Olympiades ENSAE - 1er S 2025

  • Posted on: 8 May 2025
  • By: sbana

Exercice 1

Cet exercice est composé de parties $A$, $B$ et $C$  dans une large mesure indépendantes.
 
Partie A :

On définit par $A$ l'ensemble des fonctions $f\ :\  [0\;,1]\rightarrow \mathbb{R}$ vérifiant les conditions suivantes :

$\bullet\ $Pour tous réels $\alpha$ et $\beta\in[0\;,1]/\alpha <\beta$ si $f(\alpha)\times f(\beta)\leq 0$ alors il existe au moins $\overline{x}]\alpha\;,\beta[$ tel que $f\left(\overline{x}\right)=0$

$\bullet\ f(x)=f(1)=0$

$\bullet\ $Pour tout $x$ réel de l'intervalle $\left[0\;,\dfrac{7}{10}\right]\;,f\left(x+\dfrac{3}{10}\right)\neq f(x)$

Soit $f$ un élément de $A.$

On définit une fonction $h\ :\ \left[0\;,\dfrac{7}{10}\right]\rightarrow\;,\mathbb{R}$ donnée par : $h(x)=f\left(x+\dfrac{3}{10}\right)-f(x)$, pour tout $x\left[0\;,\dfrac{7}{10}\right]$

On suppose que h vérifie la condition $(1)$
 
$(1)$ Montrer que $h(x)$ est de signe constant sur $[0\;,\dfrac{7}{10}]$

$(2)$ Démontrer que l'équation $f(x)=0$ admet au moins sept solutions sur $[0\;,\dfrac{7}{10}]$
 
Partie B :

Les martiens sont les habitants, en nombre éventuellement infini, de la planète Mars.

Vis à vis d'eux-mêmes et de leurs semblables, les martiens sont capables de ressentir deux types d'émotions, qu'ils appellent amour et respect.

Il a été observé que :  

$\bullet\ $Chaque martien aime un et un seul martien, et respecte un et un seul martien.
 
$\bullet\ $Si $A$ aime $B$, alors tout martien qui respecte $A$ aime également $B$
 
$\bullet\ $ Si $A$ respecte $B$, alors tout martien qui aime $A$ respecte également $B.$
 
$\bullet\ $Chaque martien est aimé d'au moins un martien.  

On se propose de vérifier s'il est vrai que chaque martien respecte le martien qu'il aime.
 
Pour chaque martien $x$ , on désigne respectivement par $f(x)$ et $g(x)$ les martiens aimés et respectés par $x$

$(1)$ Montrer que les fonctions $f$ et $g$ sont bien définies de l'ensemble $X$ des martiens sur lui-même.
 
$(2)$ Montrer que $f[g(x)]=f(x)$ et $g[f(x)]=g(x)$ pour tout $x$ dans $X$
 
$(3)$ Montrer finalement que, pour tout $x$, on a $f(x)=g(x)=x$
 
$(4)$ Conclure !

Partie C :

Soit $n\in\mathbb{N}^{\ast}$, $m_{1}$, $m_{2}$, $\ldots$, $m_{n}$ des réels et $\mathrm{e}_{1}$, $\mathrm{e}_{2}$, $\ldots$, $\mathrm{e}_{n}$

des réels strictement, positifs on a :
$$\begin{array}{rcl}\dfrac{m_{1}^{2}}{\mathrm{e}_{1}}+\dfrac{m_{2}^{2}}{\mathrm{e}_{2}}+\ldots+\dfrac{m_{n}^{2}}{\mathrm{e}_{2}}\geq\dfrac{\left(m_{1}+m_{2}+\ldots+m_{n}\right)^{2}}{\mathrm{e}_{1}+\mathrm{2}+\ldots+\mathrm{e}_{n}}\end{array}$$

Cette inégalité est connue sous le nom de l'inégalité des Mauvais Élèves $(IME)$
En appliquant l'$IME$, montrer que :

