From 00c04f59473acf77562767cb564a549a904f6e5f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?S=C3=A9bastien=20Miquel?= Date: Wed, 24 Apr 2024 19:50:11 +0200 Subject: [PATCH] Miscs --- Exercices 2022.org | 4 +- Exercices 2023.org | 211 ++++++++++++++++++++++++++----------- Exercices 2023.pdf | Bin 1109064 -> 1122995 bytes Exercices XENS MP 2023.pdf | Bin 710806 -> 719233 bytes 4 files changed, 152 insertions(+), 63 deletions(-) diff --git a/Exercices 2022.org b/Exercices 2022.org index cfac00d..76591a3 100644 --- a/Exercices 2022.org +++ b/Exercices 2022.org @@ -1,7 +1,7 @@ #+title: Exercices 2022 #+author: Sébastien Miquel #+date: 25-02-2023 -# Time-stamp: <12-02-24 12:40> +# Time-stamp: <16-04-24 18:59> * Meta :noexport: @@ -3784,7 +3784,7 @@ Soient $m,n\geq 2$, $p\in \interval]{0, 1}[$ et $q = 1-p$. Soit $(X_n)$ une suit # 294 #+call: get_exa(6042) -#+BEGIN_exercice $\bigstar$ $\bigstar$ [X 2022] +#+BEGIN_exercice $\bigstar$ [X 2022] Soit $\sigma_n$ une variable aléatoire suivant la loi uniforme sur $\mc S_n$. 1. Soit $L_n$ la longueur du cycle de $\sigma_n$ contenant $1$. Déterminer l'espérance de $L_n$. 2. Quelle est la probabilité que $1$ et $2$ soient dans un même cycle de $\sigma_n$ ? diff --git a/Exercices 2023.org b/Exercices 2023.org index 262ebf0..da23f89 100644 --- a/Exercices 2023.org +++ b/Exercices 2023.org @@ -2,7 +2,7 @@ #+title: Exercices 2023 #+author: Sébastien Miquel #+date: 02-12-2023 -# Time-stamp: <14-04-24 20:04> +# Time-stamp: <24-04-24 18:12> #+OPTIONS: @@ -53,7 +53,7 @@ #+END_SRC #+RESULTS: -| 1 | 8 | 937 | +| 2 | 27 | 945 | #+BEGIN_SRC emacs-lisp (defun find_bad_hash () @@ -337,10 +337,16 @@ On veut que les $\sin (\frac{\theta_i - \theta_j}{2})$ soient rationnels, c'est- Il suffit donc de prendre les doubles d'une infinité de points rationnels sur le cercle. #+END_proof -#+BEGIN_exercice [ENS 2023 # 20] :sup: +#+BEGIN_exercice [ENS 2023 # 20] Anneau des entiers algébriques :sup: Soit $z\in\C$ annulé par un polynôme unitaire à coefficients entiers. Soit $Q\in\Z[X]$. Montrer que $Q(z)$ est annulé par un polynôme unitaire à coefficients entiers. #+END_exercice #+BEGIN_proof +On note $P$ le polynôme unitaire qui annule $z$ (polynôme minimal, via lemme de Gauss). + +Pour $z^2$, je vois mal quoi faire, si ce n'est $P = \prod (X - x_i^2)$. + +Par dimension, on sait que $Q(z)$ admet un polynôme annulateur dans $\Q[X]$. + !! #+END_proof @@ -458,11 +464,11 @@ Soit $n\in\N^*$, $\K$ un corps, et $V$ un sous-espace vectoriel de $\M_n(\K)$ do Quelle est la dimension maximale d'un sous-espace vectoriel $V$ de $\M_n(\R)$ tel que pour tout $(X,Y)\in V^2$, on ait $\op{Tr} (XY) = 0$. #+END_exercice #+BEGIN_proof -!! - On a $X\bot X^T$, donc la dimension de $X$ est $\leq \frac{n^2}{2}$. -Réciproquement. +Réciproquement. Dans $\M_n(\C)$, on prend une diagonale, où le second coefficient est $i\times$ le premier etc. + +Dans $\M_2(\C)$ : $\begin{pmatrix}a & b \\ 0 & ia\end{pmatrix}$. On cherche une forme réelle : $ia = \ol{a}$ donne $u + \frac{\pi}{2} = - u$, c'est-à-dire $u = - \frac{\pi}{4}$. Donc $\begin{pmatrix}1 & -1 \\ 1 & 1\end{pmatrix}$ et idk, il faudrait écrire les équations pour l'autre matrice. #+END_proof @@ -490,13 +496,19 @@ Soient $n\geq 1$ et $E$ une partie de $\mc P(\db{1,n})$. #+END_proof -#+BEGIN_exercice [ENS 2023 # 39] +#+BEGIN_exercice [ENS 2023 # 39] :sup: Soient $(a_1,\dots, a_n)\in\R^n$ telle que $|a_i|\geq 2$, pour tout $i\in\db{1,n}$. 1. Soit $A\in\M_n(\R)$ telle que $\forall i,\, a_{ii} = a_i$, $a_{ij} = 1$ si $|i-j| = 1$ et $a_{ij} = 0$ sinon. Montrer que $A$ est inversible et que son déterminant a le même signe que $\prod a_k$. 