From c6cc8b66ed934981b4aa9b57fa55498a01779d49 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?S=C3=A9bastien=20Miquel?= Date: Wed, 1 May 2024 16:30:16 +0200 Subject: [PATCH] Des changements. --- Exercices 2023.org | 159 +++++++++++++++++++++++++------------ Exercices XENS MP 2023.pdf | Bin 731523 -> 737131 bytes 2 files changed, 107 insertions(+), 52 deletions(-) diff --git a/Exercices 2023.org b/Exercices 2023.org index c3556cb..d0fdb5c 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: <24-04-24 19:54> +# Time-stamp: <01-05-24 16:29> #+OPTIONS: @@ -884,7 +884,7 @@ On note $\lN \cdot\rN$ la norme d'opérateur sur $\M_n(\C)$ associée à la norm ** Analyse #+BEGIN_exercice [ENS 2023 # 79] -Soit $p\gt 1$. On pose, pour $x\in\R^n$, $\lN x\rNp = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}$. +Soit $p\gt 1$. On pose, pour $x\in\R^n$, $\lN x\rN_{p} = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}$. 1. Montrer qu'il s'agit bien d'une norme. 2. Montrer l'inégalité de Hölder. 3. Dans $\R^2$, dessiner la boule unité de la norme $p$ pour plusieurs valeurs de $p$. @@ -1300,7 +1300,7 @@ Soit $\alpha=(\alpha - {i\geq 1}$ une suite de $\Z$ nulle a partir d'un certain Dans la suite, on note $f$ la somme de cette série entière. - Expliciter $f$ si $\alpha=(-\delta_{i,1})_{i\geq 1}$. - Pour une somme $g$ de série entière sur un intervalle $]-a,a[$ non trivial, on pose $\Delta(g):z\mapsto zg'(z)$. Expliciter $P(\Delta)(g)$ lorsque $g:z\mapsto z^k$ avec $k\in\N$ et $P\in\R[X]$. - - Soit $v\in\C^{\N^*}$ une suite complexe, et $P\in\R[X]$ sans racine dans $\N^*$ tels que, pour tout $n\geq 1$, $v_{n+1}=\frac{v_n}{P(n+1)}$. Montrer que $\sum_{n\geq 1}v_nz^n$ a un rayon de convergence non nul et donner une methode simple pour trouver une équation différentielle lineaire non triviale a coefficients polynomiaux dont sa somme est solution. + - Soit $v\in\C^{\N^*}$ une suite complexe, et $P\in\R[X]$ sans racine dans $\N^*$ tels que, pour tout $n\geq 1$, $v_{n+1}=\frac{v_n}{P(n+1)}$. Montrer que $\sum_{n\geq 1}v_nz^n$ a un rayon de convergence non nul et donner une methode simple pour trouver une équation différentielle linéaire non triviale a coefficients polynomiaux dont sa somme est solution. - Résoudre le meme probleme qu'en (d) lorsqu'il existe $P$ et $Q$ dans $\R[X]$ sans racine dans $\N^*$ telles que $v_{n+1}=\frac{Q(n+1)}{P(n+1)}\,v_n$ pour tout $n\geq 1$, et en supposant cette fois-ci que $\deg(Q)\leq\deg(P)$. - Justifier que le cadre de la question - s'applique bien a la suite $(u_n)_{n\geq 1}$ lorsque $R\gt 0$. #+end_exercice @@ -1457,7 +1457,7 @@ On considère l'équation différentielle $(1)\colon X'(t)=P(t)X(t)$ où $P$ est #+begin_exercice [ENS 2023 # 146] - Soit $f\colon (x,y)\ \mapsto \left(\ln\left(x^2+y^2\right),\arctan\left(\frac{y}{x}\right)\right)$. Donner le domaine de définition $\Omega$ de $f$. Étudier la continuité et la différentiabilité de $f$. - - On identifie naturellement $\R^2$ à $\C$. Montrer que, si $(x,y)\in\Omega$, $df_{(x,y)}$ est $\C$-lineaire. + - On identifie naturellement $\R^2$ à $\C$. Montrer que, si $(x,y)\in\Omega$, $df_{(x,y)}$ est $\C$-linéaire. #+end_exercice #+BEGIN_proof - $\Om = \{(x,y)\mid x\neq 0\}$. La continuité et la différentiabilité ne posent pas de problème. @@ -1578,7 +1578,7 @@ Soit $L$ la courbe du plan complexe d'équation $|z|^2=\cos(2\arg(z))$. # 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 $\op{coVol}(L)=|\mathrm{det}(e_1,e_2)|$. +Soit $(e_1,e_2)$ une famille libre de vecteurs de $\R^2$. On pose $L=\Z e_1+\Z e_2$ 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 $x-y\in L$. @@ -1674,7 +1674,7 @@ Deux joueurs $A$ et $B$ lancent une pièce truquée donnant pile avec une probab On joue a pile ou face avec une pièce pipée qui donne pile avec probabilité $p\lt \frac{1}{2}$. On lance la pièce $2n$ fois