From 38756dd0b0530a95aa5780f272341fbeb5af4438 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?S=C3=A9bastien=20Miquel?= Date: Fri, 7 Aug 2026 16:10:13 +0200 Subject: [PATCH] Cleaned up --- Exercices 2025.org | 105 +++++++++++++++++++++++++++---------- Exercices XENS MP 2025.pdf | Bin 664981 -> 354232 bytes 2 files changed, 76 insertions(+), 29 deletions(-) diff --git a/Exercices 2025.org b/Exercices 2025.org index c8e01e7..cd63ab5 100644 --- a/Exercices 2025.org +++ b/Exercices 2025.org @@ -2,7 +2,7 @@ #+title: Exercices 2025 #+author: Sébastien Miquel #+date: 22-07-2025 -# Time-stamp: <30-06-26 11:54> +# Time-stamp: <03-08-26 22:02> #+OPTIONS: toc:t * Meta :noexport: @@ -14,7 +14,7 @@ #+RESULTS: | ? | ! | todo | unexed | unexed xens | -| 1 | 2 | 5 | 1323 | 5 | +| 0 | 0 | 0 | 1318 | 0 | ** Options @@ -1963,7 +1963,8 @@ limites. Sinon, on converge vers $F(ax + b)$, avec $a\neq 0$. #+END_proof -#+begin_exercice [ENS L 2025 # 111] :todo: +# ID:9056 +#+begin_exercice [ENS L 2025 # 111] Soit $(f_n)_{n\in\N}$ une suite de fonctions de $[0,1]$ dans $]0,1]$, convergeant simplement vers une fonction $f$. 1. Pour $n \ge 2$, on pose $t_n = \frac{1}{\ln n} \sum_{i=1}^n @@ -1982,7 +1983,7 @@ convergeant simplement vers une fonction $f$. + $\forall x \lt x_0, \ \exists C_x \gt 0, \ \exists n_0 \in \N^*, \ \forall n \ge n_0, \ f_n(x) \le e^{-C_x n};$ + $\forall x \gt x_0, \ f(x) \ge x - x_0$. #+end_exercice -#+BEGIN_proof :todo: +#+BEGIN_proof 1. Sans difficultés. 2. Soit $x \in [0, 1/2[$. @@ -1999,7 +2000,13 @@ convergeant simplement vers une fonction $f$. Le même raisonnement, en intégrant de $x$ à $y$, donne $f_n(x) \le e^{-C_x n}$ avec $C_x = \frac{y - x}{M} \gt 0$. - Soit $x \gt x_0$. + Soit $x \gt x_0$. On a $\sigma_n(x)\tend{n\ra + \i} \i$ et + $\sigma_n'(x) \geq \sum_{k=1}^{n-1} \frac{k f_k(x)}{\sigma_k(x)}\geq \sum_{k=1}^{n-1} f_k(x)\geq \sigma_n(x) - 1$. + + Donc, pour $x_0\lt y\lt x$, on a $\sigma_n(x)\geq (\sigma_n(y) - 1) e^{x-y} +1$, donc $f(x)\geq f(y) e^{x-y}$, donc $f$ ne s'annule pas. + + + #+END_proof @@ -2494,8 +2501,6 @@ $1$ aurait plusieurs antécédents par un même $\sigma_k$). Naturellement, sous la condition $N_1$, la probabilité que ces éléments soient tous distincts est $\frac{(n-1)!}{(n-1)^{n-1}}$ qui tend vers $0$. -Formellement : !! - Enfin, l'énoncé demande $\P(G \mid \bigcup_{i=1}^n N_i) = \frac{\P\big(G \cap \bigcup_{i=1}^n N_i\big)}{\P\big(\bigcup_{i=1}^n N_i\big)} = \frac{\P\big(\bigcup (N_i \cap G)\big)}{\P\big(\bigcup_{i=1}^n N_i\big)}$. Comme les évènements $N_i$ sont disjoints, et que les $\P(N_i)$ et les $\P(G\cap N_i)$ sont égaux, on obtient que $\P(G\mid N) = \P(G\mid N_1)$. #+END_proof @@ -2652,32 +2657,58 @@ Soient $(X_k)_{k \in \N^*}$ une suite de variables de Rademacher indépendantes 1. Déterminer le temps moyen d'arrêt (en $0$ ou $N$ cette fois) en partant de $k$. #+end_exercice -#+begin_exercice Inégalité de Khintchine [ENS P 2025 # 154] :todo: +#+call: get_exa(9058) +#+BEGIN_exercice Inégalité de Khintchine [ENS P 2025] On considère $n$ variables aléatoires de Rademacher indépendantes -$(\eps_i)_{1\leq i\leq n}$. Montrer que, pour tout réel $p \gt 0$, il +$(\eps_i)_{1\leq i\leq n}$. On veut montrer que, pour tout réel $p \gt 0$, il existe $(c_p, C_p) \in (\R_+^*)^2$ indépendant de $n \in \N^*$ tel que, pour tout $(z_1, \ldots z_n) \in \C^n$, $$c_p \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}} \leq \left(\mathbf{E} \left|\sum_{i=1}^n \eps_i z_i\right|^p\right)^{\frac{1}{p}} \leq C_p \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}}.$$ -#+end_exercice + 1. Établir l'inégalité pour $p = 4$. + 2. Établir l'inégalité pour tout $p \geq 2$. + 3. Établir l'inégalité pour $p = 1$. + 4. Établir l'inégalité pour tout $p\gt 0$. +#+END_exercice #+BEGIN_proof -On prend $p = 2$. C'est vide. +Notons $S = \sum_{i=1}^n \eps_i z_i$ et $\Sigma = \left(\sum_{i=1}^n |z_i|^2\right)^{\frac{1}{2}}$. +Puisque les $\eps_i$ sont indépendantes, centrées et de variance $1$, on a $\mathbf{E}[|S|^2] = \sum_{i,j} z_i \overline{z_j} \mathbf{E}[\eps_i \eps_j] = \sum_{i=1}^n |z_i|^2 = \Sigma^2$. +Ainsi, $\|S\|_2 = \Sigma$. + 1. L'inégalité de Cauchy-Schwarz donne $\mathbf{E}[|S|^2] \leq \sqrt{\mathbf{E}[|S|^4]}$. On a donc $\Sigma \leq \|S\|_4$, ce qui donne $c_4 = 1$. + Pour la majoration, on développe le moment d'ordre 4 : + $$\mathbf{E}[|S|^4] = \sum_{i,j,k,l} z_i \overline{z_j} z_k \overline{z_l} \mathbf{E}[\eps_i \eps_j \eps_k \eps_l]$$ + Par indépendance des $\eps_i$, l'espérance vaut $1$ si les indices forment des paires, et $0$ sinon. Les cas possibles non nuls se regroupent en : -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. + + $i=j=k=l$ : donne $\sum_i |z_i|^4$ + + $i=j \neq k=l$, $i=l \neq j=k$ ou $i=k \neq j=l$. + En utilisant $|z_i^2 \overline{z_j}^2| \leq |z_i|^2 |z_j|^2$ et en majorant le tout, on obtient : + $$\mathbf{E}[|S|^4] \leq 3 \sum_{i,j} |z_i|^2 |z_j|^2 = 3 \Sigma^4$$ + Ainsi, $\|S\|_4 \leq 3^{1/4} \Sigma$, d'où l'on peut choisir $C_4 = 3^{1/4}$. + 2. Pour $p \geq 2$, la croissance des normes assure que $\|S\|_p \geq \|S\|_2 = \Sigma$, donc $c_p = 1$ convient. + De plus, en développant $\mathbf{E}[|S|^{2^k}]$ de façon analogue, on peut majorer ce moment par un multiple de $\Sigma^{2^k}$, assurant l'existence d'une constante $C_{2^k}$. + Puisque $p \mapsto \|S\|_p$ est croissante (car $\phi\colon x \mapsto x^{q/p}$ est convexe), pour $p \geq 2$ quelconque, il suffit de choisir un entier $k$ tel que $2^k \geq p$ pour obtenir $\|S\|_p \leq \|S\|_{2^k} \leq C_{2^k}\Sigma$, ce qui prouve l'existence de $C_p$. + 3. La croissance des normes donne $\|S\|_1 \leq \|S\|_2 = \Sigma$, d'où $C_1 = 1$. -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. + Pour la minoration, on applique Cauchy-Schwarz à $|S|^{1/2}$ et $|S|^2$ : (En fait, on se débrouille pour