2. Montrer que la conclusion tient encore si l'on suppose $|a_{ij}|\leq 1$ si $|i-j| = 1$ au lieu de $a_{ij} = 1$. #+END_exercice #+BEGIN_proof -!! + 1. $A$ est inversible car diagonale dominante. + + Le signe du déterminant s'obtient en augmentant les coefficients, ou plutôt en diminuant les autres. + + Par récurrence ? On a $\Delta_n = a_n \Delta_{n-1} - \Delta_{n-2}$ + On montre que $|\Delta_n|\gt |\Delta_{n-1}|$ + le signe, ça passe par récurrence. + 2. … C'est clair. #+END_proof @@ -508,7 +520,10 @@ On considère $\phi:\left(\R^4\right)^2 \ra \M_4(\R)$ qui à $(u, v)$ associe la 4. Décrire l'image et le noyau d'une telle matrice. #+END_exercice #+BEGIN_proof - + 1. + 2. + 3. + 4. #+END_proof @@ -1110,13 +1125,36 @@ $B_{\epsilon}=\{t\in\R\ ;\ \forall i\in I,\op{sign}(P_i(t))\in\{ \eps(i),0\}\}$. - Si $A_{\eps}$ est non vide, montrer que $B_{\eps}$ est l'adhérence de $A_{\eps}$. Si $A_{\eps}$ est vide, montrer que $B_{\eps}$ est soit vide suit un singleton. #+end_exercice +# ID:7008 #+begin_exercice [ENS 2023 # 109] -Soit $I$ un intervalle de $\R$ et $f:I\ra\R$ de classe $\mc C^n$. - - Soient $x_0,\ldots,x_n$ des points de $I$. On note $V(x_0,\ldots,x_n)$ le déterminant de Vandermonde associe a $(x_0,\ldots,x_n)$. Montrer qu'il existe $\tau\in I$ tel que +Soit $I$ un intervalle de $\R$ et $f\colon I\ra\R$ de classe $\mc C^n$. + - Soient $x_0,\ldots,x_{n-1}$ des points de $I$. + - sV2 Soit $P$ le polynôme d'interpolation de Lagrange de $f$ aux points $x_0, \dots, x_{n-1}$. Montrer que pour tout $x\in I$, il existe $c\in I$ tel que + $$f(x) - P(x) = \frac{f^{(n)}(c)}{n!} \prod_{i=0}^{n-1} (x - x_i).$$ + - On note $V(x_0,\ldots,x_n)$ le déterminant de Vandermonde associe a $(x_0,\ldots,x_n)$. Montrer qu'il existe $\tau\in I$ tel que - $\begin{vmatrix}1&x_0&x_0^2&\cdots&x_0^{n-1}&f(x_0)\\ 1&x_1&x_1^2&\cdots&x_1^{n-1}&f(x_1)\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\ 1&x_n&x_n^2&\cdots&x_n^{n-1}&f(x_n)\\ \end{vmatrix}=\dfrac{f^{(n)}(\tau)}{n!}\,V(x_0,x_1,\ldots,x_n)$ + $$\begin{vmatrix}1&x_0&x_0^2&\cdots&x_0^{n-1}&f(x_0)\\ 1&x_1&x_1^2&\cdots&x_1^{n-1}&f(x_1)\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\ 1&x_n&x_n^2&\cdots&x_n^{n-1}&f(x_n)\\ \end{vmatrix}=\dfrac{f^{(n)}(\tau)}{n!}\,V(x_0,x_1,\ldots,x_n).$$ - On suppose que $n=2$, que $I$ est un segment et que $f$ est strictement convexe. On note $\Gamma_f=\{(x,f(x));x\in I\}\subset\R^2$ le graphe de $f$. Montrer qu'il existe une constante $C$, dépendant uniquement de $I$ et $f$, telle que le nombre de points de $\Gamma_f\cap\frac{1}{N}\,\Z^2$ soit majore par $C\,N^{2/3}$ pour tout entier $N\geq 1$. #+end_exercice +#+BEGIN_proof + 1. + 1. + 2. On part du déverminant de Vandermonde. Par des opérations sur les colonnes, on transforme la dernière en $P(x_i) - f(x_i)$, où $P$ est un polynôme de degré $\leq n-1$. + + On choisit pour $P$ le polynôme d'interpolation de $f$ en $x_0,\dots, x_{n-1}$. + + Alors le déterminant vaut $\big(f(x_n) - P(x_n)\big) V(x_0,\dots, x_{n-1})$. + + Par ailleurs, on sait que $f(x_n) - P(x_n) = \frac{\la}{n!} (x_n - x_1)\dots (x_n - x_{n-1})$ (choisir $\la$ pour que ce soit correct en $x_n$), et on obtient $\la = f^{(n)}(\tau)$. + 2. En trois points $\frac{i}{N}, \frac{j}{N}, \frac{k}{N}$ qui vérifient la condition, on a + $$\begin{vmatrix}1 & i & f(x_i) N \\ 1 & j & f(x_j)N \\ 1 & k & f(x_k)N \end{vmatrix} = \frac{f^{(2)}(\tau)}{2} \begin{vmatrix}1 & i & i^2 \\ 1 & j & j^2 \\ 1 & k & k^2 \end{vmatrix},$$ + donc $(i-j)(j-k)(i-k) = \frac{N k}{f^{(2)}(\tau)}$, si $f^{(2)}(\tau)\neq 0$, où $k$ est un entier. + + En particulier, il existe une constante $C$ telle que l'un des trois facteur soit $\geq C N^{1/3}$. Cela implique la borne en $N^{2/3}$. + + Si $f^{(2)}$ s'annule, on applique ce qui précède sur chaque deux intervalles sur lequel ce n'est pas le cas. +#+END_proof + #+begin_exercice [ENS 2023 # 110] Pour $n\in\N$, on pose $w_n=\int_0^{\pi/2}\cos^n(x)\,dx$. @@ -1158,18 +1196,31 @@ Soit $f\colon \R^+ \ra \R$ continue et presque périodique. Soit $f$ une fonction continue par morceaux et croissante de $[0,1]$ dans $\R$. Montrer que $\int_0^1f(x)e^{i\lambda x}dx\underset{\lambda\to+\i}{=}O \left(\frac{1}{\lambda}\right)$. #+end_exercice +# ID:7007 #+begin_exercice [ENS 2023 # 115] -Soient $f_1,\dots,f_n,g_1,\dots,g_n$ des fonctions de $\mc C^0([0,1],\R)$. Soit $A$ la matrice de terme général $A_{i,j}=\int_0^1f_i(x)g_j(x)\,dx$. + - Es On admet l'existence d'une notion d'intégrale multiple sur un rectangle de $\R^n$, et que pour des fonctions continues $f_1,\dots, f_n\colon\R\ra\R$, + $$\int_{[0,1]^n} f_1(x_1)\dots f_n(x_n)\dx_1\dots \dx_n = \prod_{i=1}^n \int_{0}^1 f_i(x)\dx.