et on compte le nombre de «Piles». Déterminer l'entier $n$ qui maximise la probabilité d'avoir compté au moins $n+1$ «Piles». #+end_exercice #+BEGIN_proof -On a $P(S_{2n} = n+k)\lelq P(S_{2n} = n-k)$, puis on montre que $P(S_{2n}\geq n+1) + \frac{1}{2}P(S_{2n} = n)$ est décroissante. Mais on connaît $P(S_{2n} = n)$, et il suffit de voir quand elle devient plus petite que les premières valeurs de $P(S_{2n} \geq n+1)$. +On a $P(S_{2n} = n+k)\leq P(S_{2n} = n-k)$, puis on montre que $P(S_{2n}\geq n+1) + \frac{1}{2}P(S_{2n} = n)$ est décroissante. Mais on connaît $P(S_{2n} = n)$, et il suffit de voir quand elle devient plus petite que les premières valeurs de $P(S_{2n} \geq n+1)$. #+END_proof @@ -1983,7 +1983,7 @@ $\mathbb{D}_{2n}=\{\sigma\in\mc{O}(E)\ ;\ \forall i\in\db{1,n }\,\sigma(v_i)\in\ #+end_exercice #+begin_exercice [ENS PSI 2023 # 203] -- Soit $A\in S^+_n(\R)$ de rang $k$. Montrer qu'il existe des vecteurs $U_1,\ldots,U_k$ lineairement indépendants dans $\R^n$ tels que $A=\sum_{j=1}^kU_jU_j^T$.Soient $A,B\in{\cal M}_n(\R)$. Leur produit d'Hadamard $A\circ B\in{\cal M}_n(\R)$ est la matrice de terme général $a_{ij}b_{ij}$. +- Soit $A\in S^+_n(\R)$ de rang $k$. Montrer qu'il existe des vecteurs $U_1,\ldots,U_k$ linéairement indépendants dans $\R^n$ tels que $A=\sum_{j=1}^kU_jU_j^T$.Soient $A,B\in{\cal M}_n(\R)$. Leur produit d'Hadamard $A\circ B\in{\cal M}_n(\R)$ est la matrice de terme général $a_{ij}b_{ij}$. - Montrer que, si $A$ et $B$ sont des matrices symétriques de rang $1$, alors $A\circ B$ est symétrique de rang au plus $1$. - Montrer que, si $A$ et $B$ sont symétriques positives, alors $A\circ B$ est symétrique. - Si $A$ et $B$ sont symétriques positives, montrer que $A\circ B$ est symétrique positive. @@ -2228,7 +2228,7 @@ Soient $\eps\in\bigg{]}\,0\,;\dfrac{1}{4}\,\bigg{[}$ et $M$ la matrice $M=\left( #+begin_exercice [ENS PC 2023 # 227] Soit $f\colon\R^n\to\R^n$ qui preserve le produit scalaire canonique : -$\forall(x,y)\in(\R^n)^2,\ \langle f(x),f(y)\rangle=\langle x,y\rangle$. Montrer que $f$ est une isometrie lineaire. +$\forall(x,y)\in(\R^n)^2,\ \langle f(x),f(y)\rangle=\langle x,y\rangle$. Montrer que $f$ est une isometrie linéaire. #+end_exercice #+begin_exercice [ENS PC 2023 # 228] @@ -2456,6 +2456,7 @@ On note $p(n)$ le nombre de partitions de $n$ pour $n\in\N^*$. Monter que $p(n)\ On a $p(n)\leq p(n-1) + p(n-2) + \dots + p(1) + 1$, en considérant le (un) plus grand élément de la partition. Formellement, on a une surjection $\sqcup_{k=0}^{n-1} \mc P_k \ra \mc P_n\quad (X, k)\mapsto X + (n-k)$. #+END_proof +# Relier à Lucas : 3721 #+begin_exercice [X MP 2023 # 276] :sup: Soient $e_r\gt \cdots\gt e_2\gt e_1\geq 0$ des entiers, $n=\sum_{k=1}^r2^{e_k}$ et $X=\{s\in\N;\ 2^s\,|\,n!\}$. - Montrer que $\max X=n-r$. @@ -2463,38 +2464,66 @@ Soient $e_r\gt \cdots\gt e_2\gt e_1\geq 0$ des entiers, $n=\sum_{k=1}^r2^{e_k}$ #+end_exercice #+BEGIN_proof - Formule de Legendre. - - Relié à Lucas. + - Il faut que quand $n_i = 0$, $k_i = 0$ (chiffres en base $2$). + + Avec Q1, on veut que le nombre de uns dans la décomposition de $n$ soit la somme du nombre de uns de $k$ et de $n-k$. + + Quand on fait la somme de deux nombres en binaires, le nombre de uns du résultat est $\leq$ à la somme des nombres de uns des sommandes. Avec égalité si et seulement si il n'y a aucune retenue (chaque retenue fait perdre un un). + + Donc forcément, il faut que là où $k$ a un chiffre $1$, $n$ l'ait également. #+END_proof # Relier à d'autres… #+begin_exercice [X MP 2023 # 277] - - Montrer que l'équation $a^2-2b^2=1$ admet une infinite de solutions $(a,b)\in\N^2$. + - Montrer que l'équation $a^2-2b^2=1$ admet une infinité de solutions $(a,b)\in\N^2$. Déterminer l'ensemble des solutions. - Que dire de l'ensemble des solutions de $a^2-2b^2=-1$? #+end_exercice +#+BEGIN_proof + - $(3 + 2\sqrt{2})^k$ + - $(1+\sqrt{2})^{2k+1}$ +#+END_proof #+begin_exercice [X MP 2023 # 278] Si $G$ est un groupe, les éléments d'ordre fini forment-il un sous-groupe? #+end_exercice #+BEGIN_proof -Non : cf le produit libre $\Z/2\Z * \Z/2\Z$ : l'ensemble des mots sur $\{A,B\}$ où deux $A$ d'affilée se simplifient, munit de la concaténation. Cf # 281 +Non : cf le produit libre $\Z/2\Z * \Z/2\Z$ : l'ensemble des mots sur $\{A,B\}$ où deux $A$ d'affilée, ou deux $B$ d'affilées se simplifient, muni de la concaténation. Cf # 281 #+END_proof #+begin_exercice [X MP 2023 # 279] -- Trouver deux groupes $G_1$ et $G_2$ non isomorphes de cardinal $2023=7.17^2$. - - Soit $p$ premier. Montrer qu'un groupe de cardinal $p^2$ est isomorphe a $\Z/p^2\Z$ ou $\hat{\text{a}}\,(\Z/p\Z)^2$. + - Trouver deux groupes $G_1$ et $G_2$ non isomorphes de cardinal $2023=7.17^2$. + - Soit $p$ premier. Montrer qu'un groupe de cardinal $p^2$ est isomorphe à $\Z/p^2\Z$ ou à $(\Z/p\Z)^2$. - Soient $G,H$ deux groupes finis et $\psi\colon G\to H$ un morphisme surjectif. -Montrer que $|G|=|H|\times|\op{Ker}\psi|$. - - On suppose que $G$ est un groupe de cardinal $2023$, que $H=\Z/7\Z$ et que $\phi\colon G\to H$ est un morphisme surjectif. Montrer que $G$ est isomorphe a $\Z/7\Z\times\op{Ker}\phi$. + Montrer que $|G|=|H|\times|\op{Ker}\psi|$. + - On suppose que $G$ est un groupe de cardinal $2023$, que $H=\Z/7\Z$ et que $\phi\colon G\to H$ est un morphisme surjectif. Montrer que $G$ est isomorphe à $\Z/7\Z\times\op{Ker}\phi$. - Montrer que tout groupe de cardinal $2023$ est isomorphe a $G_1$ ou $G_2$. #+end_exercice +#+BEGIN_proof + - + - Le centre est non trivial : on regarde l'action de $G$ sur lui-même par conjugaison, les éléments du centre ont une orbite restreinte à eux mêmes. Sinon, tout élément est conjugué à un nombre d'éléments multiple de $p$, $\Phi\colon g\mapsto g x g^{-1}$ est un morphisme de groupe, donc son noyau est un sous-groupe, donc de cardinal $1, p, p^2$, donc son image est de cardinal $1, p, p^2$. + + La somme des cardinaux des orbites fait $p^2$, donc le nombre d'orbite de cardinal $p$ est un multiple de $p$. + + Soit $G$ est commutatif, auquel cas soit $G$ est cyclique, soit il admet deux éléments $h,g$ d'ordre $p$, tel que $\langle g\rangle\neq \langle h\rangle$. Alors $G\simeq \big(\Z/p\Z\big)^2$, via $(i,j)\mapsto h^i g^j$ (injective + cardinal). + + Sinon, le centre de $G$ est de cardinal $p$ engendré par $g$. On prend un élément $h$ qui n'est pas dans le centre, $(i,j)\mapsto g^i h^j$ est injectif. + - On peut écrire $G$ comme la réunion de $g_1\Ker \phi, \dots, g_7 \Ker \phi$. + + On a $g_1 n_1 g_2 n_2 = g_1 g_2 g_2^{-1} n_1 g_2 n_2$. On veut montrer que la conjugaison de $N$ par $g_2$ est triviale, ce qui vient du fait qu'en conjuguant $7$ fois de suite, on est sensé faire l'identité, et $k\mapsto (n_1\mapsto g_2^{-k} n_1 g_2)$ est un morphisme, et $N$ n'a pas d'automorphisme d'ordre $7$. + + $\Ker \phi$ est un sous-groupe normal de $G$. Tout élément de $g$ donne, par conjugaison, un automorphisme de $\Ker \phi$, qui ne dépend que de la classe de $g$ modulo $\Ker \phi$ (car $\Ker \phi$ est commutatif). + + On a un morphisme $H\ra \op{Aut}(\Ker \phi)$, qui est forcément trivial, donc $(g_1,n_1)\mapsto g_1 n_1$ est un morphisme. +#+END_proof + # Cf année précédente #+begin_exercice [X MP 2023 # 280] -Soit $G$ un groupe fini de neutre 1. Soit $\phi$ un automorphisme de $G$ sans point fixe c'est-a-dire tel que : $\forall x\in G$, $\phi(x)=x\Rightarrow x=1$. On note $n$ l'ordre de $\phi$ ; c'est le plus petit entier $n\in\N^*$ tel que $\phi^n=\op{id}$. +Soit $G$ un groupe fini de neutre $1$. Soit $\phi$ un automorphisme de $G$ sans point fixe c'est-a-dire tel que : $\forall x\in G$, $\phi(x)=x\Rightarrow x=1$. On note $n$ l'ordre de $\phi$ ; c'est le plus petit entier $n\in\N^*$ tel que $\phi^n=\op{id}$. - Montrer que $\forall x\in G$, $x\,\phi(x)\,\phi^2(x)\,\cdots\,\phi^{n-1}(x)=1$. - Si $n=2$, que peut-on dire du groupe $G$? Donner un exemple. - Si $n=3$, montrer que, pour tout $x\in G$, $x$ et $\phi(x)$ commutent. @@ -2533,23 +2562,31 @@ Soit $G$ un groupe fini d'ordre $n$. On note, pour tout diviseur positif $d$ de - Montrer que, si pour tout diviseur positif $d$ de $n$, $|\{x\in G,\ x^d=1\}|\leq d$, alors $G$ est cyclique. - Soient $\mathbb{K}$ un corps et $G$ un sous-groupe fini de $\mathbb{K}^*$. Montrer que $G$ est cyclique. #+end_exercice -# Cf un précédent +#+call: get_exo(4216) #+begin_exercice [X MP 2023 # 284] -On pose $\Q[i]=\{a+ib\;;\ a,b\in\Q\}$. +On pose $\Q[i]=\{a+ib,\,a,b\in\Q\}$. - Montrer que $\Q[i]$ est un sous-corps de $\C$. - Déterminer les éléments de $\Q[i]\setminus\{0\}$ qui sont d'ordre fini. #+end_exercice +# ID:7023 #+begin_exercice [X MP 2023 # 285] -- Soient $\mathbb{K}$ un corps, $(a,b)\in\mathbb{K}^2$, $P=X^2-aX-b$. On considère la $\mathbb{K}$-algèbre $A$ admettant une base sur $\mathbb{K}$ de la forme $(1,x)$ avec $x^2=ax+b$. À quelle condition cette algèbre est-elle un corps? + - Soient $\mathbb{K}$ un corps, $(a,b)\in\mathbb{K}^2$, $P=X^2-aX-b$. On considère la $\mathbb{K}$-algèbre $A$ admettant une base sur $\mathbb{K}$ de la forme $(1,x)$ avec $x^2=ax+b$. À quelle condition cette algèbre est-elle un corps? - On suppose que $\mathbb{K}=\mathbb{F}_p$ ou $p$ est un nombre premier. Combien de $\mathbb{F}_p$-algèbres non isomorphes peut-on obtenir ainsi? #+end_exercice +#+BEGIN_proof + - À condition que $P$ soit irréductible. + - Si c'est un corps : $(x - a/2)^2 = c$ on obtient $x'$ avec $x'^2 = c$, avec $c$ n'est pas un carré, et il y une unique classe de non carré. + + Si c'est pas un corps, $x^2 = 1$, l'algèbre est isomorphe à $\m F_p \times \m F_p$. +#+END_proof + #+BEGIN_exercice [X MP 2023 # 286] Soit $p$ un nombre premier. On suppose que, pour toute $\mathbb{F}_p$-algèbre $A$, il existe un endomorphisme $u_A$ de $A$ de sorte que, pour tout couple $(A, B)$ de $\mathbb{F}_p$-algèbres et tout morphisme $\tau$ de $\mathbb{F}_p$-algèbres de $A$ dans $B$, on ait $\tau \circ u_A=u_B \circ \tau$. Que dire des $u_A$ ? #+END_exercice #+BEGIN_proof -Pour tout isomorphisme $\tau\colon A\ra \A$, $u_A$ commute avec $\tau$. +Pour tout morphisme d'algèbre $\tau\colon A\ra A$, $u_A$ commute avec $\tau$. Donc $u_A(PMP^{-1}) = Pu_A(M)P^{-1}$, donc $u_A = \op{Id}$, ou $u_A = 0$. #+END_proof #+begin_exercice [X MP 2023 # 287] :sup: @@ -2583,15 +2620,15 @@ On pose $F\colon z\in\C^*\mapsto\prod_{k=1}^n(1+q^{2k-1}z)(1+q^{2k-1}z^{-1})$. Ind. Exprimer $F(q^2z)$ en fonction de $F(z)$. #+end_exercice #+BEGIN_proof - - Existence claire, unicité via l'unicité polynomiale. - - On a $F(q^2z)$. + - Existence claire, via $f(z) = f(1/z)$. Unicité via l'unicité polynomiale. + - On a $F(q^2z) = F(z) \frac{1 + q^{2n+1}z}{1 + qz} \frac{1 + z^{-1}q^{-1}}{1 + q^{2n-1} z^{-1}}$. Multiplier par le dénominateur, identifié. #+END_proof #+begin_exercice [X MP 2023 # 290] Soit $p$ un nombre premier. Trouver tous les entiers $n\in\N$ tels que $(X+Y)^n$ soit congru a $X^n+Y^n$ modulo $p$. #+end_exercice #+BEGIN_proof -Pour $n=p$ ok. Sinon, cf Lucas pour les coefficients binomiaux, on veut que $n$ divise tous les ${n\choose k}$. +Pour $n=p$ ok, si $n = p^{\a}$ aussi. Sinon, cf Lucas pour les coefficients binomiaux : on choisit un $p$ qui divise $n$, on veut $\forall u,\, \lfloor \frac{n}{p^u}\rfloor = \lfloor \frac{k}{p^u}\rfloor + \lfloor \frac{n-k}{p^u}\rfloor$, il faut écrire la décomposition en base $p$. #+END_proof @@ -2599,7 +2636,7 @@ Pour $n=p$ ok. Sinon, cf Lucas pour les coefficients binomiaux, on veut que $n$ Soit $f\in\C[X]$ tel que $f(0)\neq 0$. Soit $(k,n)\in(\N^*)^2$. Montrer qu'il existe $P\in\C[X]$ tel que $X^n$ divise $P^k-f$. #+end_exercice #+BEGIN_proof -C'est des DLs. +C'est des DLs : faire le DL de $\sqrt[k]{f}$. #+END_proof #+begin_exercice [X MP 2023 # 292] @@ -2611,7 +2648,7 @@ Soit $p$ un nombre premier. Pour deux polynômes $P,Q$ dans $\Z[X,Y]$, on note $ #+BEGIN_proof - $T^p$ - $T^p$ - - + - La première dit que $P$ est un monôme, puisque $P(X)P(Y)$ est une somme de sommandes que l'on peut identifier. La seconde que c'est un $X^{p^{\a}}$. #+END_proof @@ -2624,14 +2661,17 @@ Interpolation de Hermite. #+begin_exercice [X MP 2023 # 294] - - Soient $N_1,\ldots,N_r$ des entiers premiers entre eux deux a deux, et $f_1,\ldots,f_r$ des entiers. Montrer qu'il existe un entier $F$ tel que $F\equiv f_i\ [N_i]$ pour tout $i\in\db{1,r}$. - - Soient $N_1,\ldots,N_r$ des éléments de $\C[X]$ premiers entre eux deux a deux, et $f_1,\ldots,f_r$ des éléments de $\C[X]$. Montrer qu'il existe $F\in\C[X]$ tel que $N_i$ divise $F-f_i$ pour tout $i\in\db{1,r}$. + - Soient $N_1,\ldots,N_r$ des entiers premiers entre eux deux à deux, et $f_1,\ldots,f_r$ des entiers. Montrer qu'il existe un entier $F$ tel que $F\equiv f_i\ [N_i]$ pour tout $i\in\db{1,r}$. + - Soient $N_1,\ldots,N_r$ des éléments de $\C[X]$ premiers entre eux deux à deux, et $f_1,\ldots,f_r$ des éléments de $\C[X]$. Montrer qu'il existe $F\in\C[X]$ tel que $N_i$ divise $F-f_i$ pour tout $i\in\db{1,r}$. - Soient $f,g$ deux éléments de $\C[X]$ premiers entre eux, et $n\in\N^*$. Montrer qu'il existe $h\in\C[X]$ tel que $g$ divise $h^n-f$. #+end_exercice #+BEGIN_proof - - - - Se ramener au cas de $g = X^n$, via ce qui précède, peut-être. + - Se ramener au cas de $g = X^k$, qui se fait par DL via ce qui précède, peut-être. + + $g = \prod (X-\a_i)^{\b_i}$. On veut que $\forall i,\, h^n \equiv f [(X-\a_i)^{\b_i}]$, + Via des DLs, il existe $h_i$ tel qu'il suffise que $h\equiv h_i [(X-\a_i)^{\b_i}]$ (ici on utilise $f\wedge g = 1$, $h_i$ est obtenu en prenant une racine $n$-ième de $f$). #+END_proof @@ -2639,7 +2679,13 @@ Interpolation de Hermite. Soit $n\in\N$. Le polynôme $X^{n+1}-nX^n+1$ est-il irréductible dans $\Z[X]$? #+end_exercice #+BEGIN_proof -!! Pour $n=2$, $1$ est racine :) +Pour $n=2$, $1$ est racine :) + +Critère de Perron : si $|a_{n-1}|\gt 1 + |a_0| + \dots + |a_{n-2}|$ et $a_0\neq 0$. + +D'une part, il n'y a pas de racines de module $1$. Donc au moins une racine complexe est $\gt 1$. + +On écrit $P = (X-x_1) Q$. Et on obtient que $\sum_{i=0}^{n-2} |q_i|\lt 1$. Donc toutes les racines de $Q$ sont de module $\lt 1$. #+END_proof @@ -2655,8 +2701,10 @@ Soit $P\in\Z[X]$ possedant $n$ racines distinctes $x_1,\ldots,x_n\in\Z$. On écr - En déduire qu'il existe $i\in\{1,\ldots,r\}$ tel que $\deg(Q_i)\geq 2\left\lfloor\frac{n+1}{2}\right\rfloor$. #+end_exercice #+BEGIN_proof - - - - C'est juste le degré de $R$. + - $R' = \sum Q_i' Q_i$ ; $2 P P' = (\prod Q_i)\sum \frac{Q_i'}{Q_i}$ + - $R$ est de degré $\geq 2n$. Donc un des $Q_i$ est de degré $\geq n$. Si $n$ pair c'est bon. + + Si $n$ est impair, il faut montrer qu'un des $Q_i$ est de degré $n+1$. Cela vient du fait que $P^2 + 1 = \prod Q_i$, donc les $Q_i$ ne peuvent pas avoir de racines réelles. #+END_proof @@ -2719,7 +2767,7 @@ Donc on est point fixe si et seulement si $(ad - bc)(bc - ad) = adbc \ssi (ad-bc #+END_proof -# À relier… +# Relier à 6486 #+begin_exercice [X MP 2023 # 302] Soit $R\in\M_n(\Z)$ non nulle et $M=I_n+3R$. Montrer que, pour tout $k\in\N^*$, $M^k\neq I_n$. #+end_exercice @@ -2762,19 +2810,26 @@ Pour une matrice de projecteur $P\in\M_n(\mathbb{K})$, on pose $R_P=\det(I_n+(X- #+begin_exercice [X MP 2023 # 306] -Soient $E$ un $\C$-espace vectoriel de dimension finie $n\geq 1$ et $V$ un sous-espace vectoriel de $\mc{L}(E)$. On suppose qu'il existe une application $q:V\to\C$ telle que $u^2=q(u)\,\mathrm{id}$ pour tout $u\in V$. - - Monter que, pour tous $u,v\in