \in G$ telles que f(A) = f(B). On pose $C = AB^{-1}$. + Soient $A, B \in G$ telles que $f(A) = f(B)$. On pose $C = AB^{-1}$. -i) Montrer que $\forall k \in \N$, $\operatorname{tr}(C^k) = n$. -ii) En déduire que $C I_n$ est nilpotente. -c) Montrer que $G$ est fini. + a) Montrer que $\forall k \in \N$, $\op{tr}(C^k) = n$. + b) En déduire que $C - I_n$ est nilpotente. + 3. Montrer que $G$ est fini. #+end_exercice #+begin_exercice [Mines MP 2025 # 563] @@ -6284,6 +6998,7 @@ c) Le résultat subsiste-t-il si $A$ n'est pas de classe $C^1$ ? d) Le résultat subsiste-t-il si $A$ est valeurs dans l'ensemble des matrices trigonalisables à spectre inclus dans $\mathbb{R}^{+*}$ ? #+end_exercice +# ID:8880 #+begin_exercice [Mines MP 2025 # 568] Soient $(E, \langle , \rangle)$ un espace euclidien, $X$ une partie de $E$. Montrer que $X$ est finie si et seulement si $\{\langle x,y\rangle,\ (x,y)\in X^2\}$ est fini. #+end_exercice @@ -6312,7 +7027,7 @@ b) Montrer que $x_0$ est l'unique solution de $A^T A x = A^T b$. #+end_exercice #+begin_exercice [Mines MP 2025 # 573] -Montrer l'existence et calculer $\min_{P \in \mathbb{R}_{n-1}[X]} \sum_{i=1}^{n} (i^n P(i))^2$. +Montrer l'existence et calculer $\min_{P \in \mathbb{R}_{n-1}[X]} \sum_{i=1}^{n} (i^n - P(i))^2$. #+end_exercice #+begin_exercice [Mines MP 2025 # 574] @@ -6570,7 +7285,7 @@ b) Déterminer l'adhérence de $\mathcal{Z}_p$ pour n'importe quel $p$ dans $\N^ #+end_exercice #+begin_exercice [Mines MP 2025 # 613] -Soit $f: \mathbb{R}^2 \to \mathbb{R}$ continue. Montrer que la restriction de $f$ à un cercle de rayon strictement positif n'est pas injective. +Soit $f\colon \mathbb{R}^2 \to \mathbb{R}$ continue. Montrer que la restriction de $f$ à un cercle de rayon strictement positif n'est pas injective. #+end_exercice #+begin_exercice [Mines MP 2025 # 614] @@ -6609,7 +7324,7 @@ Justifier l'existence de $k_X = \min\{k \in \N, ||u_X a_k||_{\infty} \leq b\}$. #+end_exercice #+begin_exercice [Mines MP 2025 # 618] -Soient $n$ et $p$ deux entiers naturels. Soit $f \in \mathcal{L}(\mathbb{R}^n, \mathbb{R}^p)$. Montrer que $f$ est surjective si et seulement si l'imageqdirecte par $f$ de tout ouvert de $\mathbb{R}^n$ est un ouvert de $\mathbb{R}^p$. +Soient $n$ et $p$ deux entiers naturels. Soit $f \in \mathcal{L}(\mathbb{R}^n, \mathbb{R}^p)$. Montrer que $f$ est surjective si et seulement si l'image directe par $f$ de tout ouvert de $\mathbb{R}^n$ est un ouvert de $\mathbb{R}^p$. #+end_exercice #+begin_exercice [Mines MP 2025 # 619] @@ -6689,7 +7404,7 @@ b) On note $x_n$ la racine réelle de $P_{2n+1}$. Montrer que $(x_n)_{n\geq 0}$ #+end_exercice #+begin_exercice [Mines MP 2025 # 631] -Soient $a,b\in\N$ avec $b>a\geq 1$. Déterminer la limite de la suite de terme général $\sum_{n=0}^{b} \ln\left(1-\frac{k}{n^2}\right)$. +Soient $a,b\in\N$ avec $b>a\geq 1$. Déterminer la limite de la suite de terme général $\sum_{n=a}^{b} \ln\left(1-\frac{k}{n^2}\right)$. #+end_exercice #+begin_exercice [Mines MP 2025 # 632] @@ -6708,9 +7423,9 @@ $\boldsymbol{b}$ ) Montrer que $u$ est décroissante à partir d'un certain rang #+begin_exercice [Mines MP 2025 # 635] Soit $\alpha \in ]1, +\infty[$. -a) Montrer que, pour tout $n \in \N^*$, l'équation $\prod_{k=1}^n (kx+n^2) = \alpha n^{2n}$ possède une unique solution strictement positive, que l'on notera $x_n$. -b) Montrer que $\forall n \in \N^*, x_n < 2\alpha$. -c) Montrer que $(x_n)$ converge et déterminer sa limite. + 1. Montrer que, pour tout $n \in \N^*$, l'équation $\prod_{k=1}^n (kx+n^2) = \alpha n^{2n}$ possède une unique solution strictement positive, que l'on notera $x_n$. + 2. Montrer que $\forall n \in \N^*, x_n < 2\alpha$. + 3. Montrer que $(x_n)$ converge et déterminer sa limite. #+end_exercice #+begin_exercice [Mines MP 2025 # 636] @@ -6720,9 +7435,14 @@ b) Étudier la monotonie et la limite éventuelle de $(k_j)_{j\in\N^*}$. c) Montrer que $\frac{k_{j+1}}{k_i} \xrightarrow{j \to +\infty} e$. #+end_exercice +# ID:8972 #+begin_exercice [Mines MP 2025 # 637] -Soit $(a_n)$ une suite strictement monotone telle que $a_n \to +\infty$ et $(a_{n+1}-a_n)$ est bornée. Soit $f \in \mathcal{C}^1(\mathbb{R}^{+*},\mathbb{R})$ telle que $f'(x) \underset{x \to +\infty}{\longrightarrow} 0$. -Montrer que, si $f(a_n) \longrightarrow \ell \in \mathbb{R}$, alors $f(x) \underset{x \to +\infty}{\longrightarrow} \ell$.- +Soit $(a_n)$ une suite strictement monotone telle que $a_n \to ++\infty$ et $(a_{n+1}-a_n)$ est bornée. Soit $f \in +\mathcal{C}^1(\mathbb{R}^{+*},\mathbb{R})$ telle que $f'(x) +\underset{x \to +\infty}{\longrightarrow} 0$. Montrer que, si $f(a_n) +\longrightarrow \ell \in \mathbb{R}$, alors $f(x) \underset{x \to ++\infty}{\longrightarrow} \ell$. #+end_exercice #+begin_exercice [Mines MP 2025 # 638] @@ -6834,9 +7554,11 @@ d) Montrer que $\arctan(x) = \sum_{n=0}^{+\infty} \frac{(-1)^n}{2n+1} x^{2n+1}$ #+begin_exercice [Mines MP 2025 # 652] On pose $a_n = \frac{n! e^n}{n^n \cdot \sqrt{n}}$. -a) Montrer que $\sum (\ln(a_{n+1})- \ln(a_n))$ converge. -b) Donner un équivalent de $\sum_{k=0}^{+\infty} \frac{1}{k^2}$ quand $n$ tend vers $+\infty$. -c) Démontrer que $n! \sim \sqrt{2\pi n} \, n^n e^{-n} \left( 1 + \frac{1}{12n} + o\left(\frac{1}{n}\right) \right)$. + 1. Montrer que $\sum (\ln(a_{n+1})- \ln(a_n))$ converge. + 2. Donner un équivalent de $\sum_{k=n}^{+\infty} \frac{1}{k^2}$ quand + $n$ tend vers $+\infty$. + 3. Démontrer que $n! \sim \sqrt{2\pi n} \, n^n e^{-n} \left( 1 + + \frac{1}{12n} + o\left(\frac{1}{n}\right) \right)$. #+end_exercice # ID:8803 @@ -7365,13 +8087,13 @@ b) Montrer que $f$ n'est pas développable en série entière au voisinageqde 0. #+begin_exercice [Mines MP 2025 # 722] Soit $(u_n)$ une suite définie par $u_0 \in ]0, \pi/2[$ et $u_{n+1} = \sin(u_n)$ pour $n \in \N$. Pour $n \in \N$, on pose $a_n = \int_0^{u_n} \frac{\mathrm{d}t}{1+\sin t}$. -Pour -$n \in \N$ -, on pose $a_n = \int_0^{u_n} \frac{\mathrm{d}t}{1 + \sin t}$. +Pour $n \in \N$, on pose $a_n = \int_0^{u_n} \frac{\mathrm{d}t}{1 + \sin t}$. -a) Déterminer la limite de la suite $(u_n)$ et étudier le comportement de la suite $(a_n)$. -b) Montrer la divergence de $\sum u_n^2$ et la convergence de $\sum u_n^3$. -c) Déterminer le rayon de convergence de la série entière $\sum a_n x^n$. Préciser le comportement de sa somme en 1 et en -1. + 1. Déterminer la limite de la suite $(u_n)$ et étudier le + comportement de la suite $(a_n)$. + 2. Montrer la divergence de $\sum u_n^2$ et la convergence de $\sum u_n^3$. + 3. Déterminer le rayon de convergence de la série entière $\sum a_n + x^n$. Préciser le comportement de sa somme en $1$ et en $-1$. #+end_exercice #+begin_exercice [Mines MP 2025 # 723] @@ -7381,9 +8103,9 @@ unité ouvert. #+end_exercice #+begin_exercice [Mines MP 2025 # 724] -Pour $n \in \N$, on note p(n) le cardinal de l'ensemble $\{(x,y,z) \in \N^3 : x+2y+3z=n\}$. -Soit $G: t \mapsto \sum_{n=0}^{+\infty} p(n) t^n$. a) Montrer que le rayon de $G$ est $\geq 1$ et que $\forall t \in ]-1, 1[\ , G(t) = \frac{1}{(1-t)(1-t^2)(1-t^3)}$ -b) Expliciter p(n) et en déterminer un équivalent. +Pour $n \in \N$, on note $p(n)$ le cardinal de l'ensemble $\{(x,y,z) \in \N^3 : x+2y+3z=n\}$. Soit $G\colon t \mapsto \sum_{n=0}^{+\infty} p(n) t^n$. + 1. Montrer que le rayon de $G$ est $\geq 1$ et que $\forall t \in ]-1, 1[\ , G(t) = \frac{1}{(1-t)(1-t^2)(1-t^3)}$ + 2. Expliciter $p(n)$ et en déterminer un équivalent. #+end_exercice #+begin_exercice [Mines MP 2025 # 725] @@ -7731,23 +8453,25 @@ d) Étudier les extrema de $f$. #+end_exercice #+begin_exercice [Mines MP 2025 # 775] -Soit -$$f:(x,y)\in\mathbb{R}^2\mapsto\sqrt{1+x^2}+\sqrt{1+y^2}-\frac{xy}{\sqrt{2}}.