$$ -On pose $B(x_1,\dots,x_n)=\det\big{(}f_i(x_j)\big{)}$ et $C(x_1,\dots,x_n)=\det\big{(}g_i(x_j)\big{)}$.Montrer que $\int_{[0,1]^n}B(x_1,\ldots,x_n)\,C(x_1,\ldots,x_n)\,dx_1\ldots dx_n=n!\det(A)$. + - Soient $f_1,\dots,f_n,g_1,\dots,g_n$ des fonctions de $\mc C^0([0,1],\R)$. Soit $A$ la matrice de terme général $A_{i,j}=\int_0^1f_i(x)g_j(x)\dx$. + + On pose $B(x_1,\dots,x_n)=\det\big(f_i(x_j)\big)$ et $C(x_1,\dots,x_n)=\det\big(g_i(x_j)\big)$. Montrer que $\int_{[0,1]^n}B(x_1,\ldots,x_n)\,C(x_1,\ldots,x_n)\,dx_1\ldots dx_n=n!\det(A)$. + - s Montrer que si $\det A = 0$, alors la famille $(f_1,\dots, f_n)$ est liée. + - s En déduire que si $(f_1,\dots,f_n)$ est libre, il existe $(x_1,\dots, x_n)\in\R^n$ tels que $\big(f_i(x_j)\big)$ soit inversible. #+end_exercice +#+BEGIN_proof +Le produit des déterminants est $\sum_{\sigma,\sigma'} \eps(\sigma) \eps(\sigma')\prod_{i=1}^n f_i(x_{\sigma(i)}) \prod_{i=1}^n g_i(x_{\sigma'(i)}) = \sum_{\sigma,\sigma'} \eps(\sigma) \eps(\sigma') \prod_{i=1}^n f_{\sigma(i)}(x_i) g_{\sigma'(i)}(x_i)$. -#+begin_exercice [ENS 2023 # 116 - La fonction $f:x\in[1,+\i[\,\mapsto\frac{\sin(x^2)}{x}$ est-elle uniformement continue?] -- Soit $f$ une fonction de classe $C^1$ de $\R^+$ dans $\R$ admettant une limite en $+\i$ et telle que $f'$ est uniformément continue. Est-ce que $f'$ a une limite en $+\i$? +Quand on l'intègre, on obtient $\sum_{\sigma,\sigma'} \eps(\sigma') \eps(\sigma) \prod_{i=1}^n \int_0^1 f_{\sigma(i)}(x) g_{\sigma'(i)}(x_i)$, et l'intégrale ne dépend que de $\sigma^{-1}\circ \sigma$, ce qui permet d'obtenir le résultat. +#+END_proof + + +#+begin_exercice [ENS 2023 # 116] + - La fonction $f:x\in[1,+\i[\,\mapsto\frac{\sin(x^2)}{x}$ est-elle uniformement continue? + - Soit $f$ une fonction de classe $C^1$ de $\R^+$ dans $\R$ admettant une limite en $+\i$ et telle que $f'$ est uniformément continue. Est-ce que $f'$ a une limite en $+\i$? #+end_exercice #+begin_exercice [ENS 2023 # 117] -[Rennes sur dossier] Soient $d,N\in\N$ tels que $N\gt d$. Soient $(P - {n\in\N}$ une suite de polynômes a coefficients réels de degré au plus $d$ et $x_1,...,x_N$ des réels distincts. On suppose que pour tout $j\in\{1,...,N\}$, la suite $(P_n(x_j))_{n\in\N}$ est bornée. Montrer que l'on peut extraire de $(P - {n\in\N}$ une suite $(Q - {n\in\N}$ qui converge uniformément sur $[0,1]$ vers un polynôme de degré au plus $d$. +Soient $d,N\in\N$ tels que $N\gt d$. Soient $(P - {n\in\N}$ une suite de polynômes a coefficients réels de degré au plus $d$ et $x_1,...,x_N$ des réels distincts. On suppose que pour tout $j\in\{1,...,N\}$, la suite $(P_n(x_j))_{n\in\N}$ est bornée. Montrer que l'on peut extraire de $(P - {n\in\N}$ une suite $(Q - {n\in\N}$ qui converge uniformément sur $[0,1]$ vers un polynôme de degré au plus $d$. #+end_exercice #+begin_exercice [ENS 2023 # 118] @@ -1426,7 +1477,7 @@ On considère l'équation différentielle $(1)\colon X'(t)=P(t)X(t)$ où $P$ est #+begin_exercice [ENS 2023 # 149] -[Rennes sur dossier] Soient $q\in\R^+$, $D=\{(x,y)\in\R^2\,;\,x\geq 0,\,y\geq 0,\,x+y=1\}$, Déterminer $\min_{(x,y)\in D}(x^q+y^q)$. +Soient $q\in\R^+$, $D=\{(x,y)\in\R^2\,;\,x\geq 0,\,y\geq 0,\,x+y=1\}$, Déterminer $\min_{(x,y)\in D}(x^q+y^q)$. #+end_exercice #+begin_exercice [ENS 2023 # 150] @@ -1502,7 +1553,7 @@ Les éléments de $G$ sont de la forme $z\mapsto az + b$, ou $z\mapsto a\ol{z} + On considère $G^+$ (isométries affines qui préservent l'orientation), dont les éléments ont un unique point fixe. Si il existe deux éléments qui ne commutent pas dans $G^+$, on s'en sort : conjuguer, puis multiplier par l'inverse. Par ailleurs, deux éléments commutent si et seulement si ils ont le même point fixe. -Si tous les éléments de $G^+$ ont le même point fixe, tout élément de $G^{-}$ (qui a une droite de points fixes) doit