V$, il existe $B(u,v)\in\C$ tel que $uv+vu=2B(u,v)\,\mathrm{id}_E$. - - Montrer que $B$ est une forme bilineaire. +Soient $E$ un $\C$-espace vectoriel de dimension finie $n\geq 1$ et $V$ un sous-espace vectoriel de $\mc{L}(E)$. On suppose qu'il existe une application $q\colon V\to\C$ telle que $u^2=q(u)\op{id}_E$ pour tout $u\in V$. + - Monter que, pour tous $u,v\in V$, il existe $B(u,v)\in\C$ tel que $uv+vu=2B(u,v)\op{id}_E$. + - Montrer que $B$ est une forme bilinéaire. - Soient $d\geq 1$ et $u_1,\ldots,u_d\in V$ tels que $B(u_i,u_j)=-\delta_{ij}$ pour tous $i,j\in\db{1,n}$. Montrer que $(u_1,\ldots,u_d)$ est libre. - Soient $d\geq 2$ et $u_1,\ldots,u_d\in V$ tels que $B(u_i,u_j)=-\delta_{ij}$ pour tous $i,j\in\db{1,n}$. Montrer que les $u_i$ sont de trace nulle, et que $\dim E$ est paire. #+end_exercice +#+BEGIN_proof + - $(u+v)^{2}$ + - Se vérifie. + - Probablement comme si ils étaient orthogonaux… + - $u_{i}^{2} = -\op{Id}_{E}$ et $u_{i}u_{j} + u_{j} u_{i} = 0$ et les endomorphismes anti-commutent, donc $u_{j}$ envoie $E_{i}$ sur $E_{-i}$. +#+END_proof -# À relier à un classique. + +# À relier à un classique : 5692 #+begin_exercice [X MP 2023 # 307] Soit $n\in\N$ avec $n\geq 2$. Soit $\phi\in\mc{L}\left(\M_n(\C)\right)$. On suppose que $\phi(I_n)$ est inversible et que $\forall A,B\in\M_n(\C)$, $\phi(AB)=\phi(A)\,\phi(B)$. Montrer qu'il existe $P\in\mathrm{GL}_n(\C)$ tel que : $\forall A\in\M_n(\C)$, $\phi(A)=PAP^{-1}$. #+end_exercice -# Relier : automorphismes d'algèbres de $\C(X)$ +# Relier : automorphismes d'algèbres de $\C(X)$ : 4136. #+begin_exercice [X MP 2023 # 308] - Caracteriser les endomorphismes $\phi$ de $\C(X)$ vérifiant $(*)$ : $\forall F_1,\,F_2\in\C(X)$, $\phi(F_1F_2)=\phi(F_1)\,\phi(F_2)$. - Déterminer les automorphismes de $\C(X)$ vérifiant $(*)$. @@ -3313,7 +3368,7 @@ Pour $x$ réel, on pose $f(x)=\sum_{n=1}^{+\i}\dfrac{x^n}{1-x^n}$ sous reserve d # Relier à un précédent #+begin_exercice [X MP 2023 # 364] - Soient $U$ un voisinage de $0$ dans $\C$, et $f:U\to\C$ somme d'une série entière. Soit $k\in\N^*$ tel que $f(z)=O(z^k)$ quand $z$ tend vers $0$. Montrer que, pour $r$ voisin de $0^+$, il existe au moins $2k$ nombres complexes $z$ de module $r$ tels que $f(z)$ soit un nombre réel. - - Soient $A$ et $B$ deux polynômes a coefficients réels dont toute combinaison lineaire a coefficients réels est scindée ou nulle. Soient $x\lt y$ deux racines de $A$. Montrre que $[x,y]$ contient au moins une racine de $B$. + - Soient $A$ et $B$ deux polynômes a coefficients réels dont toute combinaison linéaire a coefficients réels est scindée ou nulle. Soient $x\lt y$ deux racines de $A$. Montrre que $[x,y]$ contient au moins une racine de $B$. #+end_exercice @@ -3412,7 +3467,7 @@ Soient $\alpha,\beta\gt 0$. Pour $x\gt 0$, on pose $I(x)=\int_0^{+\i}t^{\beta-1} #+begin_exercice [X MP 2023 # 375] - Soient $K$ un segment et $f:K\to K$ une fonction continue croissante. Montrer que $f$ admet un point fixe. - - On considère l'équation différentielle non lineaire $(E):\ x'=\cos(x)+\cos(t)$. On admet que pour tout $a\in\R$ il existe une unique solution $\phi_a$ de $(E)$ sur $\R$ vérifiant $\phi(0)=a$, et que, pour tous $a,b$ réels distincts, les fonctions $\phi_a$ et $\phi_b$ ne coincident en aucun point. Montrer que $(E)$ possede une solution $2\pi$-périoddique. + - On considère l'équation différentielle non linéaire $(E):\ x'=\cos(x)+\cos(t)$. On admet que pour tout $a\in\R$ il existe une unique solution $\phi_a$ de $(E)$ sur $\R$ vérifiant $\phi(0)=a$, et que, pour tous $a,b$ réels distincts, les fonctions $\phi_a$ et $\phi_b$ ne coincident en aucun point. Montrer que $(E)$ possede une solution $2\pi$-périoddique. #+end_exercice # ID:6896 @@ -3794,7 +3849,7 @@ On pose$M=\left(\begin{array}{cccc}a&-b&-c&-d\\ b&a&d&-c\\ c&-d&a&b\\ d&c&-b&a\e #+end_exercice #+begin_exercice [X PSI 2023 # 422] -On considère deux variables aléatoires $X$ et $Y$ indépendantes a valeurs dans $\db{0,n]\!]