avoir $l^c \leq (l^1)(l^d)$, avec $c,d\geq 2$). -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 $\sqrt{\E(|U|^2)} = \sqrt{\eps} G$. + $$\mathbf{E}[|S|^{5/2}] = \mathbf{E}[|S|^{1/2} |S|^2] \leq \sqrt{\mathbf{E}[|S|] \mathbf{E}[|S|^4]}$$ + Comme $\frac{5}{2} \geq 2$, la croissance des normes $L^q$ assure que $\|S\|_2 \leq \|S\|_{5/2}$, c'est-à-dire $\mathbf{E}[|S|^2]^{5/4} \leq \mathbf{E}[|S|^{5/2}]$. En élevant au carré, il vient : + $$\Sigma^5 = \mathbf{E}[|S|^2]^{5/2} \leq \mathbf{E}[|S|] \mathbf{E}[|S|^4]$$ + Or $\mathbf{E}[|S|^4] \leq C_4^4 \Sigma^4 = 3\Sigma^4$. En simplifiant par $\Sigma^4$, on en déduit $\Sigma \leq 3 \mathbf{E}[|S|]$, d'où $c_1 = \frac{1}{3}$. -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. + Une alternative plus classique consiste à appliquer Hölder : $L^2 + \le (L^1)^{1/3} (L^4)^{2/3}$. + 4. Le cas $p \geq 2$ a été traité à la question 2. Pour $p \in ]0, 2[$, on a trivialement $\|S\|_p \leq \|S\|_2 = \Sigma$, donc $C_p = 1$. + Pour la minoration, on généralise l'astuce de la question 3 en appliquant Cauchy-Schwarz à $|S|^{p/2}$ et $|S|^2$ : + $$\mathbf{E}[|S|^{\frac{p+4}{2}}] = \mathbf{E}[|S|^{p/2} |S|^2] \leq \sqrt{\mathbf{E}[|S|^p] \mathbf{E}[|S|^4]}$$ + Puisque $p \gt 0$, on a $c = \frac{p+4}{2} \gt 2$. La croissance des normes $L^q$ garantit $\mathbf{E}[|S|^2]^{c/2} \leq \mathbf{E}[|S|^c]$, ce qui donne en élevant au carré : + $$\Sigma^{p+4} = \mathbf{E}[|S|^2]^{\frac{p+4}{2}} \leq \mathbf{E}[|S|^p] \mathbf{E}[|S|^4]$$ + En injectant $\mathbf{E}[|S|^4] \leq C_4^4 \Sigma^4$, on obtient $\Sigma^{p+4} \leq \mathbf{E}[|S|^p] C_4^4 \Sigma^4$, soit en simplifiant : + $\Sigma^p \leq C_4^4 \mathbf{E}[|S|^p]$ + On obtient $\Sigma \leq C_4^{4/p} \|S\|_p$, ce qui permet de choisir $c_p = C_4^{-4/p}$. #+END_proof +#+BEGIN_indication +#+END_indication # ID:8866 #+begin_exercice [ENS L 2025 # 155] @@ -4349,7 +4380,8 @@ Soit $E$ un $\R$ espace vectoriel de dimension finie. Montrer que tout convexe n #+end_exercice -#+begin_exercice [X MP 2025 # 297] :todo: +# ID:nil +#+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 $\Z^2$. @@ -4390,7 +4422,7 @@ euclidien de rayon $r$ contenant au moins $k$ points de $\Z^2$. Cette fonction vérifie une propriété de symétrie. - Super dur !! + Super dur. 4. Simple. #+END_proof @@ -4686,7 +4718,8 @@ Soit $s\colon \R \ra \R$ telle que $(*): \forall x \in \R, s(x+1) = s(x) + \frac #+END_proof -#+begin_exercice [X MP 2025 # 316] :todo: +# ID:9053 +#+begin_exercice [X MP 2025 # 316] 1. Soit $f \in \mc C^0(\R, \R)$. Montrer que $f$ est affine si et seulement si, pour tout réel $x$, on a $\frac{f(x+h) + f(x-h) - 2f(x)}{h^2} \underset{h \ra 0^+}{\longrightarrow} 0$. 