$$ Déterminer les extrema de $f$ et préciser leur nature. +Soit $f\colon (x,y)\in\mathbb{R}^2\mapsto\sqrt{1+x^2}+\sqrt{1+y^2}-\frac{xy}{\sqrt{2}}$. Déterminer les extrema de $f$ et préciser leur nature. #+end_exercice #+begin_exercice [Mines MP 2025 # 776] Soit $D = \{(x, y) \in [0, \pi]^2 : x + y \le \pi\}$. -Déterminer les extrema de $\phi: (x,y) \in D \mapsto \sin(x) \sin(y) \sin(x+y)$. +Déterminer les extrema de $\phi\colon (x,y) \in D \mapsto \sin(x) \sin(y) \sin(x+y)$. #+end_exercice +# ID:8909 #+begin_exercice [Mines MP 2025 # 777] -Soit $f$ définie sur $\mathbb{R}^2$ par : $\forall (x,y) \in \mathbb{R}^2$, $f(x,y) = x^2 xy^2$. -a) Déterminer les points critiques de $f$. -b) Soit $D$ une droite passant par (0,0). Montrer que la restriction de $f$ à $D$ admet un minimum local en (0,0). -c) La fonction $f$ possède-t-elle un extremum local en (0,0)? +Soit $f$ définie sur $\mathbb{R}^2$ par : $\forall (x,y) \in \mathbb{R}^2$, $f(x,y) = x^2 - xy^2$. + 1. Déterminer les points critiques de $f$. + 2. Soit $D$ une droite passant par $(0,0)$. Montrer que la restriction + de $f$ à $D$ admet un minimum local en $(0,0)$. + 3. La fonction $f$ possède-t-elle un extremum local en (0,0)? #+end_exercice +# ID:8908 #+begin_exercice [Mines MP 2025 # 778] Soit $E$ un espace vectoriel normé non nul. Soit $N$ une norme quelconque sur $E$. Montrer que $N$ n'est pas différentiable en 0. #+end_exercice @@ -7770,8 +8494,9 @@ On pose $g: x \in E \mapsto f(x) e^{-\|x\|^2}$. Montrer que $g$ admet un minimum b) On prend $E = \mathbb{R}^n$ muni de sa structure euclidienne canonique. Soient $a \in \mathbb{R}^n \setminus \{0\}$ et $g: x \in E \mapsto \langle a, x \rangle \ e^{-\|x\|^2}$. Déterminer le minimum et le maximum de $g$ et indiquer les points en lesquels ils sont atteints. #+end_exercice +# ID:8986 #+begin_exercice [Mines MP 2025 # 781] -Déterminer les extrema de $f:(x,y)\in\mathbb{R}^2\mapsto xe^y+ye^x$. +Déterminer les extrema globaux et locaux de $f\colon (x,y)\in\mathbb{R}^2\mapsto xe^y+ye^x$. #+end_exercice #+begin_exercice [Mines MP 2025 # 782] @@ -7932,13 +8657,13 @@ e) Montrer que $(u_k)$ converge. #+end_exercice #+begin_exercice [Mines MP 2025 # 802] -a) Soit $U$ un ouvert convexe de $\mathbb{R}^n$ et $f:U\to\mathbb{R}$ une fonction de classe $\mathcal{C}^2$. Montrer que $f$ est convexe si et seulement si $H_f(x)\in\mathcal{S}_n^+(\mathbb{R})$ pour tout $x\in U$ (où $H_f(x)$ désigne la matrice hessienne de $f$ en x). + 1. Soit $U$ un ouvert convexe de $\mathbb{R}^n$ et $f:U\to\mathbb{R}$ une fonction de classe $\mathcal{C}^2$. Montrer que $f$ est convexe si et seulement si $H_f(x)\in\mathcal{S}_n^+(\mathbb{R})$ pour tout $x\in U$ (où $H_f(x)$ désigne la matrice hessienne de $f$ en x). -b) On fixe $p \in \mathbb{R}^{+*}$. Soit + 2. On fixe $p \in \mathbb{R}^{+*}$. Soit -$$f: (x_1, \dots, x_n) \in (\mathbb{R}^{+*})^n \mapsto \left(\sum_{i=1}^n x_i^p\right)^{1/p}.$$ + $$f: (x_1, \dots, x_n) \in (\mathbb{R}^{+*})^n \mapsto \left(\sum_{i=1}^n x_i^p\right)^{1/p}.$$ -À quelle condition la fonction $f$ est-elle convexe? + À quelle condition la fonction $f$ est-elle convexe? #+end_exercice #+begin_exercice [Mines MP 2025 # 803] @@ -7961,10 +8686,10 @@ Montrer que l'ensemble des vecteurs tangents à $SL_n(\mathbb{R})$ au point $I_n #+end_exercice #+begin_exercice [Mines MP 2025 # 806] -Soit $f \in \mathcal{C}^1(\mathbb{R}^n, \mathbb{R}^n)$ telle que l'imageqde tout fermé de $\mathbb{R}^n$ par $f$ est fermée. On suppose - -de plus que, pour tout $a \in \mathbb{R}^n$, $\mathrm{d} f(a)$ est bijective. On pose $X = f(\mathbb{R}^n)$. -a) Soit $x \in X$. Montrer que l'espace tangent $T_x X$ est égal à $\mathbb{R}^n$. b) Montrer que $X = \mathbb{R}^n$. Ind. On pourra raisonner par l'absurde. +Soit $f \in \mathcal{C}^1(\mathbb{R}^n, \mathbb{R}^n)$ telle que +l'image de tout fermé de $\mathbb{R}^n$ par $f$ est fermée. On suppose de plus que, pour tout $a \in \mathbb{R}^n$, $\mathrm{d} f(a)$ est bijective. On pose $X = f(\mathbb{R}^n)$. + 1. Soit $x \in X$. Montrer que l'espace tangent $T_x X$ est égal à $\mathbb{R}^n$. + 2. Montrer que $X = \mathbb{R}^n$. Ind. On pourra raisonner par l'absurde. #+end_exercice @@ -7977,8 +8702,9 @@ a) Déterminer $\mathbf{E}(\operatorname{card}(X))$. b) Déterminer $\mathbf{E}(\operatorname{card}(X \cap Y))$. #+end_exercice +# ID:8998 #+begin_exercice [Mines MP 2025 # 808] -Soient $n, p \in \N$ avec $1 \le p < n$. On considère une urne contenant $p$ boules blanches et $n$-p boules noires. On effectue des tirages sans remise des boules de l'urne. Donner la loi et l'espérance de la variable donnant le rang de la dernière boule blanche tirée. +Soient $n, p \in \N$ avec $1 \le p < n$. On considère une urne contenant $p$ boules blanches et $n-p$ boules noires. On effectue des tirages sans remise des boules de l'urne. Donner la loi et l'espérance de la variable donnant le rang de la dernière boule blanche tirée. #+end_exercice #+begin_exercice [Mines MP 2025 # 809] @@ -7990,12 +8716,11 @@ a) Soit $p \in [0, 1[$. On suppose que $X \sim \mathcal{G}(p)$. Montrer que $X$ #+end_exercice #+begin_exercice [Mines MP 2025 # 810] -On considère une urne contenant deux fois plus de boules noires que de blanches. On y effectue des tirages avec remise, et on note $X$ la variable aléatoire donnant le nombre de tirages nécessaires pour obtenir pour la première fois deux boules noires consécutives. Pour - -tout $n \geq0$, on pose $u_n = \mathbf{P}(X > n)$. a) Montrer que $u_{n+2} = \frac{1}{3}u_{n+1} + \frac{2}{9}u_n$ pour tout $n \geq0$. -b) En déduire la loi de $X$ c) Montrer que $X$ admet des moments de tout ordre, et calculer son espérance. - -b) Calculer la fonction génératrice de $X_n$ et en déduire sa loi et son espérance. +On considère une urne contenant deux fois plus de boules noires que de blanches. On y effectue des tirages avec remise, et on note $X$ la variable aléatoire donnant le nombre de tirages nécessaires pour obtenir pour la première fois deux boules noires consécutives. Pour tout $n \geq0$, on pose $u_n = \mathbf{P}(X > n)$. + 1. Montrer que $u_{n+2} = \frac{1}{3}u_{n+1} + \frac{2}{9}u_n$ pour tout $n \geq0$. + 2. En déduire la loi de $X$. + 3. Montrer que $X$ admet des moments de tout ordre, et calculer son espérance. + 4. Calculer la fonction génératrice de $X_n$ et en déduire sa loi et son espérance. #+end_exercice #+begin_exercice [Mines MP 2025 # 811] @@ -8091,16 +8816,51 @@ b) Montrer la relation $(b+r)E_{n,b} = b + bE_{n-1,b-1} + rE_{n-1,b}$ (si b>0 et $\forall (u,v) \in ]-1,1[^2,\ v\ \partial_2 F(u,v) + r\ F(u,v) = \frac{uv}{(1-u)^2(1-v)^2}$. #+end_exercice +# ID:8970 #+begin_exercice [Mines MP 2025 # 822] -a) Montrer que le polynôme $P = X^3 X^2 X 1$ admet une unique racine réelle et deux racines complexes non réelles de module strictement inférieur à 1.b) On lance une pièce équilibrée. Pour $n \geq3$, $A_n$ est l'événement « obtenir trois pile consécutifs pour la première fois à l'instant $n$ ». + 1. Montrer que le polynôme $P = X^3 - X^2- X- 1$ admet une unique racine + réelle et deux racines complexes non réelles de module strictement + inférieur à 1. + 2. On lance une pièce équilibrée. Pour $n \geq3$, $A_n$ est + l'événement « obtenir trois pile consécutifs pour la première fois + à l'instant $n$ ». -Déterminer une relation de récurrence d'ordre 3 vérifiée par la suite $(\mathbf{P}(A_n))_{n\geq 3}$ et en déduire sa limite. + Déterminer une relation de récurrence d'ordre $3$ vérifiée par la + suite $(\mathbf{P}(A_n))_{n\geq 3}$ et en déduire sa limite. #+end_exercice +#+begin_proof + 1. Étudions la fonction $x \mapsto P(x)$ sur $\R$. On a $P'(X) = 3X^2 - 2X - 1$, dont les racines sont $1$ et $-1/3$. + Le maximum local de $P$ est atteint en $-1/3$ et vaut $P(-1/3) = -22/27 < 0$. + Ainsi, $P(x) < 0$ pour tout $x \leq 1$. Sur $[1, +\infty[$, $P$ est strictement croissante, avec $P(1) = -2$ et $P(2) = 1$. + D'après le théorème de la bijection, $P$ admet une unique racine réelle $\alpha \in ]1, 2[$. + + Comme $P$ est à coefficients réels et de degré $3$, il admet deux autres racines