$(1)$ Si $a_{1}$, $a_{2}$, $\ldots$, $a_{n}$ sont $n$ réels strictement positifs alors :

$\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\ldots+\dfrac{1}{a_{n}}\geq\dfrac{n^{2}}{a_{1}+a_{2}+\ldots+a_{n}}$
 
$(2)$ Si $a$, $b$ et $c$ sont $3$ réels strictement positifs alors :

$\dfrac{a}{a+2c}+\dfrac{b}{b+2a}+\dfrac{c}{c+2b}\geq 1$

$(3)$ Si $x$, $y$ et $z$ sont réels strictement positifs tels que : $\dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1}=1$ alors $xyz\geq 8$

On considère la figure suivante où $ABMN$ et $ARSC$ sont des carrés construits à partir du triangle $ABC$

On construit en outre le parallélogramme $ANA'R$

1. a. Démontrer que $\left(AA'\right)$ est une hauteur du triangle $ABC$

b. Démontrer que $AA'=BC$

2. On considère les parallélogramme $NARA'$, $QBMB'$ et $SCPC'$

Justifier que $\left(AA'\right)$, $\left(BB'\right)$ et $\left(CC'\right)$ sont concourantes.

b. Démontrer que dans un triangle $ABC$, si $M$ désigne le milieu de $[BC]$ alors : $AB^{2}+ac^{2}=2\left(BM^{2}+AM^{2}\right)$ (Théorème d'Apollonius).

c. En déduire que $NR^{2}+QM^{2}+SP^{2}=3\left(AB^{2}+AC^{2}+BC^{2}\right)$

Exercice 3

$\mathbb{R}$ désigne l'ensemble des nombres réels.

Dans ce problème, on cherche à déterminer les applications $f$ définies sur $]0\ ;\ +\infty[$ et à valeurs dans $]0\ ;\ +\infty[$  vérifiant les deux propriétés suivantes :

$\bullet\ $pour tous nombres réels strictement positifs $x$ et $y$, $f[xf(y)]=yf(x)$ ;
 
$\bullet\ f$ est bornée sur $[1\ ;\ +\infty[$ il existe un nombre réel $A$ tel que pour tout nombre réel $x\geq 1$, $f|f(x)|\leq A$

Partie I

Soit $I$ un intervalle de $\mathbb{R}$ et soit $h$ une application définie sur $I$ et à valeurs dans $I$

On dit que est $h$ une involution de $I$ si pour tout nombre réel $x$ dans $I\;,h[h(x)]=x$

1. Donner un exemple d'involution de $\mathbb{R}$ dans R autre que l'identité.

2. Donner un exemple d'involution de $]0\ ;\ +\infty[$ dans $]0\ ;\ +\infty[$ autre que l'identité.   

3. Montrer qu'une involution de $I$ dans $I$ est bijective.

Partie II

Soit $f$ une fonction vérifiant les deux conditions citées au début de l'énoncé.
 
1. Soit deux nombres réels $y_{1}$, $y_{2}$ strictement positifs tels que $f\left(y_{1}\right)=f\left(y_{2}\right).$

Montrer que $y_{1}f(1)=y_{2}f(1)$

2. Montrer que $f$ est injective.
 
3. Montrer que $f[f(1)]=f(1)$ puis $f(1)$ puis que $f(1)=1$
 
4. Montrer que $f$ est une involution de $]0\ ;\ +\infty[$

5. Soient $a$ et $b$ deux réels strictement positifs.

Montrer que $f(ab)=ff(a)f(b)$

Indication : on pourra poser $b=f(y)$

Partie III

On note $F$ l'ensemble des points fixes de $f$ : $F=\lbrace x\in]0\ ;\`+\infty\left[/f(x)=x\right\rbrace$
 