préserver ce point, sinon on créerai d'autres éléments de $G^+$ avec un point fixe différent. +Si tous les éléments de $G^+$ ont le même point fixe, tout élément de $G^{-}$ (qui a une droite de points fixes) doit préserver ce point, sinon on créerai d'autres éléments de $G^+$ avec un point fixe différent. #+END_proof @@ -1519,29 +1570,43 @@ Sinon, il existe une suite $(a_n)$ qui s'accumule. On peut supposer qu'elle s'ac #+begin_exercice [ENS 2023 # 159] Soit $L$ la courbe du plan complexe d'équation $|z|^2=\cos(2\arg(z))$. - Trouver une équation cartésienne réelle définissant $L$. - - En déduire une paramétrisation de $L\cap(\R^+)^2$ sous la forme $\{(x(r),y(r)),\ r\in[0,1]\}$. - Montrer que la longueur de la courbe $L$ entre le point $(0,0)$ et le point $(x(r),y(r))$ s'écrit : $A(r)=\int_0^r\frac{1}{\sqrt{1-t^2}}dt$. - - Montrre que $A$ définit une bijection de $[-1,1]$ dans un intervalle de la forme $[-w,w]$ ou $w\gt 0$. + - En déduire une paramétrisation de $L\cap(\R^+)^2$ sous la forme $\{(x(r),y(r)),\ r\in[0,1]\}$. + - Montrer que la longueur de la courbe $L$ entre le point $(0,0)$ et le point $(x(r),y(r))$ s'écrit : $A(r)=\int_0^r\frac{1}{\sqrt{1-t^2}}dt$. + - Montrer que $A$ définit une bijection de $[-1,1]$ dans un intervalle de la forme $[-w,w]$ ou $w\gt 0$. - On définit $B=A^{-1}$. Montrer que $B$ vérifie une équation différentielle du second ordre. #+end_exercice +# ID:7009 #+begin_exercice [ENS 2023 # 160] -Soit $(e_1,e_2)$ une famille libre de vecteurs de $\R^2$. On pose $L=\Ze_1+\Ze_2$ et on note $\mathrm{Vol}(L)=|\mathrm{det}(e_1,e_2)|$. - - Soit $A$ un disque ferme de $\R^2$, d'aire strictement supérieure a $\mathrm{Vol}(L)$. Montrer qu'il existe deux éléments distincts $x$ et $y$ de $A$ tels que $x-y\in L$. - - Soit $\eps\gt 0$. Montrer qu'il existe dans $L\setminus\{0\}$ un élément $\ell$ tel que $\|\ell\|\leq\frac{2+\eps}{\sqrt{\pi}}\ \sqrt{\mathrm{Vol}(L)}$. +Soit $(e_1,e_2)$ une famille libre de vecteurs de $\R^2$. On pose $L=\Ze_1+\Ze_2$ et on note $\op{coVol}(L)=|\mathrm{det}(e_1,e_2)|$. + - sV2 Pour $e_1 = \vv{2}{0}$ et $e_2 = \vv{1}{1}$, représenter $L$ dans $\R^2$. Que représente géométriquement le covolume $\op{coVol}(L)$ ? + - Soit $A$ un disque fermé de $\R^2$, d'aire strictement supérieure a $\op{coVol}(L)$. Montrer qu'il existe deux éléments distincts $x$ et $y$ de $A$ tels que $x-y\in L$. + + Indication : Les parallélogramme du réseau $L$ découpent le disque $A$ en un nombre fini de morceaux $A_i$. Considérer les translatés $A_i'$ des $A_i$, déplacés dans le parallélogramme à l'origine de $\R^2$, appliquer l'hypothèse. + - Soit $\eps\gt 0$. Montrer qu'il existe dans $L\setminus\{0\}$ un élément $\ell$ tel que $\|\ell\|\leq\frac{2+\eps}{\sqrt{\pi}}\ \sqrt{\op{coVol}(L)}$. - Soit $p$ un nombre premier congru a $1$ modulo $4$. - - Montrer qu'il existe $\omega\in\Z$ tel que $p$ divise $1+\omega^2$. - - Montrer qu'il existe $(a,b)\in\Z^2$ tel que $p=a^2+b^2$. + - En considérant $m = (p-1)!$, montrer qu'il existe $\omega\in\Z$ tel que $p$ divise $1+\omega^2$. + - En utilisant la question précédente, montrer qu'il existe $(a,b)\in\Z^2$ tel que $p=a^2+b^2$. #+end_exercice +#+BEGIN_proof + - + - + - + - + - + - +#+END_proof + #+begin_exercice [ENS 2023 # 161] -- On note $D$ le disque unite du plan euclidien $\R^2$. Démontrer qu'il existe une suite $(C - {i\in\N}$ de parties de $D$ telle que : - - pour tout $i\in\N$, l'ensemble $C_i$ soit un carré de $\R^2$ dont les cotes sont parallèles aux axes ; - - les $C_i$ soient d'intérieurs disjoints ; - - $\sum_{i\in\N}\mathrm{Aire}(C_i)=\pi$. - - On note $C=[-1,1]^2$. Démontrer qu'il existe une suite $(D - {i\in\N}$ de parties de $C$ telle que : - - pour tout $i\in\N$, l'ensemble $D_i$ soit un disque ferme de $\R^2$ ; - - les $D_i$ soient d'intérieurs disjoints ; - - $\sum_{i\in\N}\mathrm{Aire}(D_i)=4$. + - On note $D$ le disque unite du plan euclidien $\R^2$. Démontrer qu'il existe une suite $(C_i)_{i\in\N}$ de parties de $D$ telle que : + + pour tout $i\in\N$, l'ensemble $C_i$ soit un carré