$ qui suivent la meme loi. Trouver les lois de $X$ possibles pour que $X+Y$ suive la loi uniforme sur $[\![0,n}$. +On considère deux variables aléatoires $X$ et $Y$ indépendantes a valeurs dans $\db{0,n}$ qui suivent la meme loi. Trouver les lois de $X$ possibles pour que $X+Y$ suive la loi uniforme sur $\db{0,n}$. #+end_exercice #+begin_exercice [X PSI 2023 # 423] @@ -3831,7 +3886,7 @@ Caracteriser les matrices $A\in\M_n(\C)$ telles que l'ensemble des matrices semb #+end_exercice #+begin_exercice [X PC 2023 # 440] -Soit $G$ une partie de $\mathrm{GL}_2(\R)$ qui contient $I_2$ et qui est stable par produit et passage a l'inverse. On note $\mathrm{Vect}(G)$ l'ensemble des combinaisons lineaires d'éléments de $G$. Montrer que $\mathrm{Vect}(G)\neq\M_2(\R)$ si et seulement si une des conditions suivantes est vérifiée : +Soit $G$ une partie de $\mathrm{GL}_2(\R)$ qui contient $I_2$ et qui est stable par produit et passage a l'inverse. On note $\mathrm{Vect}(G)$ l'ensemble des combinaisons linéaires d'éléments de $G$. Montrer que $\mathrm{Vect}(G)\neq\M_2(\R)$ si et seulement si une des conditions suivantes est vérifiée : (i) il existe $P\in\mathrm{GL}_2(\R)$ telle que, pour toute $M\in G$, la matrice $P^{-1}MP$ est triangulaire supérieure, @@ -3935,7 +3990,7 @@ Soient $A\in\M_{m,n}(\R)$ non nulle, $X\in\R^n$ et $Y\in\R^m$. On munit $\R^n$ e - Montrer que (i) $\Longleftrightarrow$ (i)'. - On suppose (i) vérifie. Montrer qu'alors (ii) $\Longleftrightarrow$ (ii)'. - Montrer l'unicite de $X$ vérifiant (i) et (ii). Notons $BY$ ce vecteur. - - Montrer que $B$ est lineaire. Montrer que, pour tout $Y\in\R^m$, $\|BY\|\leq\frac{\|Y\|}{\sqrt{\lambda_1}}$ ou $\lambda_1$ est la plus petite valeur propre non nulle de $A^TA$, et qu'il y a des cas d'égalité non triviaux. + - Montrer que $B$ est linéaire. Montrer que, pour tout $Y\in\R^m$, $\|BY\|\leq\frac{\|Y\|}{\sqrt{\lambda_1}}$ ou $\lambda_1$ est la plus petite valeur propre non nulle de $A^TA$, et qu'il y a des cas d'égalité non triviaux. #+end_exercice #+begin_exercice [X PC 2023 # 457] @@ -3950,7 +4005,7 @@ Soit $\alpha\in\,]0,1[$. #+end_exercice #+begin_exercice [X PC 2023 # 459] -Soit $f\colon\mc{S}_n(\R)\to\R$ une forme lineaire. Montrer l'equivalence des trois assertions suivantes : +Soit $f\colon\mc{S}_n(\R)\to\R$ une forme linéaire. Montrer l'equivalence des trois assertions suivantes : i) $\forall A\in\M_n(\R),\ f\big{(}AA^T\big{)}\geq 0$; @@ -4152,8 +4207,8 @@ Posons $ A=\Q\cap\left[\,0\,;\,1\,\right]$. Existe-t-il une suite $(f_n)$ de fon #+begin_exercice [X PC 2023 # 494] On considère l'ensemble $E$ des applications continues $ f\colon\R\mapsto\R$ telles qu'il existe $ M\gt 0$ vérifiant $\forall x,y\in\R,\left|f(x+y)-f(x)-f(y)\right|\leq M$. - - Montrer que $E$ est un espace vectoriel contenant le sous-espace des applications lineaires et celui des applications bornées. - - Soit $ f\in E$. Pour $n\in\N$, on pose $ g_n:x\in\R\mapsto 2^{-n}f(2^nx)$. Montrer que la suite $(g_n)$ converge uniformément vers une application lineaire $g$. En déduire que $f$ s'écrit, de facon unique, comme somme d'une application lineaire et d'une application bornée. + - Montrer que $E$ est un espace vectoriel contenant le sous-espace des applications linéaires et celui des applications bornées. + - Soit $ f\in E$. Pour $n\in\N$, on pose $ g_n:x\in\R\mapsto 2^{-n}f(2^nx)$. Montrer que la suite $(g_n)$ converge uniformément vers une application linéaire $g$. En déduire que $f$ s'écrit, de facon unique, comme somme d'une application linéaire et d'une application bornée. #+end_exercice #+begin_exercice [X PC 2023 # 495] @@ -5235,7 +5290,7 @@ Soient $(E,\|\ \|)$ un espace norme réel de dimension finie, $k\in]0,1[$, $f$ u #+begin_exercice [Mines 2023 # 680] Soit $E=\mc C^0([-1,1],\R)$ muni de la norme de la convergence uniforme. Pour $f\in E$ on pose $\phi(f)=\int_0^1f(t)\,dt-\int_{-1}^0f(t)\,dt$. - - Montrer que $\phi$ est une forme lineaire continue sur $E$ et calculer $\|\phi\|$. + - Montrer que $\phi$ est une forme linéaire continue sur $E$ et calculer $\|\phi\|$. - Existe-t-il $f$ unitaire telle que $|\phi(f)|=\|f\|\,?