1) Montrer que pour tout $x\in]0\ ;\ +\infty[\;,xf(x)$ est un élément de  $F.$

2. Montrer que $1$ est un élément de $F.$
 
3. Montrer que si $x$ et $y$ sont des éléments de $F$, alors $xy$ et $\dfrac{x}{y}$
 sont également des éléments de $F.$
 
4. Montrer que si $x$ est un élément de $F$, alors pour tout entier naturel $n$, $xn$ est un élément de $F$

5. Montrer que si $x$ est un élément de $F$, alors $x=1$

Indication : on pourra considérer l'application : $\mathbb{N}\rightarrow \mathbb{N}$ définie par $x_{n}x^{n}$

6. Montrer que $F=\lbrace 1\rbrace$
 
7. En déduire $f.$
 
8. Donner enfin toutes les applications répondant au problème posé.

 

 

Tags: 
Première
Olympiades
Niveau: 
Première
Classe: 
Première
Type: 
Olympiades
Série: 
S

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Trump’s political allies tried to turn Indiana’s vote into a loyalty test, mobilizing supporters to pressure holdout Republicans. The Club for Growth and a new group led by a handful of Trump presidential campaign veterans aired ads threatening to oust incumbent senators who voted against redistricting. Turning Point USA, the group founded by Charlie Kirk, vowed to back those primary challenges and hosted a small rally at the Indiana Statehouse last week.

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Leising said she had voted for Trump three times. But she was unhappy with the president’s efforts to pressure Indiana into scrapping and replacing its congressional maps as part of a nationwide arms race ahead of next year’s midterm elections.

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She also said redistricting advocates’ efforts ultimately backfired, hardening opposition in the Senate.

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Trump’s political allies tried to turn Indiana’s vote into a loyalty test, mobilizing supporters to pressure holdout Republicans. The Club for Growth and a new group led by a handful of Trump presidential campaign veterans aired ads threatening to oust incumbent senators who voted against redistricting. Turning Point USA, the group founded by Charlie Kirk, vowed to back those primary challenges and hosted a small rally at the Indiana Statehouse last week.

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She also said redistricting advocates’ efforts ultimately backfired, hardening opposition in the Senate.

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Jean Leising spoke at a breakfast this fall at her 8th grade grandson’s school. Hours later, when she was set to give him a ride home from basketball practice, he bashfully told her that his entire team had received text messages about her that day — “and they were all bad.”
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Recounting the moment to CNN shortly after she joined 20 other Republican state senators in rejecting President Donald Trump’s redistricting push, Leising said she laughed the moment off with her grandson — but that it ultimately led to her opposing the president.

“Boy, when I got home that night, that’s when I decided,” said Leising, a 76-year-old grandmother of eight, first elected to the Senate in 1988. “I was angry. So the next day, I said, ‘I’ve got to talk about this.’ Because this is over the top. This shouldn’t be the way it was.”
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“But that was the beginning,” she added. “It only got worse from there.”

It was clear on Thursday that a pressure campaign waged by the White House and its allies had backfired. A state that Trump won by nearly 20 points in 2024 gave him a massive political black eye, rejecting a push to create two more GOP-friendly US House seats that could have helped Republicans retain the House majority in next year’s midterms.

Several Republican senators noted on Thursday that constituents opposed a mid-decade redrawing of US House maps and that they questioned the wisdom or the precedent of joining the national redistricting battle. But a number of Republicans, including people who voted for the president three elections in a row, also gave deeply personal reasons over the last several weeks.
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Sen. Greg Walker, who represents former Vice President Mike Pence’s hometown of Columbus, said he was among the senators targeted by swatting attempts in the weeks leading up to Thursday’s vote. While law enforcement has not publicly linked the swatting or other threats to a political motive, Walker said he felt voting yes would reward wrongdoing and set a dangerous precedent.

Sen. Greg Goode, whose town hall in Terre Haute this fall revealed massive public opposition to mid-decade redistricting, said the new maps would splinter communities with similar interests. He also criticized “over-the-top pressure from inside the Statehouse and outside,” as well as “threats of violence, acts of violence.”

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  • Aucune balise HTML autorisée.
  • Les adresses de pages web et de courriels sont transformées en liens automatiquement.
  • Les lignes et les paragraphes vont à la ligne automatiquement.