complexes conjuguées $z_1$ et $z_2 = \overline{z_1}$. + Le produit des racines de $P$ vaut $1$, donc $\alpha z_1 z_2 = 1$, soit $\alpha |z_1|^2 = 1$. + Puisque $\alpha > 1$, on en déduit que $|z_1|^2 < 1$, c'est-à-dire que les deux racines complexes non réelles de $P$ sont de module strictement inférieur à $1$. + + 2. Notons $p_n = \mathbf{P}(A_n)$. Pour $n \geq 4$, on peut conditionner selon les premiers lancers, qui forment un système complet d'événements : + + Si les premiers lancers sont Pile-Pile-Pile (probabilité $1/8$), le jeu s'arrête à l'instant $3$. Comme $n \geq 4$, $A_n$ n'est pas réalisé. + + Si le premier lancer est Face (probabilité $1/2$), la séquence de trois Pile doit apparaître pour la première fois $n-1$ lancers plus tard, ce qui se produit avec probabilité $p_{n-1}$. + + Si les premiers lancers sont Pile-Face (probabilité $1/4$), la séquence doit apparaître $n-2$ lancers plus tard (probabilité $p_{n-2}$). + + Si les premiers lancers sont Pile-Pile-Face (probabilité $1/8$), la séquence doit apparaître $n-3$ lancers plus tard (probabilité $p_{n-3}$). + + On obtient donc la relation de récurrence linéaire d'ordre $3$ : + $$p_n = \frac{1}{2} p_{n-1} + \frac{1}{4} p_{n-2} + \frac{1}{8} p_{n-3}.$$ + + L'équation caractéristique associée est $r^3 - \frac{1}{2}r^2 - \frac{1}{4}r - \frac{1}{8} = 0$. + En multipliant par $8$, on remarque que cela équivaut à $(2r)^3 - (2r)^2 - (2r) - 1 = 0$, soit $P(2r) = 0$. + Les racines de l'équation caractéristique sont donc $\alpha/2$, $z_1/2$ et $z_2/2$. + D'après la question 1, on a $\alpha \in ]1, 2[$, donc $|\alpha/2| < 1$, et $|z_1/2| = |z_2/2| < 1/2 < 1$. + Toutes les racines de l'équation caractéristique étant de module strictement inférieur à $1$, la suite $(p_n)_{n\geq 3}$, qui est combinaison linéaire de leurs puissances $n$-ièmes, converge vers $0$. +#+end_proof + #+begin_exercice [Mines MP 2025 # 823] On lance simultanément $n \in \N^*$ fois deux pièces équilibrées. Soit $E_n$ l'événement « les deux pièces donnent le même nombre de pile ». -a) i) Pour $a,b,n\in\N$ tels que $a+b\leqslant n$, montrer que $\sum_{k=0}^n \binom{a}{k}\binom{b}{n-k}=\binom{a+b}{n}$. -ii) Calculer $P(E_n)$. b) On note $N$ le nombre de fois où les pièces ont donné le même nombre de pile au cours des $n$ lancers. Calculer E(N). + 1. + 1. Pour $a,b,n\in\N$ tels que $a+b\leqslant n$, montrer que $\sum_{k=0}^n \binom{a}{k}\binom{b}{n-k}=\binom{a+b}{n}$. + 2. Calculer $P(E_n)$. + 2. On note $N$ le nombre de fois où les pièces ont donné le même nombre de pile au cours des $n$ lancers. Calculer $E(N)$. #+end_exercice #+begin_exercice [Mines MP 2025 # 824] @@ -8119,7 +8879,7 @@ Une pièce tombe sur pile avec probabilité $p \in ]0,1[$. On la lance jusqu'à #+end_exercice #+begin_exercice [Mines MP 2025 # 826] -On dispose de $n$ chapeaux et $n$ tiroirs. On rangeqaléatoirement chaque chapeau dans un des tiroirs (chaque tiroir pouvant contenir jusqu'à $n$ chapeaux). On note $X_k$ la variable aléatoire donnant le numéro du tiroir dans lequel est rangé le chapeau numéro $k$. On note $Z_n$ la variable aléatoire donnant le nombre de tiroirs vides à l'issue du rangement. +On dispose de $n$ chapeaux et $n$ tiroirs. On range aléatoirement chaque chapeau dans un des tiroirs (chaque tiroir pouvant contenir jusqu'à $n$ chapeaux). On note $X_k$ la variable aléatoire donnant le numéro du tiroir dans lequel est rangé le chapeau numéro $k$. On note $Z_n$ la variable aléatoire donnant le nombre de tiroirs vides à l'issue du rangement. a) Calculer l'espérance et la variance de $Z_n$. b) Déterminer un équivalent de $\mathbf{E}(Z_n)$ et de $\mathbf{V}(Z_n)$ lorsque $n$ tend vers $+\infty$. #+end_exercice @@ -8172,8 +8932,9 @@ Soient $X_1, \ldots, X_n$ des variables aléatoires $i$.i.d. suivant la loi géo Pour $n \geq 2$, soit $X_n$ une variable aléatoire à valeurs dans $\db{1,n}$ telle que, pour tout $k \in \db{1,n}$, $\mathbf{P}(X_n=k) = \frac{\ln(k)}{\ln(n!)}$. Déterminer un équivalent de $\mathbf{E}(X_n)$ et $\mathbf{E}(X_n^2)$ lorsque $n$ tend vers $+\infty$. #+end_exercice +# ID:9003 #+begin_exercice [Mines MP 2025 # 836] -Soient a et $b$ dans $\mathbb{R}$ avec a < $b$. Quelle est la variance maximale d'une variable aléatoire à valeurs dans [a,b]? +Soient a et $b$ dans $\mathbb{R}$ avec $a < b$. Quelle est la variance maximale d'une variable aléatoire à valeurs dans $[a,b]$ ? #+end_exercice #+begin_exercice [Mines MP 2025 # 837] @@ -8247,21 +9008,26 @@ définie. c) Exprimer S(x,h) à l'aide de la variable aléatoire $X_h$, où $X_h #+end_exercice #+begin_exercice [Mines MP 2025 # 845] -Soient $(p_n)_{n\geq 1}$ une suite d'éléments de ]0,1[ et $(X_n)_{n\geq 1}$ une suite de variables aléatoires indépendantes telle que, pour tout $n, X_n \sim \mathcal{G}(p_n)$. Montrer que $A = \{\omega, X_n(\omega) \xrightarrow[n \to +\infty]{}$ $+\infty$ } est un événement, et calculer sa probabilité en fonction de $(p_n)$. +Soient $(p_n)_{n\geq 1}$ une suite d'éléments de $]0,1$[ et +$(X_n)_{n\geq 1}$ une suite de variables aléatoires indépendantes +telle que, pour tout $n, X_n \sim \mathcal{G}(p_n)$. Montrer que $A = +\{\omega, X_n(\omega) \xrightarrow[n \to +\infty]{}$ $+\infty$ } est +un événement, et calculer sa probabilité en fonction de $(p_n)$. #+end_exercice #+begin_exercice [Mines MP 2025 # 846] Soit $(X_k)_{k\geq 1}$ une suite de variables de Rademacher idépendantes. Pour $n\in\N^*$, soit $S_n = X_1 + \dots + X_n$. -a) Calculer $\mathbf{E}(e^{tS_n})$ si $n \in \N^*$ et $t \in \mathbb{R}$. -b) Montrer que, pour $n \in \N^*$ et $a \in \mathbb{R}^{+*}$, $\mathbf{P}(|S_n| \geq na) \leqslant 2 \exp\left(-\frac{na^2}{2}\right)$. -c) Montrer que le résultat de la question précédente subsiste si on suppose que $(X_k)_{k \geq1}$ est une suite $i$.i.d. de variables aléatoires centrées et bornées par 1. + 1. Calculer $\mathbf{E}(e^{tS_n})$ si $n \in \N^*$ et $t \in \mathbb{R}$. + 2. Montrer que, pour $n \in \N^*$ et $a \in \mathbb{R}^{+*}$, $\mathbf{P}(|S_n| \geq na) \leqslant 2 \exp\left(-\frac{na^2}{2}\right)$. + 3. Montrer que le résultat de la question précédente subsiste si on + suppose que $(X_k)_{k \geq1}$ est une suite i.i.d. de variables + aléatoires centrées et bornées par $1$. #+end_exercice #+begin_exercice [Mines MP 2025 # 847] -Soient $a, b \in \mathbb{R}$ avec $a \leq b, X$ une variable aléatoire à valeurs dans $[a, b], (X_n)_{n \geq 1}$ une suite de variables $i$.i.d. suivant la loi de $X$. Pour $n \in \N^*$, on pose $S_n = X_1 + \cdots + X_n$. +Soient $a, b \in \mathbb{R}$ avec $a \leq b, X$ une variable aléatoire à valeurs dans $[a, b], (X_n)_{n \geq 1}$ une suite de variables i.i.d. suivant la loi de $X$. Pour $n \in \N^*$, on pose $S_n = X_1 + \cdots + X_n$. -Soit -$$f \in \mathcal{C}^0([a,b],\mathbb{R}).