$ #+end_exercice @@ -5340,7 +5395,7 @@ Pour $n\in\N^*$, on pose $ u_n=\sum_{k=1}^n\sin\left(\frac{\sqrt{k}}{n}\right)$. #+begin_exercice [Mines 2023 # 696] Soit $\mc{B}$ le sous-espace de $\C^{\Z}$ forme des suites $(u - {n\in\Z}$ bornées. Soit $T$ l'endomorphisme de $\mc{B}$ qui a $(u - {n\in\Z}$ associe $(u_{n+1})_{n\in\Z}$. - - Montrer que $T$ est lineaire. Déterminer ses valeurs propres et ses sous-espaces propres. + - Montrer que $T$ est linéaire. Déterminer ses valeurs propres et ses sous-espaces propres. - Déterminer les sous-espaces de dimension finie de $\mc{B}$ stables par $T$. #+end_exercice @@ -5917,7 +5972,7 @@ Soit $P\in\R[X]$ de degré $p\in\N^*$. #+begin_exercice [Mines 2023 # 805] Soit $f:x\mapsto(\arcsin(x))^2$. - - Montrer que $f$ est solution d'une équation différentielle lineaire d'ordre $2$, sur un intervalle que l'on precisera. + - Montrer que $f$ est solution d'une équation différentielle linéaire d'ordre $2$, sur un intervalle que l'on precisera. - Montrer que $f$ est développable en série entière au voisinage de $0$. Experimer les coefficients de ce développement en série entière et donner son rayon de convergence. #+end_exercice @@ -6154,7 +6209,7 @@ Déterminer les fonctions $y$ de $\R$ dans $\R$ dérivables et telles que $y'(x) #+begin_exercice [Mines 2023 # 844] Soit $f$ la fonction de $\R$ dans $\R$ telle que $f(0)=0$ et que $\forall x\in\R^*,f(x)=e^{-1/x^2}$. - Montr per que $f$ est de classe $C^{\i}$ sur $\R$. - - La fonction $f$ est-elle solution d'une équation différentielle lineaire homogéné? + - La fonction $f$ est-elle solution d'une équation différentielle linéaire homogéné? #+end_exercice #+begin_exercice [Mines 2023 # 845] @@ -6166,7 +6221,7 @@ Soient $n\in\N^*$ et $\omega\in\C$ tel que $\omega^n=1$. Trouver les fonctions $ #+end_exercice #+begin_exercice [Mines 2023 # 847] -On considère la fonction $f\colon\R\to\R$ définie par : $f(x)=\exp(-x^{-2})$ si $x\neq 0$ et $f(0)=0$. Montr per que $f$ n'est solution d'aucune équation différentielle lineaire homogéné a coefficients constants (d'ordre quelconque). +On considère la fonction $f\colon\R\to\R$ définie par : $f(x)=\exp(-x^{-2})$ si $x\neq 0$ et $f(0)=0$. Montr per que $f$ n'est solution d'aucune équation différentielle linéaire homogéné a coefficients constants (d'ordre quelconque). #+end_exercice #+begin_exercice [Mines 2023 # 848] @@ -6267,7 +6322,7 @@ Soit $f:(\R^+)^2\to\R$ définie par $f(0,0)=0$ et $f(x,y)=\frac{xy}{(x+1)(y+1)(x #+end_exercice #+begin_exercice [Mines 2023 # 865] -Soient $E$ un espace vectoriel normé de dimension finie, $f$ une forme lineaire sur $E$. +Soient $E$ un espace vectoriel normé de dimension finie, $f$ une forme linéaire sur $E$. Montrer que l'application $g\colon x\in E\mapsto f(x)\,e^{-\lN x\rN^2}$ admet un minimum et un maximum, puis déterminer ce maximum et ce minimum. #+end_exercice @@ -7061,7 +7116,7 @@ Un espace norme réel est dit separable lorsqu'il contient une partie denombrabl #+begin_exercice [Centrale 2023 # 1245] Soit $E$ l'espace des fonctions polynomiales de $\R$ dans $\R$. Pour tout $f\in E$, on note $\phi(f)$ la primitive de $f$ d'intégrale nulle sur l'intervalle $[0,1]$. - - Justifier la définition de $\phi$ puis etablir qu'il s'agit d'une application lineaire sur $E$. + - Justifier la définition de $\phi$ puis etablir qu'il s'agit d'une application linéaire sur $E$. 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