de $\R^2$ dont les cotes sont parallèles aux axes ; + + les $C_i$ soient d'intérieurs disjoints ; + + $\sum_{i\in\N}\mathrm{Aire}(C_i)=\pi$. + - On note $C=[-1,1]^2$. Démontrer qu'il existe une suite $(D_i)_{i\in\N}$ de parties de $C$ telle que : + + pour tout $i\in\N$, l'ensemble $D_i$ soit un disque ferme de $\R^2$ ; + + les $D_i$ soient d'intérieurs disjoints ; + + $\sum_{i\in\N}\mathrm{Aire}(D_i)=4$. #+end_exercice ** Probabilités @@ -1568,7 +1633,7 @@ On peut alors transformer $(i_1 i_2) (j_1 j_2)$ en $(i_1 j_2) (j_1 i_2)$, qui pr # ID:6834 -#+begin_exercice [ENS 2023 # 164] +#+begin_exercice [ENS 2023 # 164] :sup: Un derangement est une permutation $\sigma\in\mc{S}_n$ sans point fixe. On note $D_n$ le sous-ensemble de $\mc{S}_n$ formé des derangements. - Soit $X$ une variable aléatoire suivant la loi uniforme sur $D_n$. Calculer la probabilité que $X$ soit une permutation paire. @@ -2137,7 +2202,7 @@ ii) $\forall x\gt 0,\ P_2(x)\lt P_4(x)\lt P_1(x)\lt P_3(x)$. #+end_exercice #+begin_exercice [ENS PC 2023 # 222] -[PL] Soit $E=\M_n(\R)$. Calculer le déterminant de l'application $\Phi\colon M\in E\mapsto M^T\in E$. +Soit $E=\M_n(\R)$. Calculer le déterminant de l'application $\Phi\colon M\in E\mapsto M^T\in E$. #+end_exercice #+begin_exercice [ENS PC 2023 # 223] @@ -2847,12 +2912,19 @@ Soit $t_1, \ldots, t_n$ des réels. #+begin_exercice [X MP 2023 # 323] On munit $\R^n$ de son produit scalaire standard et on note $\|A\|=\sup_{X\in B_f(0,1)}\|AX\|$ pour $A\in\M_n(\R)$. - - Montrer que $\|\!|\!|\!|\!|\!|$ définit une norme sur $\M_n(\R)$. - - Montrer que $\|\!|\!|A|\!|\!|=\sup_{(X,Y)\in B_f(0,1)^2}|\left\langle AX,Y\right\rangle|$. - - On prend $A=\Big{(}\dfrac{1}{i+j+1}\Big{)}_{0\leq i,j\leq n}$ dans $\M_{n+1}(\R)$. Pour $X=(x_0\cdots x_n)^T$ et $Y=(y_0\cdots y_n)^T$ dans $\R^{n+1}$, donner une interpretation de $\left\langle AX,Y\right\rangle$ a l'aide d'une intégrale faisant intervenir $P:t\in[0,2\pi]\mapsto\sum_{k=0}^nx_ke^{ikt}$ et $Q:t\in[0,2\pi]\mapsto\sum_{k=0}^ny_ke^{ikt}$. + - Montrer que $\lN \cdot\rN$ définit une norme sur $\M_n(\R)$. + - Montrer que $\lN A\rN=\sup_{(X,Y)\in B_f(0,1)^2}|\left\langle AX,Y\right\rangle|$. + - On prend $A=\Big(\dfrac{1}{i+j+1}\Big)_{0\leq i,j\leq n}$ dans $\M_{n+1}(\R)$. Pour $X=(x_0\cdots x_n)^T$ et $Y=(y_0\cdots y_n)^T$ dans $\R^{n+1}$, donner une interpretation de $\left\langle AX,Y\right\rangle$ a l'aide d'une intégrale faisant intervenir $P:t\in[0,2\pi]\mapsto\sum_{k=0}^nx_ke^{ikt}$ et $Q:t\in[0,2\pi]\mapsto\sum_{k=0}^ny_ke^{ikt}$. - En déduire que $\|\!|\!|A|\!|\leq 2\pi$. - Montrer que l'on a meme $\|\!|A|\!|\!|\leq\pi$. #+end_exercice +#+BEGIN_proof + - + - + - $\langle AX, Y\rangle = $ +#+END_proof + + ** Analyse @@ -4186,7 +4258,7 @@ Déterminer les extrema de $f\colon (x,y)\mapsto 3x^2+2xy+2y^2-x^4$ sur le disqu *** Probabilités #+begin_exercice [X PC 2023 # 511] -On a un de equilibre a $N$ faces numerotées de 1 a $N$, et on effectue une suite de lancers indépendants. Le jeu s'arrete lorsque le résultat du lancer $n+1$ est strictement inférieur a celui du lancer $n$. +On a un de equilibre a $N$ faces numérotées de 1 a $N$, et on effectue une suite de lancers indépendants. Le jeu s'arrete lorsque le résultat du lancer $n+1$ est strictement inférieur a celui du lancer $n$. - Calculer la probabilité $\pi_k$ que le jeu s'arrete apres le rang $k$. - Montrer que $\pi_k$ tend vers 0 pour $k\to+\i$. #+end_exercice @@ -4451,11 +4523,16 @@ Soit $E$ un $\mathbb{K}$-espace vectoriel de dimension finie. On dit que $h\in\m Soient $A\in\M_n(\R)$ et $M\in\M_n(\R)$. On suppose que toutes les matrices semblables a $A$ appartiennent au commutant de $M$. Déterminer $M$. Meme question dans $\M_n(\C)$. #+end_exercice -#+begin_exercice [Mines 2023 # 556] +# ID:7010 +#+begin_exercice [Mines 2023 # 556] :sup: Soient $p,q\in\C$. On note $x_1$, $x_2$ et $x_3$ les racines (non nécessairement distinctes) du polynôme $X^3+pX+q$. Pour $j\in\N$, on pose $N_j=x_1^j+x_2^j+x_3^j$. Calculer, pour $n\in\N^*$, le déterminant de la matrice $M_n=(N_{i+j-2})_{1\leq i,j\leq n}$. #+end_exercice +#+BEGIN_proof +Il suffit de calculer les trois premières valeurs, $N_0,N_1,N_2$, puis ils vérifient une récurrence d'ordre $2$. +#+END_proof + #+begin_exercice [Mines 2023 # 557] ${}^{\bigstar}$ Soit $n\in\N^*$. Calculer $\det((i\wedge j)_{1\leq i,j\leq n})$. @@ -6305,7 +6382,7 @@ On tire au hasard un élément $A$ de $P(\db{1,n})$. Calculer la probabilité qu #+end_exercice #+begin_exercice [Mines 2023 # 881] -Soient $m,n\in\N^*$ tel que $m\leq\frac{n}{2}$. On se donne deux urnes contenant chacune des boules numerotées de $1$ a $n$. On tire $m$ boules dans chaque urne et l'on note $X$ le nombre de doublons. Calculer la loi de $X$ puis sa variance. +Soient $m,n\in\N^*$ tel que $m\leq\frac{n}{2}$. On se donne deux urnes contenant chacune des boules numérotées de $1$ a $n$. On tire $m$ boules dans chaque urne et l'on note $X$ le nombre de doublons. Calculer la loi de $X$ puis sa variance. #+end_exercice #+begin_exercice [Mines 2023 # 882] @@ -6316,7 +6393,7 @@ Un couple met au monde quatre enfants. Chaque enfant a la probabilité $p\in]0,1 # ID:6955 #+begin_exercice [Mines 2023 # 883] -Soit $p\in]0,1[$. Dans un sac contenant $n$ jetons numerotes de $1$ a $n$, on tire $S$ jetons ou $S$ est une variable aléatoire suivant la loi binomiale de paramètre $n$ et $p$. Quelle est la probabilité d'obtenir $n$ jetons de numéros consécutifs? +Soit $p\in]0,1[$. Dans un sac contenant $n$ jetons numérotes de $1$ a $n$, on tire $S$ jetons ou $S$ est une variable aléatoire suivant la loi binomiale de paramètre $n$ et $p$. Quelle est la probabilité d'obtenir $n$ jetons de numéros consécutifs? #+end_exercice #+begin_exercice [Mines 2023 # 884] @@ -6333,27 +6410,33 @@ On considère une urne qui contient une proportion $p\in\,]0,1[$ de boules blanc #+end_exercice #+begin_exercice [Mines 2023 # 887] -On considère une urne remplie avec des boules numerotées de $1$ a $2n$. On procede a une suite de tirages sans remise. +On considère une urne remplie avec des boules numérotées de $1$ a $2n$. On procede a une suite de tirages sans remise. - Calculer la probabilité que les boules impaires soient tires exactement dans l'ordre $1,3,\ldots,2n-1$. - Soit $X$ la variable correspondant au nombre de tirages nécessaires pour obtenir toutes les boules impaires. Déterminer la loi et l'espérance de $X$. #+end_exercice #+begin_exercice [Mines 2023 # 888] -Soit $n\geq 2$. On place $n$ boules numerotées de $1$ a $n$ dans une urne et l'on realise des tirages successifs avec remise. On note $X$ le rang du tirage donnant pour la premiere fois un numero supérieur ou égal aux precedents. +Soit $n\geq 2$. On place $n$ boules numérotées de $1$ a $n$ dans une urne et l'on realise des tirages successifs avec remise. On note $X$ le rang du tirage donnant pour la premiere fois un numéro supérieur ou égal aux precedents. - Déterminer la loi de $X$. - Calculer l'espérance et la variance de $X$. #+end_exercice #+begin_exercice [Mines 2023 # 889] -Une urne contient $n+1$ boules numerotées de 0 a $n$. On y effectue des tirages avec remise. On pose $X_1=1$. Pour $i\geq 2$, $X_i$ est la variable de Bernoulli égale a 1 si le numero de la boule titée au $i$-eme tirage n'avait jamais ete obtenu avant. On pose, pour $i\in\N^*$, $Y_i=X_1+\cdots+X_i$. +Une urne contient $n+1$ boules numérotées de $0$ à $n$. On y effectue des tirages avec remise. On pose $X_1=1$. Pour $i\geq 2$, $X_i$ est la variable de Bernoulli égale à $1$ si le numéro de la boule tirée au $i$-eme tirage n'avait jamais été obtenu avant. On pose, pour $i\in\N^*$, $Y_i=X_1+\cdots+X_i$. - Déterminer la loi des $X_i$. - Calculer l'espérance et la variance de $Y_i$. Donner un équivalent de $\mathbf{E}(Y_n)$. - Pour $(i,j)\in(\N^*)^2$, calculer $\mathbf{P}(X_i=1,X_j=1)$. - Étudier l'indépendance des $X_i$. #+end_exercice +#+BEGIN_proof + 1. $P(X_i = 1) = \sum_{k=0}^n P(X_i = 1 \cap T_i = k)$ et $P(X_i = 1 \cap T_i = k) = \frac{(k-1)^{i-1}}{k^{i-1}} \frac{1}{n+1}$. + 2. On peut trouver directement la loi de $Y_i$. + 