$$ Montrer que $\mathbf{E}\left(f\left(\frac{S_n}{n}\right)\right) \underset{n \to +\infty}{\longrightarrow} f\left(\mathbf{E}(X)\right)$. +Soit $f \in \mathcal{C}^0([a,b],\mathbb{R})$. Montrer que $\quad\displaystyle \mathbf{E}\left(f\left(\frac{S_n}{n}\right)\right) \underset{n \to +\infty}{\longrightarrow} f\left(\mathbf{E}(X)\right)$. #+end_exercice #+begin_exercice [Mines MP 2025 # 848] @@ -8296,7 +9062,6 @@ d) Soient $Y$ et $Z$ deux variables aléatoires à valeurs dans $\N$. Montrer que $\forall t \in [0, 1], |G_Y(t) G_Z(t)| \leq 2 \mathbf{P}(Y \neq Z)$. e) Montrer que $S \sim \mathcal{P}(\theta)$ pour $\theta = \sum_{i=1}^{+\infty} \lambda_n$. #+end_exercice - * Mines - PSI ** Algèbre @@ -8319,7 +9084,8 @@ g) En déduire que $\sum_{m=0}^{+\infty} \frac{1}{m^2} = \frac{\pi^2}{6}$. #+begin_exercice [Mines PSI 2025 # 852] On considère des entiers $N \geq1$ et $n \geq2$, et une famille de réels $(a_i)_{1 \le i \le n}$ tous distincts. On pose $\phi: P \in \mathbb{R}_N[X] \mapsto (P(a_1), P'(a_1), P(a_2), P'(a_2), \dots, P(a_n), P'(a_n))$. -a) Quel est le rang de $\phi$ ? b) À quelle condition, $\phi$ est-il un isomorphisme? Cette condition étant remplie, déterminer l'imageqréciproque de $(x_1, y_1, \dots, x_n, y_n) \in \mathbb{R}^{2n}$ par $\phi$. + 1. Quel est le rang de $\phi$ ? + 2. À quelle condition $\phi$ est-il un isomorphisme? Cette condition étant remplie, déterminer l'image réciproque de $(x_1, y_1, \dots, x_n, y_n) \in \mathbb{R}^{2n}$ par $\phi$. #+end_exercice #+begin_exercice [Mines PSI 2025 # 853] @@ -8335,9 +9101,9 @@ Soient $I_0, \ldots, I_n$ des segments de $\mathbb{R}$ non réduits à des point #+begin_exercice [Mines PSI 2025 # 855] Soient $A, B \in \mathcal{M}_n(\C)$. On pose $M = \begin{pmatrix} A & A \\ A & B \end{pmatrix}$. -a) Déterminer la rang de $M$. b) $A$ quelle condition la matrice $M$ est-elle inversible? - -c) Cette condition étant vérifiée, déterminer l'inverse de $M$. + 1. Déterminer la rang de $M$. + 2. À quelle condition la matrice $M$ est-elle inversible? + 3. Cette condition étant vérifiée, déterminer l'inverse de $M$. #+end_exercice #+begin_exercice [Mines PSI 2025 # 856] @@ -8395,7 +9161,8 @@ e) Montrer finalement que $\chi_{AB} = \chi_{BA}$. #+end_exercice #+begin_exercice [Mines PSI 2025 # 864] -Soient $A$ et $B \in \mathcal{M}_n(\mathbb{R})$ telles que AB BA = $B$.a) Montrer que $\forall X, Y \in \mathcal{M}_n(\mathbb{R}), \operatorname{tr}(XY) = \operatorname{tr}(YX)$. +Soient $A$ et $B \in \mathcal{M}_n(\mathbb{R})$ telles que AB BA = $B$. + a) Montrer que $\forall X, Y \in \mathcal{M}_n(\mathbb{R}), \operatorname{tr}(XY) = \operatorname{tr}(YX)$. b) Montrer que $\forall k \in \N$, $AB^k - B^k A = kB^k$. @@ -8405,9 +9172,7 @@ c) En déduire que $B$ est nilpotente (on pourra utiliser $\theta: X \mapsto AX #+begin_exercice [Mines PSI 2025 # 865] Montrer que $$A = \begin{pmatrix} 0 & \dots & 0 & 1 \\ \vdots & \vdots & \vdots & \vdots \\ 0 & \dots & 0 & 1 \\ 1 & \dots & 1 & 1 \end{pmatrix} \in \mathcal{M}_n(\mathbb{R})$$ - est diagonalisable puis diagonali- - -ser A. + est diagonalisable puis diagonaliser A. #+end_exercice #+begin_exercice [Mines PSI 2025 # 866] @@ -8527,16 +9292,14 @@ c) Exprimer les valeurs propres et les espaces propres de $f_A$ en fonction de c #+end_exercice #+begin_exercice [Mines PSI 2025 # 880] -a) On considère le polynôme $P=X^5-4X^4+2X^3+8X^2-8X$. Montrer que P(2)=P'(2)=0 et en déduire une factorisation de $P$ en polynômes irréductibles de $\mathbb{R}[X]$. b) Trouver les matrices $M\in\mathcal{M}_n(\mathbb{R})$ telles que $M^5-4M^4+2M^3+8M^2-8M=0$ et - - $\operatorname{tr}(M) = 0$. + a) On considère le polynôme $P=X^5-4X^4+2X^3+8X^2-8X$. Montrer que P(2)=P'(2)=0 et en déduire une factorisation de $P$ en polynômes irréductibles de $\mathbb{R}[X]$. + b) Trouver les matrices $M\in\mathcal{M}_n(\mathbb{R})$ telles que $M^5-4M^4+2M^3+8M^2-8M=0$ et +$\operatorname{tr}(M) = 0$. #+end_exercice #+begin_exercice [Mines PSI 2025 # 881] Soit $E$ un $\mathbb{K}$ -espace vectoriel de dimension supérieure ou égale à 2. Soient $f_1, \ldots, f_n$ des endomorphismes de $E$ tels que $f_1 + \cdots + f_n = \operatorname{id}$ et, si $1 \leq i, j \leq n$ avec $i \neq j$, alors -des endomorphismes de -$E$ tels que $f_1 + \cdots + f_n = \operatorname{id}$ et, si $1 \leqslant i, j \leqslant n$ $f_i \circ f_j = 0$. Soient $\lambda_1, \ldots, \lambda_n \in \mathbb{K}$. On pose $g = \sum_{i=1}^n \lambda_i f_i$. Diagonaliser $g$. #+end_exercice @@ -8564,15 +9327,16 @@ c) Trouver une condition nécessaire et suffisante pour que $\phi$ soit un proje #+end_exercice #+begin_exercice [Mines PSI 2025 # 885] -a) Soient $E$ un espace vectoriel et $p$ un projecteur de $E$. + 1. Soient $E$ un espace vectoriel et $p$ un projecteur de $E$. -Montrer que $Ker(p - id) \cap Im(p - id) = \{0\}$. + Montrer que $\Ker(p - id) \cap \Im(p - id) = \{0\}$. -b) Soient $E$ un espace vectoriel normé et $u \in \mathcal{L}(E)$ tel que : $\forall x \in E, ||u(x)|| \leq ||x||$. - - i) Soit $x \in E$ tel que u(x) = $x$ et il existe $y \in E$ tel que $x$ = u(y) y. + 2. Soient $E$ un espace vectoriel euclidien et $u \in \mathcal{L}(E)$ tel que : $\forall x \in E, ||u(x)|| \leq ||x||$. + 1. Soit $x \in E$ tel que $u(x) = x$ et il existe $y \in E$ tel que $x = u(y) - y$. -Montrer par récurrence que pour tout $n \in \N$, $||nx + y|| \le ||y||$. -ii) Montrer que $Ker(u id) \cap Im(u id) = \{0\}$. c) Soient $E$ un espace euclidien et $p$ un projecteur de $E$ tel que $\forall x \in E, \|p(x)\| \leq \|x\|$. Montrer que $p$ est un projecteur orthogonal. + Montrer par récurrence que pour tout $n \in \N$, $\|nx + y\| \le \|y\|$. + 2. Montrer que $Ker(u - id) \cap Im(u id) = \{0\}$. + 3. Soient $E$ un espace euclidien et $p$ un projecteur de $E$ tel que $\forall x \in E, \|p(x)\| \leq \|x\|$. Montrer que $p$ est un projecteur orthogonal. #+end_exercice #+begin_exercice [Mines PSI 2025 # 886] @@ -8629,7 +9393,7 @@ b) Soient $A, B \in \mathcal{S}_n^{++}(\mathbb{R})$. Montrer que $I_n + AB$ et A #+end_exercice #+begin_exercice [Mines PSI 2025 # 892] -Soient $U$ et $V$ sans $\mathcal{S}_n^+(\mathbb{R})$. Montrer que $\det(U+V) \geq \det(U) + \det(V)$. +Soient $U$ et $V$ dans $\mathcal{S}_n^+(\mathbb{R})$. Montrer que $\det(U+V) \geq \det(U) + \det(V)$. #+end_exercice #+begin_exercice [Mines PSI 2025 # 893] @@ -8692,10 +9456,11 @@ c) Montrer que l'adhérence des matrices diagonalisables dans $\mathcal{M}_n(\ma #+end_exercice #+begin_exercice [Mines PSI 2025 # 901] -Pour $n \geq2$, on considère le polynôme $P_n = X^n + X^{n-1} + \cdots + X 1$. +Pour $n \geq2$, on considère le polynôme $P_n = X^n + X^{n-1} + \cdots + X - 1$. a) Montrer que $P_n$ admet une unique racine dans $\mathbb{R}^+$, notée $x_n$. b) Montrer que $x_n \in [1/2, 1]$. -c) Montrer que $(x_n)$ converge vers un réel $\ell$. d) Donner un équivalent de $x_n - \ell$. +c) Montrer que $(x_n)$ converge vers un réel $\ell$. +d) Donner un équivalent de $x_n - \ell$. #+end_exercice #+begin_exercice [Mines PSI 2025 # 902] @@ -8707,9 +9472,7 @@ b) Étudier la réciproque. #+begin_exercice [Mines PSI 2025 # 903] Soit $a \in \mathbb{R}^{+*}$. -a) Si $n \in \N$, montrer que l'équation $\sum_{k=0}^{n} \frac{1}{x-k} = a$ admet une unique solution dans - - $n, +\infty$, que l'on notera $x_n$. +a) Si $n \in \N$, montrer que l'équation $\sum_{k=0}^{n} \frac{1}{x-k} = a$ admet une unique solution dans $n, +\infty$, que l'on notera $x_n$. b) Étudier la monotonie de la suite $(x_n)$. c) Trouver un équivalent simple de $x_n$. @@ -8916,7 +9679,8 @@ c) Trouver les limites de $f$ et $g$ en $+\infty$. d) En déduire que $\int_{-t} #+begin_exercice [Mines PSI 2025 # 934] On pose $\cos: z \in \C \mapsto \frac{e^{iz} + e^{-iz}}{2}$. a) Exprimer $\cos z$ en fonction de $\Re \mathfrak{e}(z)$ et $\operatorname{Im}(z)$. -b) En déduire $\max_{|z| \le 1} |\cos z|^2$. c) On pose $f:(x,y)\in\mathbb{R}^2\mapsto\cos(x+iy)$. +b) En déduire $\max_{|z| \le 1} |\cos z|^2$. +c) On pose $f:(x,y)\in\mathbb{R}^2\mapsto\cos(x+iy)$. Montrer que la fonction $r \in \mathbb{R}^+ \mapsto \int_0^{2\pi} f(r\cos\theta, r\sin\theta) \,d\theta$ est constante. #+end_exercice @@ -9021,21 +9785,22 @@ On dispose d'une station d'appels. Le nombre d'appels entre 10 $h$ et 11 $h$ est #+end_exercice #+begin_exercice [Mines PSI 2025 # 947] -On considère un immeuble de trois étages avec un rez-de-chaussée). Cinq personnes prennent l'ascenseur. On considère que chacune va aller à un étageqde manière équiprobable et indépendamment des quatre autres. L'ascenseur ne fait pas demi-tour, il ne fait qu'une montée pour déposer les personnes. On note $X_i$ le nombre de personnes qui descendent à l'étageq$i$. -a) Donner la loi de $X_1$, $\mathbf{E}(X_1)$, $\mathbf{V}(X_1)$. -b) Que dire de $X_2$ et $X_3$ ? -On considère $Y_i$ qui vaut 1 si l'ascenseur s'arrête au $i$-ème étage, 0 sinon. -c) Déterminer $P(Y_i = 0)$ et $P(Y_i = 1)$. -d) En déduire $\mathbf{E}(Z)$ avec $Z$ la variable aléatoire