2. Et si $f$ n'est pas continue ? #+end_exercice @@ -4699,7 +4732,14 @@ Soit $s\colon \R \ra \R$ telle que $(*): \forall x \in \R, s(x+1) = s(x) + \frac Si $f$ est $\mc C^2$, cela donne bien que $f''$ s'annule sur un ensemble dense. + Si $f$ admet un minimum en un point, on peut montrer que sa dérivée s'annule. + Soient $a,b$. On suppose $f(a) = f(b) = 0$. S'il existe un point + $x_0$ en lequel $f\gt 0$. Alors, c'est encore vrai en retirant à + $f$ une expression quadratique $g(x) = f(x) - \eps (x-a) (b-x)$. + Celle-ci admet donc un maximum. + + En ce point, on obtient $\frac{g(x+h) + g(x-h) - 2g(x)}{h^2}\leq 0$, or c'est $\eps + \frac{f(x+h) + f(x-h) - 2f(x)}{h^2}$, contradiction. 2. Il suffit de prendre $f(x) = \a$ si $x\gt 0$ et $-\a$ sinon. #+END_proof @@ -4913,13 +4953,20 @@ Quand tu intègres,… #+END_proof -#+begin_exercice [X MP 2025 # 333] :todo: +# ID:9057 +#+begin_exercice [X MP 2025 # 333] 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. + 1. On a $s_n = \sum_{i=0}^{n-1} a_i \geq 2^n - 1$, peut-être que cela suffit ? + + Si $(a_i)$ vérifie cela, et a une somme d'inverse $\sum_{i=0}^{n_0} \frac{1}{a_i}\gt 2$, il existe $a_0,\dots, a_{n_0}$ qui vérifie les $\sum_{i=0}^{n-1} a_i \geq 2^n - 1$ et pour laquelle la somme d'inverse est maximale, mais alors, toute les inégalités doivent être des égalités. + 2. Si on avait une contradiction, on aurait une contradiction finie, + et on pourrait minimiser $a_n$. On se ramène donc au cas où chaque + $a_i$ est le premier élément qui ne soit pas une somme des + précédents : NON, faux, en diminuant $a_n$, un autre entier + pourrait devenir somme de $a_n$ et d'un autre. #+END_proof @@ -11813,7 +11860,7 @@ b) Montrer que $\sum_{k=1}^{n} \ln(k) = n \ln(n) n + O(\ln(n))$. c) Soient $(a_k)_{k\geq 2}$ une suite réelle et $b:[2,+\infty[\to\mathbb{R} \text{ de classe } \mathcal{C}^1. \text{ On pose } A(t)=\sum_{k=1}^{\lfloor t\rfloor}a_k$ pour $t\geq 2$. Montrer que $\sum_{k=2}^n a_k b(k)=A(n)b(n)-\int_2^n b'(t)A(t)\,\mathrm{d}t$ pour tout entier $n\geq 2$. d) On note $\mathcal{P}$ l'ensemble des nombres premiers. On pose $R: t>1 \mapsto \sum_{p\in\mathcal{P},\ p\leqslant t} \frac{\ln p}{p} \ln(t)$. -Montrer que $\sum_{p \in \mathcal{P}, p \leqslant n} \frac{1}{p} = 1 + \ln(\ln n) \ln(\ln 2) + \frac{R(n)}{\ln n} + \int_2^n \frac{R(t)}{t(\ln t)^2} \, \mathrm{d}t \text{ pour tout entier } t = 1 + \ln(\ln n) \ln(\ln 2) + \frac{R(n)}{\ln n} + \frac{R(n)}{t(\ln t)^2} + \frac{R(n)}{t(\ln t)^2} + \frac{R(n)}{t(\ln t)^2} + \frac{R(n)}{??}$ +Montrer que $\sum_{p \in \mathcal{P}, p \leqslant n} \frac{1}{p} = 1 + \ln(\ln n) \ln(\ln 2) + \frac{R(n)}{\ln n} + \int_2^n \frac{R(t)}{t(\ln t)^2} \, \mathrm{d}t \text{ pour tout entier } t = 1 + \ln(\ln n) \ln(\ln 2) + \frac{R(n)}{\ln n} + \frac{R(n)}{t(\ln t)^2} + \frac{R(n)}{t(\ln t)^2} + \frac{R(n)}{t(\ln t)^2} + \frac{R(n)}{?}$ e) Montrer qu'il existe une constante $C$ telle que $\sum_{p \in \mathcal{P}, n \leq p} \frac{1}{p} = \ln(\ln n) + C + O\left(\frac{1}{\ln n}\right)$ quand $n \to +\infty$. #+end_exercice diff --git a/Exercices XENS MP 2025.pdf 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