3. +#+END_proof + #+begin_exercice [Mines 2023 # 890] -Soit $(J - {n\in\N}$ une suite de joueurs. Le joueur $J_0$ affronte le joueur $J_1$ ; le gagnant affronte $J_2$, puis le gagnant de ce nouveau match affronte $J_3$ et ainsi de suite. Lors d'un match, le joueur entrant a une probabilité $p\in]0,1[$ de remporter le match. Le jeu termine lorsqu'un meme joueur remporte trois victoires. Pour $n\in\N$, on note $A_n$ l'évènement \lt \lt le $n$-ième match est joue \gt \gt . Déterminer la limite de $\mathbf{P}(A_n)$ quand $n\to+\i$. +Soit $(J_n)_{n\in\N}$ une suite de joueurs. Le joueur $J_0$ affronte le joueur $J_1$ ; le gagnant affronte $J_2$, puis le gagnant de ce nouveau match affronte $J_3$ et ainsi de suite. Lors d'un match, le joueur entrant a une probabilité $p\in]0,1[$ de remporter le match. Le jeu termine lorsqu'un meme joueur remporte trois victoires. Pour $n\in\N$, on note $A_n$ l'évènement \lt \lt le $n$-ième match est joue \gt \gt . Déterminer la limite de $\mathbf{P}(A_n)$ quand $n\to+\i$. #+end_exercice #+begin_exercice [Mines 2023 # 891] @@ -6412,11 +6495,17 @@ On considère une variable aléatoire $X$ suivant la loi de poisson de paramètr - Comparer les probabilités des évènements $(X\in 2\N)$ et $(X\in 2\N+1)$. #+end_exercice + +# ID:6985 #+begin_exercice [Mines 2023 # 901] -Soit $X$ une variable aléatoire a valeurs dans $[a,b]$, d'espérance $\mathbf{E}(X)=m$. - - Montrere que $\mathbf{V}(X)\leq(m-a)(b-m)$. - - Montrere que cette inégalité est optimale. +Soit $X$ une variable aléatoire à valeurs dans $[a,b]$, d'espérance $\mathbf{E}(X)=m \in [a,b]$. + - Montrer que $\mathbf{V}(X)\leq (m-a)(b-m)$. + - Montrer que cette inégalité est optimale. #+end_exercice +#+BEGIN_proof + - Écrire $(X-a)(b-X)\leq 0$. +#+END_proof + #+begin_exercice [Mines 2023 # 902] Soient $A\in\mc{S}_n(\R)$ et $b\in\M_{n,1}(\R)$. On pose $M=\begin{pmatrix}A&b\\ b^T&c\end{pmatrix}$ et on suppose que les racines du polynôme caracteristique de $M$ ne sont pas toutes simples. @@ -6712,7 +6801,7 @@ On pose $u=2+\sqrt{3}$, $v=2-\sqrt{3}$. Pour $n\in\N$, on note $M_n=2^n-1$ et $s_n=u^{2^n}+v^{2^n}$. - Montrer que, si $M_n$ est premier, alors $n$ est premier. - - Montrer que, pour $n\in\N$, $s_{n+1}=s_n^2-2$. Qu'en déduire sur le suite $(s - {n\in\N}$? + - Montrer que, pour $n\in\N$, $s_{n+1}=s_n^2-2$. Qu'en déduire sur le suite $(s_n)_{n\in\N}$? - Soit $q$ un nombre premier. On munit l'ensemble $B=(\Z/q\Z)^2$ des deux lois de composition interne définies par : $(x,y)+(x',y')=(x+x',y+y')$ et $(x,y)\cdot(x',y')=(xx'+3yy',xy'+x^{ '}y)$. @@ -6740,7 +6829,7 @@ Ind. Considèrer $\phi:x\mapsto ax$. #+end_exercice #+begin_exercice [Centrale 2023 # 1216] -Soit $(T - {n\in\N}$ la suite de polynômes réels définie par $T_0(X)=1,\ T_1(X)=X$ et pour $n\in\N,\,T_{n+2}(X)=2XT_{n+1}(X)-T_n(X)$. +Soit $(T_n)_{n\in\N}$ la suite de polynômes réels définie par $T_0(X)=1,\ T_1(X)=X$ et pour $n\in\N,\,T_{n+2}(X)=2XT_{n+1}(X)-T_n(X)$. - Montrer que, pour $n\in\N$, $\forall\theta\in\R$, $T_n(\cos\theta)=\cos(n\theta)$. - Montrer que $T_n\circ T_m=T_m\circ T_n$ pour $(m,n)\in\N^2$. - Montrre que, pour $n\geq m$, $2T_nT_m=T_{n+m}+T_{n-m}$. @@ -6773,7 +6862,7 @@ Soit $f\colon\R\to\R$ une fonction continue telle que, pour toute matrice $A\in\ - Rappeler la formule de développement d'un déterminant par rapport a une ligne ou une colonne. En déduire, pour $A\in\M_n(\R)$, une relation entre $\mathrm{Com}\,A$, $A$ et $\det A$. - Soit $A=(a_{i,j})_{1\leq i,j\leq n}$ la matrice de $\M_n(\R)$ définie par : $a_{i,i}=2$, $a_{i,j}=-1$ si $|i-j|=1$ et $a_{i,j}=0$ dans tout autre cas. Calculer le déterminant de $A$. - Soit $A\in\M_n(\R)$ une matrice dont les coefficients diagonaux sont strictement positifs, dont les autres coefficients sont negatifs et telle que $\sum_{j=1}^na_{i,j}\gt 0$ pour tout $i$. Montrre que $A$ est inversible. - - Montrre que les coefficients de $A^{-1}$ sont positifs. + - Montrer que les coefficients de $A^{-1}$ sont positifs. #+end_exercice #+begin_exercice [Centrale 2023 # 1221] @@ -6927,7 +7016,7 @@ Soient $(E,N)$ et $(E',N')$ deux espaces vectoriels normes. Soit $d\in\N$. Pour $P(X)=p_0+p_1X+\cdots+p_dX^d\in\R_d[X]$ on pose $\|P\|=\max(|p_0|,...,|p_d|)$. - Vérifier que l'application $\|\ \|$ est une norme sur $\R_d[X]$. -_b) i)_ Soit $(y - {n\in\N}$ une suite d'éléments de $E$, convergeant vers $\ell\in E$. +_b) i)_ Soit $(y_n)_{n\in\N}$ une suite d'éléments de $E$, convergeant vers $\ell\in E$. Montrer que l'ensemble $Y=\{y_n,\ n\in\N\}\cup\{\ell\}$ est compact. - Soit $f:E\to E'$ continue telle que, pour tout compact $K$ de $E'$, $f^{-1}(K)$ est un compact de $E$. Montrer que, si $F$ est un ferme de $E$, alors $f(F)$ est un ferme de $E'$. @@ -7133,11 +7222,11 @@ $$\prod_{k=0}^n(1+x^{2^k}+x^{2^{k+1}})=\left(\sum_{k=0}^{2^{n+1}-1}(-j)^{n-s_k}x $\forall(x,y)\in\R^2,\ f(x+y)=f(x)+f(y)$. -On dit qu'une suite réelle $(a - {n\in\N}$ vérifie la propriété $(P)$ si :1. La série entière $\sum a_nx^n$ a un rayon de convergence supérieur ou égal a $1$, +On dit qu'une suite réelle $(a_n)_{n\in\N}$ vérifie la propriété $(P)$ si :1. La série entière $\sum a_nx^n$ a un rayon de convergence supérieur ou égal a $1$, 2. La somme $S_a$ de cette série entière admet une limite réelle en $1^-$. - 3. - Montrer que, si la série $\sum a_n$ converge absolument, alors la suite $(a - {n\in\N}$ vérifie $(P)$, + 3. - Montrer que, si la série $\sum a_n$ converge absolument, alors la suite $(a_n)_{n\in\N}$ vérifie $(P)$, 4. Étudier la réciproque. - 5. Trouver toutes les suites $(a - {n\in\N}$ périodiques qui vérifient $(P)$. + 5. Trouver toutes les suites $(a_n)_{n\in\N}$ périodiques qui vérifient $(P)$. #+end_exercice #+begin_exercice [Centrale 2023 # 1265] @@ -7178,7 +7267,7 @@ Montrer que la suite $(f - {n\in\N^*}$ converge simplement sur $\R^+$ vers une f - Rappeler le théorème de convergence dominée. Le démontrer sous l'hypothèse supplémentaire d'une convergence uniforme sur tout segment. - - Soit $(f - {n\in\N}\in\left(\R^{\N}\right)^{\N}$ une suite de fonctions qui converge simplement sur $\N$ vers une suite $f\in\R^{\N}$. On suppose l'existence d'une suite sommable positive $\phi\in\R^{\N}$ telle que $\forall n\in\N,\;\forall t\in\N,\;|f_n(t)|\leq\phi(t)$. + - Soit $(f_n)_{n\in\N}\in\left(\R^{\N}\right)^{\N}$ une suite de fonctions qui converge simplement sur $\N$ vers une suite $f\in\R^{\N}$. On suppose l'existence d'une suite sommable positive $\phi\in\R^{\N}$ telle que $\forall n\in\N,\;\forall t\in\N,\;|f_n(t)|\leq\phi(t)$. Montrer que les suites $f_n$ et $f$ sont sommables et que $\lim_{n\ra+\i}\sum_{k=0}^{+\i}f_n(k)=\sum_{k=0}^{+\i}f (k)$. #+end_exercice @@ -7256,7 +7345,7 @@ Sa somme est notée $S(x,y)$. #+begin_exercice [Centrale 2023 # 1277] On melange les cartes d'un jeu de $2n$ cartes. -Avec quelle probabilité les cartes de numero impair sont-elles correctement ordonnées? +Avec quelle probabilité les cartes de numéro impair sont-elles correctement ordonnées? #+end_exercice #+begin_exercice [Centrale 2023 # 1278] @@ -7278,7 +7367,7 @@ Soient $X_1,\ldots,X_n$ des variables aléatoires i.i.d. suivant la loi de Poiss - Calculer $\int_0^{+\i}\mathbf{P}(T_n\geq x)\,dx$. #+end_exercice -#+begin_exercice [Centrale 2023 # 1280.] +#+begin_exercice [Centrale 2023 # 1280] - Rappeler les formules des probabilités totales et composées. On fixe $d\in\N^*$ et $(U_n)_{n\geq 1}$ une suite de variables aléatoires indépendantes uniformément distribuées sur $\db{1,d}$. Soit $N_d=\inf\{n\geq 2,\ U_n\in\{U_1,\ldots,U_{n-1}\}\}$. @@ -7287,7 +7376,7 @@ On fixe $d\in\N^*$ et $(U_n)_{n\geq 1}$ une suite de variables aléatoires indé - Pour tout réel $x\gt 0$, calculer $\lim_{d\to+\i}\mathbf{P}\left(\dfrac{N_d}{\sqrt{d}}\gt x\right)$. #+end_exercice -#+begin_exercice [Centrale 2023 # 1281.] +#+begin_exercice [Centrale 2023 # 1281] - Soient $x\gt 0$ et $X_x$ une variable de Poisson de paramètre $x$. Calculer l'espérance de $X_x$. Montrer que $\mathbf{P}(|X_x-\mathbf{E}(X_x)|\geq\eps x)=O\left(\dfrac{1}{x }\right)$ quand $x\to+\i$. 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