représentant le nombre d'arrêts de l'ascenseur. +On considère un immeuble de trois étages avec un rez-de-chaussée). Cinq personnes prennent l'ascenseur. On considère que chacune va aller à un étageqde manière équiprobable et indépendamment des quatre autres. L'ascenseur ne fait pas demi-tour, il ne fait qu'une montée pour déposer les personnes. On note $X_i$ le nombre de personnes qui descendent à l'étage $i$. On considère $Y_i$ qui vaut 1 si l'ascenseur s'arrête au $i$-ème étage, 0 sinon. + 1. Donner la loi de $X_1$, $\mathbf{E}(X_1)$, $\mathbf{V}(X_1)$. + 2. Que dire de $X_2$ et $X_3$ ? + 3. Déterminer $P(Y_i = 0)$ et $P(Y_i = 1)$. + 4. En déduire $\mathbf{E}(Z)$ avec $Z$ la variable aléatoire représentant le nombre d'arrêts de l'ascenseur. #+end_exercice #+begin_exercice [Mines PSI 2025 # 948] Soit $(X_n)_{n\geq 2}$ une suite de variables aléatoires indépendantes suivant la loi de Bernoulli de paramètre $p_n=n^{-\alpha}$ où $\alpha$ est un réel strictement positif. -On pose $\tau = \min\{n \geq2, X_n(\omega) = 1\} \in \N^* \cup \{+\infty\}$. +On pose $\tau = \min\{n \geq2, X_n(\omega) = 1\} \in \N^* \cup \{+\infty\}$. On définit $\P(\tau = +\infty)$ comme la limite, quand $n\ra +\i$, de $\P(\tau\geq n)$. -a) Trouver une condition nécessaire et suffisante sur $\alpha$ pour que $\mathbf{P}(\tau = +\infty) = 0$. -b) Trouver une condition nécessaire et suffisante sur $\alpha$ pour que $\mathbf{E}(\tau) < +\infty$. + 1. Trouver une condition nécessaire et suffisante sur $\alpha$ pour + que $\mathbf{P}(\tau = +\infty) = 0$. + 2. Trouver une condition nécessaire et suffisante sur $\alpha$ pour + que $\mathbf{E}(\tau) < +\infty$. #+end_exercice #+begin_exercice [Mines PSI 2025 # 949] @@ -10671,7 +11436,7 @@ b) Montrer, par récurrence, que, pour tout $n \in \N^*$, $\mathbf{V}(D_n) = n!$ #+begin_exercice [Centrale MP 2025 # 1173] a) Donner la définition de la signature et calculer celle de la permutation $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 2 & 5 & 8 & 3 & 6 & 1 & 7 & 4 \end{pmatrix}$. -$$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 2 & 5 & 8 & 2 & 6 & 1 & 7 & 4 \end{pmatrix}$$ + $$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 2 & 5 & 8 & 2 & 6 & 1 & 7 & 4 \end{pmatrix}$$ b) Pour tout $\sigma \in \mathcal{S}_n$, on note $\eps(\sigma)$ la signature de $\sigma$, et $\nu(\sigma)$ le nombre de ses points fixes. @@ -10692,18 +11457,17 @@ Montrer que $G$ est isomorphe à $\mathbb{Z}/n\mathbb{Z}$ et que $\widehat{G}$ e c) Dans cette question, on suppose $G$ abélien. Montrer que, si $H$ est un sous-groupe de $G$ et $\xi \in \widehat{H}$, il existe $\chi \in \widehat{G}$ tel que $\chi_{|H} = \xi$. #+end_exercice +# ID:8997 #+begin_exercice [Centrale MP 2025 # 1175] -On note, pour -$n\in\N^*$ -, $F_n=\prod_{\substack{\xi\in\mathbb{U}_n\\\omega(\xi)=n}}(X-\xi)$, où $\omega(\xi)$ est l'ordre de la racine $n$ -ième $\xi$ +On note, pour $n\in\N^*$, $F_n=\prod_{\substack{\xi\in\mathbb{U}_n\\\omega(\xi)=n}}(X-\xi)$, où $\omega(\xi)$ est l'ordre de la racine $n$-ième $\xi$ comme élément du groupe $\C^*$, c'est-à-dire le plus petit entier $n$ telle que $\xi^n = 1$. -comme élément du groupe $\C^*$. - -a) Montrer que $\omega(\xi)$ divise $n$ pour tout $\xi \in \mathbb{U}_n$. -b) Exprimer $(X-1)F_n$ et $F_n$ dans le cas où $n$ est premier. -c) Soient $A, B \in \mathbb{Q}[X]$ tels que $AB \in \mathbb{Z}[X]$ et $A \in \mathbb{Z}[X]$ est unitaire. Montrer que $B \in \mathbb{Z}[X]$. -c) Montrer que, pour tout $n \in \N^*$, $F_n \in \mathbb{Z}[X]$. -d) Soit $n \in \N^*$. Calculer $F_n(1)$. + 1. Montrer que $\omega(\xi)$ divise $n$ pour tout $\xi \in \mathbb{U}_n$. + 2. Exprimer $(X-1)F_n$ et $F_n$ dans le cas où $n$ est premier. + 3. Soient $A, B \in \mathbb{Q}[X]$ tels que $AB \in \mathbb{Z}[X]$ et + $A \in \mathbb{Z}[X]$ est unitaire. Montrer que $B \in + \mathbb{Z}[X]$. + 4. Montrer que, pour tout $n \in \N^*$, $F_n \in \mathbb{Z}[X]$. + 5. Soit $n \in \N^*$. Calculer $F_n(1)$. #+end_exercice #+begin_exercice [Centrale MP 2025 # 1176] @@ -10732,11 +11496,13 @@ Montrer que $\mu \geq \deg(A) + \deg(B) + \deg(C) - \deg(W)$. d) En déduire que $\mu > \max(\deg A, \deg B, \deg C)$. #+end_exercice +# ID:8971 #+begin_exercice [Centrale MP 2025 # 1178] -a) Rappeler la définition d'un polynôme irréductible sur un corps $\mathbb{K}$ et l'énoncé du théorème de d'Alembert-Gauss.- b) Soit $P \in \mathbb{R}[X]$ non nul tel que $\forall x \in \mathbb{R}, \ P(x) \geq 0$. Montrer que toute racine réelle de $P$ est de multiplicité paire et que le coefficient dominant de $P$ est positif. En déduire qu'il existe $(A,B) \in \mathbb{R}[X]^2$ tel que $P = A^2 + B^2$. -c) Soit $Q \in \mathbb{R}[X]$ non nul tel que $\forall x \in [-1, 1], \ Q(x) \geq0$. -i) Montrer que si $\deg(Q) \leqslant 2$ alors il existe $(a,b) \in (\mathbb{R}^+)^2$ et $\lambda \in [-1,1]$ tels que $Q = (X \lambda)^2 + b(1 X^2)$ -$a(X-\lambda)^2+b(1-X^2)$. ii) Montrer plus généralement qu'il existe $(A,B)\in\mathbb{R}[X]^2$ tel que $Q=A^2+(1-X^2)B^2$. + 1. Rappeler la définition d'un polynôme irréductible sur un corps $\mathbb{K}$ et l'énoncé du théorème de d'Alembert-Gauss. + 2. Soit $P \in \mathbb{R}[X]$ non nul tel que $\forall x \in \mathbb{R}, \ P(x) \geq 0$. Montrer que toute racine réelle de $P$ est de multiplicité paire et que le coefficient dominant de $P$ est positif. En déduire qu'il existe $(A,B) \in \mathbb{R}[X]^2$ tel que $P = A^2 + B^2$. + 3. Soit $Q \in \mathbb{R}[X]$ non nul tel que $\forall x \in [-1, 1], \ Q(x) \geq0$. + 1. Montrer que si $\deg(Q) \leqslant 2$ alors il existe $(a,b) \in (\mathbb{R}^+)^2$ et $\lambda \in [-1,1]$ tels que $Q = a(X -\lambda)^2 + b(1 -X^2)$ + 2. Montrer plus généralement qu'il existe $(A,B)\in\mathbb{R}[X]^2$ tel que $Q=A^2+(1-X^2)B^2$. #+end_exercice #+begin_exercice [Centrale MP 2025 # 1179] @@ -10751,10 +11517,10 @@ $$\begin{vmatrix} 1 + a_1 & 1 & \cdots & 1 \\ 1 & \ddots & \ddots & \vdots \\ \v #+end_exercice #+begin_exercice [Centrale MP 2025 # 1180] -a) Soit $A \in \mathcal{M}_n(\mathbb{R})$. Montrer que $A \operatorname{Com}(A)^T = \det(A)I_n$. -b) On définit $GL_n(\mathbb{Z})$ comme l'ensemble des matrices de $GL_n(\mathbb{R})$ à coefficients entiers, dont l'inverse est également à coefficients entiers. -Soit $A \in \mathcal{M}_n(\mathbb{Z})$. Montrer que $A \in \mathrm{GL}_n(\mathbb{Z})$ si et seulement si $\det(A) = \pm 1$. -c) Soit $P \in \mathbb{Q}[X]$ un polynôme irréductible sur $\mathbb{Q}$. Montrer que les racines complexes de $P$ sont simples. + 1. Soit $A \in \mathcal{M}_n(\mathbb{R})$. Montrer que $A \operatorname{Com}(A)^T = \det(A)I_n$. + 2. On définit $GL_n(\mathbb{Z})$ comme l'ensemble des matrices de $GL_n(\mathbb{R})$ à coefficients entiers, dont l'inverse est également à coefficients entiers. + Soit $A \in \mathcal{M}_n(\mathbb{Z})$. Montrer que $A \in \mathrm{GL}_n(\mathbb{Z})$ si et seulement si $\det(A) = \pm 1$. + 3. Soit $P \in \mathbb{Q}[X]$ un polynôme irréductible sur $\mathbb{Q}$. Montrer que les racines complexes de $P$ sont simples. #+end_exercice #+begin_exercice [Centrale MP 2025 # 1181] @@ -10817,9 +11583,10 @@ b) i) Donner la matrice de $\Phi$ dans la base canonique. ii) Déterminer les é #+end_exercice #+begin_exercice [Centrale MP 2025 # 1188] -Soit $u$ un endomorphisme d'un $\mathbb{R}$ -espace vectoriel de dimension $n$. a) Montrer que les valeurs propres de $u$ sont exactement les racines de $\chi_u$.b) Exprimer les coefficients des termes en $X^{n-1}$ et $X^{n-2}$ de $\chi_u$ en fonction de tr(u) et $tr(u^2)$. - -c) On suppose $u$ de rang 2. Donner une condition nécessaire et suffisante sur tr(u) et $tr(u^2)$ pour que $u$ soit diagonalisable. +Soit $u$ un endomorphisme d'un $\mathbb{R}$-espace vectoriel de dimension $n$. + 1. Montrer que les valeurs propres de $u$ sont exactement les racines de $\chi_u$. + 2. Exprimer les coefficients des termes en $X^{n-1}$ et $X^{n-2}$ de $\chi_u$ en fonction de $\tr(u)$ et $\tr(u^2)$. + 3. On suppose $u$ de rang $2$. Donner une condition nécessaire et suffisante sur $\tr(u)$ et $tr(u^2)$ pour que $u$ soit diagonalisable. #+end_exercice #+begin_exercice [Centrale MP 2025 # 1189] @@ -10891,14 +11658,14 @@ c) Soient $A \in \mathcal{M}_n(\mathbb{R})$ et $\lambda \in \operatorname{Sp}(A) #+begin_exercice [Centrale MP 2025 # 1194] L'espace $\mathbb{R}^n$ est muni de sa structure euclidienne canonique. -a) Soit $f:[a,b]\to\mathbb{R}$ une fonction continue, positive et telle que $\int_a^b f=0$. Montrer que $f$ est nulle. -b) Montrer que la matrice $M_n = \left(\frac{1}{i+j-1}\right)_{1\leqslant i,j\leqslant n}$ est symétrique définie positive. Indi- + 1. Soit $f\colon [a,b]\to\R$ une fonction continue, positive et telle + que $\int_a^b f=0$. Montrer que $f$ est nulle. + 2. Montrer que la matrice $M_n = \left(\frac{1}{i+j-1}\right)_{1\leqslant i,j\leqslant n}$ est symétrique définie positive. -cation: $\frac{1}{i+j-1} = \int_0^1 t^{i+j-2} dt$. - -c) On note $\lambda_{\min}$ (resp. $\lambda_{\max}$ ) la plus petite (resp. grande) valeur propre de $M_n$. Montrer que $\lambda_{\min} ||x||^2 \leqslant x^T M_n x \leqslant \lambda_{\max} ||x||^2$ pour tout $x \in \mathbb{R}^n$. -d) On note $F$ le sous-espace propre de $M_n$ associé à $\lambda_{\max}$. Montrer que, si $x \in F$, toutes les coordonnées de $x$ sont de même signe. -e) Déterminer $\dim F$. + Indication: $\frac{1}{i+j-1} = \int_0^1 t^{i+j-2} dt$. + 3. On note $\lambda_{\min}$ (resp. $\lambda_{\max}$ ) la plus petite (resp. grande) valeur propre de $M_n$. Montrer que $\lambda_{\min} ||x||^2 \leqslant x^T M_n x \leqslant \lambda_{\max} ||x||^2$ pour tout $x \in \mathbb{R}^n$. + 4. On note $F$ le sous-espace propre de $M_n$ associé à $\lambda_{\max}$. Montrer que, si $x \in F$, toutes les coordonnées de $x$ sont de même signe. + 5. Déterminer $\dim F$. #+end_exercice #+begin_exercice [Centrale MP 2025 # 1195] @@ -10943,15 +11710,13 @@ que $f(C) = \overline{B(0,1)}$ et $f(C \setminus \mathring{C}) = S(0,1)$. #+end_exercice #+begin_exercice [Centrale MP 2025 # 1198] -a) Soit $u$ un endomorphisme d'un espace vectoriel normé $E$. Montrer que la continuité de $u$ est équivalente à son caractère lipschitzien, et aussi à sa continuité en 0. -b) On munit $\ell^2(\mathbb{Z})$, espace vectoriel des familles de réels de carré sommable indexées par $\mathbb{Z}$ (on admet qu'il s'agit d'un sous-espace vectoriel de $\mathbb{Z}^{\N}$ ), de $(u,v)\mapsto \sum_{n}u_{n}v_{n}$, dont on + 1. Soit $u$ un endomorphisme d'un espace vectoriel normé $E$. Montrer que la continuité de $u$ est équivalente à son caractère lipschitzien, et aussi à sa continuité en 0. + 2. On munit $\ell^2(\mathbb{Z})$, espace vectoriel des familles de réels de carré sommable indexées par $\mathbb{Z}$ (on admet qu'il s'agit d'un sous-espace vectoriel de $\mathbb{Z}^{\N}$ ), de $(u,v)\mapsto \sum_{n}u_{n}v_{n}$, dont on admet qu'il s'agit d'un produit scalaire, et on munit $\ell^2(\mathbb{Z})$ de la norme associée. On pose $T: u \in \ell^2(\mathbb{Z}) \mapsto (2u_n - u_{n+1} - u_{n-1})_{n \in \mathbb{Z}}$. -admet qu'il s'agit d'un produit scalaire, et on munit $\ell^2(\mathbb{Z})$ de la norme associée. On pose $T: u \in \ell^2(\mathbb{Z}) \mapsto (2u_n - u_{n+1} - u_{n-1})_{n \in \mathbb{Z}}$. + Montrer que $T$ est un endomorphisme continu de $\ell^2(\mathbb{Z})$. -Montrer que $T$ est un endomorphisme continu de $\ell^2(\mathbb{Z})$. - -c) Montrer que $T$ est injectif mais non surjectif. -d) Montrer que $T$ + id est surjectif. + 3. Montrer que $T$ est injectif mais non surjectif. + 4. Montrer que $T + \op{id}$ est surjectif. #+end_exercice #+begin_exercice [Centrale MP 2025 # 1199] @@ -10989,7 +11754,8 @@ e) Montrer qu'il existe une constante $C$ telle que $\sum_{p \in \mathcal{P}, n #+end_exercice #+begin_exercice [Centrale MP 2025 # 1203] -a) Énoncer et démontrer le théorème des bornes atteintes.b) Montrer que l'on définit une norme sur $\mathbb{R}[X]$ en posant $\|P\| = \sup_{x \in [-1,1]} |P(x)|$ pour tout $P \in \mathbb{R}[X]$. +a) Énoncer et démontrer le théorème des bornes atteintes. +b) Montrer que l'on définit une norme sur $\mathbb{R}[X]$ en posant $\|P\| = \sup_{x \in [-1,1]} |P(x)|$ pour tout $P \in \mathbb{R}[X]$. c) Montrer que, pour tout $d \in \N^*$, il existe un unique polynôme $T_d \in \mathbb{R}[X]$ unitaire de degré $d$ tel que $\cos(d\theta) = 2^{d-1}T_d(\cos\theta)$ pour tout $\theta \in \mathbb{R}$. @@ -11100,10 +11866,7 @@ On suppose que $S$ a une limite réelle $\ell$ en $+\infty$. On souhaite montrer 2. Montrer que $S$ est bien définie sur $\mathbb{R}$. 2. On suppose dans cette question que la série $\sum a_n$ converge absolument et que $\ell=0$. - Montrer que $S$ est continue. - - Soit $m \in \N^*$. On pose $I\colon T \in \mathbb{R}^{+*} \mapsto \frac{1}{T} \int_0^T S(x) \cos\left(\frac{x}{m}\right) dx$. - - Montrer que $\lim_{T \to +\infty} I(T) = 0$. - + - Soit $m \in \N^*$. On pose $I\colon T \in \mathbb{R}^{+*} \mapsto \frac{1}{T} \int_0^T S(x) \cos\left(\frac{x}{m}\right) dx$. Montrer que $I(T)\tend{T\ra \i} 0$. - Montrer que $a_m = 0$. 3. Traiter le cas général. #+end_exercice @@ -11288,26 +12051,36 @@ b) Soit $f: U \to F$ où $F$ est un espace euclidien. On suppose $f$ différenti c) Montrer que l'inégalité est encore vérifiée si $F$ est un espace vectoriel normé de dimension finie. #+end_exercice +** Probabilités + #+begin_exercice [Centrale MP 2025 # 1234] -a) Soient $a, b \in \mathbb{R}$ avec a < $b$ et $f : [a, b] \to \mathbb{R}$ continue. + 1. Soient $a, b \in \mathbb{R}$ avec a < $b$ et $f\colon [a, b] \to \mathbb{R}$ continue. -Montrer que -$$\sum_{k=1}^{n} \frac{b-a}{n} f\left(a+k\frac{b-a}{n}\right) \xrightarrow[n \to +\infty]{} \int_{a}^{b} f(t) dt.$$ + Montrer que + $$\sum_{k=1}^{n} \frac{b-a}{n} f\left(a+k\frac{b-a}{n}\right) \xrightarrow[n \to +\infty]{} \int_{a}^{b} f(t) dt.$$ -b) On suppose que $n \in \N^*$ candidats se présentent à un poste de secrétaire. Le recruteur les rencontrent successivement et pour chacun, il doit décider s'il l'engageqou pas. Si oui, il termine le processus de recrutement sans voir les candidats suivants. Sinon, le candidat est définitivement éliminé. + 2. On suppose que $n \in \N^*$ candidats se présentent à un poste de + secrétaire. Le recruteur les rencontrent successivement et pour + chacun, il doit décider s'il l'engageqou pas. Si oui, il termine + le processus de recrutement sans voir les candidats suivants. + Sinon, le candidat est définitivement éliminé. -La valeur de chaque candidat correspond à un score et on note $s_1 < \cdots < s_n$ la liste croissante des scores obtenus. On note $\sigma \in \mathcal{S}_n$ une permutation aléatoire telle que le candidat qui passe devant le recruteur en position numéro $j$ a obtenu le score $s_{\sigma(j)}$ pour tout $j \in [1, n]$. + La valeur de chaque candidat correspond à un score et on note $s_1 + < \cdots < s_n$ la liste croissante des scores obtenus. On note + $\sigma \in \mathcal{S}_n$ une permutation aléatoire telle que le + candidat qui passe devant le recruteur en position numéro $j$ a + obtenu le score $s_{\sigma(j)}$ pour tout $j \in [1, n]$. -i) Déterminer la loi de $S_j$, variable aléatoire du score du $j^{\text{ème}}$ candidat. -ii) Déterminer la loi de $R_j$, variable aléatoire du rang du meilleur candidat parmi les $j$ premiers. -c) On choisit la stratégie de refuser les $m_n$ premiers candidats, et de choisir le premier candidat dont le score est supérieur à l'un des scores précédemment rencontrés. - - i) Soit $p_n$ la probabilité d'embaucher le meilleur candidat. + 1. Déterminer la loi de $S_j$, variable aléatoire du score du $j^{\text{ème}}$ candidat. + 2. Déterminer la loi de $R_j$, variable aléatoire du rang du meilleur candidat parmi les $j$ premiers. + 3. On choisit la stratégie de refuser les $m_n$ premiers candidats, et de choisir le premier candidat dont le score est supérieur à l'un des scores précédemment rencontrés. + 1. Soit $p_n$ la probabilité d'embaucher le meilleur candidat. -Montrer que $p_n = \frac{m_n}{n} \sum_{j=m_n+1}^n \frac{1}{j-1}$. + Montrer que $p_n = \frac{m_n}{n} \sum_{j=m_n+1}^n \frac{1}{j-1}$. -ii) On suppose que $\frac{m_n}{n} \underset{n \to +\infty}{\longrightarrow} x \in \mathbb{R}$. Montrer que $p_n \underset{n \to +\infty}{\longrightarrow} x \int_x^1 \frac{\mathrm{d}t}{t}$. + 2. On suppose que $\frac{m_n}{n} \underset{n \to +\infty}{\longrightarrow} x \in \mathbb{R}$. Montrer que $p_n \underset{n \to +\infty}{\longrightarrow} x \int_x^1 \frac{\mathrm{d}t}{t}$. -Optimiser alors $x$ pour maximiser la probabilité de recruter le meilleur candidat. + Optimiser alors $x$ pour maximiser la probabilité de recruter le meilleur candidat. #+end_exercice #+begin_exercice [Centrale MP 2025 # 1235] @@ -11318,6 +12091,7 @@ c) i) Montrer que $Y_N$ est d'espérance finie, puis que $\mathbf{E}(Y_N) = \sum - ii) En utilisant $f(x) = \sum_{k=1}^{n} x^k (1-x)^k$, donner un équivalent de $\mathbf{E}(Y_N)$ quand $N \to +\infty$. #+end_exercice +# ID:8977 #+begin_exercice [Centrale MP 2025 # 1236] Soient $E$ un espace préhilbertien réel, $(v_1, \ldots, v_n)$ une famille de vecteurs unitaires de $E$, et $(X_1, \ldots, X_n)$ une famille de variables aléatoires $i$.i.d. suivant la loi uniforme sur $\{-1,1\}$. a) Que dire d'une variable aléatoire réelle, positive et d'espérance nulle?- b) On pose $U = \sum_{i=1}^{n} X_i v_i$. Calculer $\mathbf{E}(||U||^2)$. @@ -11379,20 +12153,25 @@ i) il existe un réel $\delta > 0$ tel que $\forall t \in ]-\delta, \delta[, e^{ ii) $\forall n \in \N, X^n \in L^1 \text{ et } \sum_{n \in \N} \frac{\mathbf{E}(X^n)}{n!} t^n \text{ a un rayon de convergence non nul.}$ #+end_exercice +# ID:8984 #+begin_exercice [Centrale MP 2025 # 1243] -Soit $m \in \N^*$. On munit $\mathbb{R}^m$ de sa structure euclidienne habituelle. On note $S$ sa sphère unité. Pour une famille $e=(e_i)_{i\in I}$ d'éléments de $S$ (éventuellement infinie), on note $Coh(e) = \sup_{(i,j) \in I^2} |\langle e_i, e_j \rangle|$. -a) Rappeler sans démonstration l'inégalité de Cauchy-Schwarz. -b) Soit $e \in S^I$. Que signifie l'égalité Coh(e) = 0?c) Soit $e \in S^I$ telle que Coh(e) < 1. Montrer que $I$ est fini. +Soit $m \in \N^*$. On munit $\mathbb{R}^m$ de sa structure euclidienne habituelle. On note $S$ sa sphère unité. Pour une famille $e=(e_i)_{i\in I}$ d'éléments de $S$ (éventuellement infinie), on note $\op{Coh}(e) = \sup_{(i,j) \in I^2} |\langle e_i, e_j \rangle|$. + 1. Rappeler sans démonstration l'inégalité de Cauchy-Schwarz. + 2. Soit $e \in S^I$. Que signifie l'égalité $\op{Coh}(e) = 0$ ? + 3. Soit $e \in S^I$ telle que $\op{Coh}(e) < 1$. Montrer que $I$ est fini. -Pour $t \in \mathbb{R}$, montrer que $\mathbf{E}(e^{t\langle X,Y\rangle}) \leq e^{t^2/2m}$. -d) Soit $Z$ une variable aléatoire réelle bornée. Montrer que $e^{tZ} \in L^1$ pour tout réel $t$. -d) Soient $X = (X_1, \dots, X_n)$ et $Y = (Y_1, \dots, Y_n)$ deux vecteurs aléatoires indépendants à + 4. Soient $X = (X_1, \dots, X_n)$ et $Y = (Y_1, \dots, Y_n)$ deux + vecteurs aléatoires indépendants à valeurs dans $S$, tels que pour + tout $i \in \db{1,m}$ les variables $\sqrt{n}X_i$ et $\sqrt{n}Y_i$ + soient de Rademacher, et $X_1,\ldots,X_n$ soient indépendantes + d'une part, $Y_1, \ldots, Y_n$ indépendantes d'autre part. -valeurs dans $S$, tels que pour tout $i \in [1, m]$ les variables $\sqrt{n}X_i$ et $\sqrt{n}Y_i$ soient de Rademacher (i.e., suivant la loi uniforme sur $\{-1,1\}$ ), et $X_1,\ldots,X_n$ soient indépendantes d'une + Pour $t \in \mathbb{R}$, montrer que $\mathbf{E}(e^{t\langle X,Y\rangle}) \leq e^{t^2/2m}$. -part, $Y_1, \ldots, Y_n$ indépendantes d'autre part. -e) Soit $\eps > 0$. Démontrer qu'il existe un ensemble fini $I$ de cardinal $\lfloor e^{m\eps^2/4} \rfloor$ et une famille $e \in S^I$ telle que $Coh(e) < \eps$. + 4. Soit $\eps > 0$. Démontrer qu'il existe un ensemble fini $I$ de + cardinal $\lfloor e^{m\eps^2/4} \rfloor$ et une famille $e \in + S^I$ telle que $\op{Coh}(e) < \eps$. #+end_exercice #+begin_exercice [Centrale MP 2025 # 1244] @@ -11422,20 +12201,19 @@ c) Pour n=3, $P_3(X)=X^3-X+1$. On note $\eta_1,\eta_2,\eta_3$ les racines de $P_ #+end_exercice #+begin_exercice [Centrale PSI 2025 # 1247] -Soient $n \geq1$, $a, b \in \mathbb{R}$ tels que a < $b$. On cherche à prouver l'existence et l'unicité de $(\alpha_0,\ldots,\alpha_n)\in\mathbb{R}^{n+1}$ tel que (1):$$\forall P \in \mathbb{R}_n[X], \int_a^b P(x) dx = \frac{b-a}{n} \sum_{k=0}^n \alpha_k P\left(a + k \frac{b-a}{n}\right).$$ +Soient $n \geq1$, $a, b \in \mathbb{R}$ tels que $a < b$. On cherche à prouver l'existence et l'unicité de $(\alpha_0,\ldots,\alpha_n)\in\mathbb{R}^{n+1}$ tel que $$(1)\colon \forall P \in \mathbb{R}_n[X], \int_a^b P(x) dx = \frac{b-a}{n} \sum_{k=0}^n \alpha_k P\left(a + k \frac{b-a}{n}\right).$$ -a) On suppose l'existence de $(\alpha_0, \dots, \alpha_n)$. Montrer que (1) est équivalent à (2) + 1. On suppose l'existence de $(\alpha_0, \dots, \alpha_n)$. Montrer que (1) est équivalent à $$(2)\colon \forall Q \in \mathbb{R}_n[X], \ \int_0^n Q(x) \, \mathrm{d}x = \sum_{k=0}^n \alpha_k Q(k).$$ -$$\forall Q \in \mathbb{R}_n[X], \ \int_0^n Q(x) \, \mathrm{d}x = \sum_{k=0}^n \alpha_k Q(k).$$ + En déduire que les $\alpha_k$ sont indépendants de $(a, b)$. -En déduire que les $\alpha_k$ sont indépendants de (a, b). + 2. Pour $i \in [0,n]$, soit $B_i = \prod_{\substack{0 \leqslant k + \leqslant n \\ k \neq i}} (X-k)$. Montrer que $(B_0,\ldots,B_n)$ + est une base de $\mathbb{R}_n[X]$. Montrer l'existence et + l'unicité de $(\alpha_0, \ldots, \alpha_n)$. -c) Pour $i \in [0,n]$, soit $B_i = \prod_{\substack{0 \leqslant k \leqslant n \\ k \neq i}} (X-k)$. Montrer que $(B_0,\ldots,B_n)$ est une base de - - $\mathbb{R}_n[X]$. Montrer l'existence et l'unicité de $(\alpha_0, \ldots, \alpha_n)$. - -d) Montrer que $\forall i \in [0, n], \alpha_i = \alpha_{n-i}$. -e) Calculer les $\alpha_i$ pour n=1, 2 et 3. + 3. Montrer que $\forall i \in [0, n], \alpha_i = \alpha_{n-i}$. + 4. Calculer les $\alpha_i$ pour $n=1, 2 et 3$. #+end_exercice #+begin_exercice [Centrale PSI 2025 # 1248] @@ -11495,7 +12273,7 @@ d) Trouver le spectre de $v$. #+begin_exercice [Centrale PSI 2025 # 1254] Soient $n \geq2$ et $E = \mathbb{R}_n[X]$. -a) Rappeler la formule de Taylor pour $P \in E$ et $a \in \mathbb{R}$ ; la démontrer dans le cas a = 0. +a) Rappeler la formule de Taylor pour $P \in E$ et $a \in \mathbb{R}$ ; la démontrer dans le cas $a = 0$. b) Soit $P \in E$. Montrer qu'il existe un unique $Q \in E$ tel que $\forall x \in \mathbb{R}$, $(x-1)Q(x) = \int_{1}^{x} P(t) dt$. @@ -11525,25 +12303,19 @@ c) Supposons $\Phi$ diagonalisable. La matrice $A$ est-elle diagonalisable? d) S #+end_exercice #+begin_exercice [Centrale PSI 2025 # 1257] -Soit -$$T = \begin{pmatrix} x & y \\ 0 & z \end{pmatrix}$$ +Soit $T = \begin{pmatrix} x & y \\ 0 & z \end{pmatrix}$ avec $(x, y, z) \in \mathbb{R}^3$. a) Pour tout $n \in \N$, donner l'expression de $T^n$. -b) Soit $E_n(T) = \sum_{k=0}^n \frac{T^k}{k!}$. Est-ce que $E_n(T)$ converge? On note $E(T)$ sa limite. Calculer - -E(T). Les valeurs propres de E(T) et $T$ peuvent-elles être égales? +b) Soit $E_n(T) = \sum_{k=0}^n \frac{T^k}{k!}$. Est-ce que $E_n(T)$ converge? On note $E(T)$ sa limite. Calculer E(T). Les valeurs propres de E(T) et $T$ peuvent-elles être égales? #+end_exercice #+begin_exercice [Centrale PSI 2025 # 1258] Soient -$I = [0, \pi/2]$ - et $E = \mathcal{C}^0(I, \mathbb{R})$ muni du produit scalaire $\langle f, g \rangle = \int_0^{\frac{\pi}{2}} f(t) g(t) dt$. +$I = [0, \pi/2]$ et $E = \mathcal{C}^0(I, \mathbb{R})$ muni du produit scalaire $\langle f, g \rangle = \int_0^{\frac{\pi}{2}} f(t) g(t) dt$. -Si -$f \in E$ -, on pose $A(f): x \in I \mapsto \int_0^x f(t) dt$ et $B(f): x \in I \mapsto \int_x^{\frac{\pi}{2}} f(t) dt$. +Si $f \in E$, on pose $A(f): x \in I \mapsto \int_0^x f(t) dt$ et $B(f): x \in I \mapsto \int_x^{\frac{\pi}{2}} f(t) dt$. a) Montrer que, pour tous $f$ et $g$ de $E$, $\langle A(f), g \rangle = \langle f, B(g) \rangle$. En déduire que les valeurs propres réelles de $B \circ A$ sont positives. @@ -11659,10 +12431,10 @@ c) On pose, pour $n \in \N^*$, $g_n = \prod_{k=1}^n \left(1 e^{-\frac{\alpha}{k} #+end_exercice #+begin_exercice [Centrale PSI 2025 # 1268] -Soit $g: x \in \mathbb{R} \mapsto x x^2$. -a) Déterminer le plus grand intervalle $I$ contenant 0 tel que $g|_I$ soit injective. -b) On pose $J$ = g(I) et $f$ la réciproque de $g|_{I}$. Déterminer l'expression de $f$. -c) Montrer que $f$ admet un développement en série entière au voisinageqde 0 et l'expliciter. +Soit $g\colon x \in \mathbb{R} \mapsto x- x^2$. + 1. Déterminer le plus grand intervalle $I$ contenant 0 tel que $g|_I$ soit injective. + 2. On pose $J = g(I)$ et $f$ la réciproque de $g|_{I}$. Déterminer l'expression de $f$. + 3. Montrer que $f$ admet un $DL_n(0)$ et l'expliciter. #+end_exercice #+begin_exercice [Centrale PSI 2025 # 1269] @@ -12324,7 +13096,7 @@ b) Déterminer l'espérance et la variance de la plus grande valeur propre de $M ** Algèbre #+begin_exercice [IMT # 1354] -Soient a et $n$ deux entiers supérieurs ou égaux à a. Montrer que si $a^n 1$ est premier, alors a = 2 et $n$ est un nombre premier. +Soient $a$ et $n$ deux entiers supérieurs ou égaux à $a$. Montrer que si $a^n - 1$ est premier, alors $a = 2$ et $n$ est un nombre premier. #+end_exercice #+begin_exercice [IMT # 1355] @@ -12334,7 +13106,8 @@ Résoudre l'équation $x^2 + x + \overline{1} = \overline{0}$ dans $\mathbb{Z}/6 #+begin_exercice [IMT # 1356] Soit $A \in \mathcal{M}_3(\mathbb{R})$ telle que $A^2 = 0$ et $A \neq 0$. -Montrer que $A$ est semblable à $\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$. + 1. Montrer que $\rg A = 1$. + 2. Montrer que $A$ est semblable à $\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$. #+end_exercice # ID:8547 @@ -12397,10 +13170,11 @@ c) Étudier la diagonalisabilité de A. #+begin_exercice [CCINP # 1365] Soient $n \in \N^*$ et $A, B \in \mathbb{R}_n[X]$ non nuls. On considère l'application $f$ qui à $P \in \mathbb{R}_n[X]$ associe le reste de la division euclidienne de AP par $B$. -a) Montrer que $f$ est un endomorphisme. Est-ce un automorphisme? -b) On note $p$ le degré de $B$ et $\lambda_1, \ldots, \lambda_n$ ses racines. - - i) Montrer que 0 est valeur propre de $f$. - - ii) Soit $\alpha \in \mathbb{R}^*$ une valeur propre de $f$. Montrer qu'il existe $i \in [1, p]$ tel que $A(\lambda_i) = \alpha$.iii) L'endomorphisme $f$ est-il diagonalisable? + 1. Montrer que $f$ est un endomorphisme. Est-ce un automorphisme? + 2. On note $p$ le degré de $B$ et $\lambda_1, \ldots, \lambda_n$ ses racines. + 1. Montrer que 0 est valeur propre de $f$. + 2. Soit $\alpha \in \mathbb{R}^*$ une valeur propre de $f$. Montrer qu'il existe $i \in [1, p]$ tel que $A(\lambda_i) = \alpha$. + 3. L'endomorphisme $f$ est-il diagonalisable? #+end_exercice #+begin_exercice [IMT # 1366] @@ -12418,11 +13192,11 @@ b) En s'intéressant à $h\mapsto h\circ g-g\circ h$, montrer que $f$ est nilpot #+begin_exercice [CCINP # 1368] Soit $A = \left(\frac{i}{j}\right)_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_n(\mathbb{R})$. -a) La matrice $A$ est-elle inversible? -b) Trouver un polynôme annulateur de A. -c) Montrer que $A$ est diagonalisable et donner ses valeurs propres. -d) Donner les sous-espaces propres de A. -e) Soit $M \in \mathcal{M}_n(\mathbb{R})$ commutant avec A. Montrer que $\operatorname{Ker}(A)$ et $\operatorname{Im}(A)$ sont stables par $M$. + 1. La matrice $A$ est-elle inversible? + 2. Trouver un polynôme annulateur de $A$. + 3. Montrer que $A$ est diagonalisable et donner ses valeurs propres. + 4. Donner les sous-espaces propres de $A$. + 5. Soit $M \in \mathcal{M}_n(\mathbb{R})$ commutant avec $A$. Montrer que $\operatorname{Ker}(A)$ et $\operatorname{Im}(A)$ sont stables par $M$. #+end_exercice #+begin_exercice [IMT # 1369] @@ -12444,10 +13218,12 @@ Soit $A \in \mathcal{M}_n(\C)$ de rang 1. Montrer que $A$ est diagonalisable si #+begin_exercice [CCINP # 1372] Soit $f \in \mathcal{L}(\mathbb{R}^3)$ vérifiant $f^3 + f = 0$ et $f \neq 0$. -a) Montrer que $\mathbb{R}^3 = \operatorname{Ker} f \oplus \operatorname{Ker} (f^2 + \operatorname{id})$. -b) Soit $x \in \text{Ker}(f^2 + \text{id})$ non nul. Montrer que (x, f(x)) est libre. -c) Montrer qu'il existe une base de $\mathbb{R}^3$ dans laquelle la matrice de $f$ est $\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & -1 \end{pmatrix}$. -d) Construire $u \in \mathcal{L}(\mathbb{R}^3)$ tel que $u^2 = f$. + 1. Montrer que $\mathbb{R}^3 = \operatorname{Ker} f \oplus \operatorname{Ker} (f^2 + \operatorname{id})$. + 2. Soit $x \in \text{Ker}(f^2 + \text{id})$ non nul. Montrer que $(x, f(x))$ est libre. + 3. Montrer qu'il existe une base de $\mathbb{R}^3$ dans laquelle la + matrice de $f$ est $\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & + 1 & 0 \end{pmatrix}$. + 4. Construire $u \in \mathcal{L}(\mathbb{R}^3)$ tel que $u^2 = f$. #+end_exercice #+begin_exercice [IMT # 1373] @@ -12478,7 +13254,8 @@ ${\it b}$ ) Calculer les sous-espaces propres éventuels de $B$ et donner leur d #+begin_exercice [Navale # 1376] On munit l'espace vectoriel $\mathcal{M}_n(\mathbb{R})$ du produit scalaire canonique. -Calculer la distance de la matrice $M=(1)_{1\leqslant i,j\leqslant n}$ à l'espace $F$ des matrices de trace nulle. +Calculer la distance de la matrice $M=(1)_{1\leqslant i,j\leqslant n}$ +à l'espace $F$ des matrices de trace nulle. #+end_exercice #+begin_exercice [Navale # 1377] @@ -12496,11 +13273,9 @@ c) Relier les polynômes caractéristiques de $MM^T$ et de $M^TM$. #+begin_exercice [CCINP # 1379] On munit $\mathcal{M}_n(\mathbb{R})$ de son produit scalaire canonique. -a) Montrer que $S_n(\mathbb{R})$ et $A_n(\mathbb{R})$ sont supplémentaires orthogonaux + 1. Montrer que $S_n(\mathbb{R})$ et $A_n(\mathbb{R})$ sont supplémentaires orthogonaux -b) Déterminer la distance de -$$M=\begin{pmatrix}0&2&1\\2&0&2\\-1&-1&0\end{pmatrix}$$ - à $\mathcal{S}_3(\mathbb{R})$. + 2. Déterminer la distance de $$M=\begin{pmatrix}0&2&1\\2&0&2\\-1&-1&0\end{pmatrix}$$ à $\mathcal{S}_3(\mathbb{R})$. #+end_exercice #+begin_exercice [CCINP # 1380] @@ -12603,7 +13378,7 @@ Soient $a, b, c \in \mathbb{R}$. Pour $n \in \N^*$, on pose $u_n = a \ln(n) + b #+end_exercice #+begin_exercice [IMT # 1395] -On pose, pour $n \geq 3$, $u_n = \ln\left(\frac{n^4 2n^3 + 2n 1}{n^4 2n^3}\right)$. Étudier la nature de la série de terme général $u_n$ et calculer sa somme en cas de convergence. +On pose, pour $n \geq 3$, $u_n = \ln\left(\frac{n^4 - 2n^3 + 2n - 1}{n^4 - 2n^3}\right)$. Étudier la nature de la série de terme général $u_n$ et calculer sa somme en cas de convergence. #+end_exercice # ID:8804 @@ -12909,7 +13684,11 @@ Une urne contient a boules blanches et $b$ boules noires. On tire simultanément #+end_exercice #+begin_exercice [CCINP # 1438] -Lors d'une compétition de saut en hauteur, un participant saute à plusieurs reprises et, à l'instant $n$, a une chance sur $n$ de réussir son saut. S'il chute, la compétition s'arrête pour lui. On note $X$ le nombre de sauts réussis. Quelle est la loi de X? Existence et valeur de $\mathbf{E}(X)$ et de $\mathbf{V}(X)$ ? +Lors d'une compétition de saut en hauteur, un participant saute à +plusieurs reprises et, à l'instant $n$, a une chance sur $n$ de +réussir son saut. S'il chute, la compétition s'arrête pour lui. On +note $X$ le nombre de sauts réussis. 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