Olympiades ENSAE - 1er S 2025
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.
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In Virginia, moderate former Rep. Abigail Spanberger turned in the strongest Democratic performance in the state’s recent history as she coasted to victory. And in New Jersey, another moderate, Rep. Mikie Sherrill, busted apart the coalition Trump and her Republican rival, former state lawmaker Jack Ciattarelli, had put together to close the Garden State’s gap in recent elections.
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In New York City, democratic socialist Zohran Mamdani’s win marked the second time this year he’d defeated former Gov. Andrew Cuomo — first in the Democratic primary, and then in the general election, with Cuomo running as an independent backed by Trump.
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The Democratic wins by candidates with sharp ideological differences will do little to settle the party’s long-raging internal debate about its way forward, with a host of competitive midterm primaries just months away and the 2028 presidential primary already looming.
But their campaigns had some things in common. Though their solutions were different, the candidates focused on the issue of affordability. And they were all fiercely critical of Trump’s performance.
“It’s not just a message about Democrats; it’s a message about our entire country. I think Americans are appalled by what they are seeing coming out of this administration,” New York Rep. Alexandria Ocasio-Cortez said on CNN from Mamdani’s victory party.
In California, voters approved by a wide margin a redistricting ballot measure intended to boost Democrats’ chances in next year’s battle for control of the House. And in Pennsylvania, Democratic state Supreme Court justices won their retention votes, allowing Democrats to keep their majority on the high court in a perennial battleground state where legal challenges over voting rules are all but certain.
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Voters up and down the East Coast delivered Democrats a sweep on Tuesday, electing candidates across the party’s ideological spectrum in a vivid show of discontent with President Donald Trump nearly a year into his second term.
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In Virginia, moderate former Rep. Abigail Spanberger turned in the strongest Democratic performance in the state’s recent history as she coasted to victory. And in New Jersey, another moderate, Rep. Mikie Sherrill, busted apart the coalition Trump and her Republican rival, former state lawmaker Jack Ciattarelli, had put together to close the Garden State’s gap in recent elections.
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In New York City, democratic socialist Zohran Mamdani’s win marked the second time this year he’d defeated former Gov. Andrew Cuomo — first in the Democratic primary, and then in the general election, with Cuomo running as an independent backed by Trump.
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The Democratic wins by candidates with sharp ideological differences will do little to settle the party’s long-raging internal debate about its way forward, with a host of competitive midterm primaries just months away and the 2028 presidential primary already looming.
But their campaigns had some things in common. Though their solutions were different, the candidates focused on the issue of affordability. And they were all fiercely critical of Trump’s performance.
“It’s not just a message about Democrats; it’s a message about our entire country. I think Americans are appalled by what they are seeing coming out of this administration,” New York Rep. Alexandria Ocasio-Cortez said on CNN from Mamdani’s victory party.
In California, voters approved by a wide margin a redistricting ballot measure intended to boost Democrats’ chances in next year’s battle for control of the House. And in Pennsylvania, Democratic state Supreme Court justices won their retention votes, allowing Democrats to keep their majority on the high court in a perennial battleground state where legal challenges over voting rules are all but certain.
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Voters up and down the East Coast delivered Democrats a sweep on Tuesday, electing candidates across the party’s ideological spectrum in a vivid show of discontent with President Donald Trump nearly a year into his second term.
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In Virginia, moderate former Rep. Abigail Spanberger turned in the strongest Democratic performance in the state’s recent history as she coasted to victory. And in New Jersey, another moderate, Rep. Mikie Sherrill, busted apart the coalition Trump and her Republican rival, former state lawmaker Jack Ciattarelli, had put together to close the Garden State’s gap in recent elections.
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In New York City, democratic socialist Zohran Mamdani’s win marked the second time this year he’d defeated former Gov. Andrew Cuomo — first in the Democratic primary, and then in the general election, with Cuomo running as an independent backed by Trump.
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The Democratic wins by candidates with sharp ideological differences will do little to settle the party’s long-raging internal debate about its way forward, with a host of competitive midterm primaries just months away and the 2028 presidential primary already looming.
But their campaigns had some things in common. Though their solutions were different, the candidates focused on the issue of affordability. And they were all fiercely critical of Trump’s performance.
“It’s not just a message about Democrats; it’s a message about our entire country. I think Americans are appalled by what they are seeing coming out of this administration,” New York Rep. Alexandria Ocasio-Cortez said on CNN from Mamdani’s victory party.
In California, voters approved by a wide margin a redistricting ballot measure intended to boost Democrats’ chances in next year’s battle for control of the House. And in Pennsylvania, Democratic state Supreme Court justices won their retention votes, allowing Democrats to keep their majority on the high court in a perennial battleground state where legal challenges over voting rules are all but certain.
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Voters up and down the East Coast delivered Democrats a sweep on Tuesday, electing candidates across the party’s ideological spectrum in a vivid show of discontent with President Donald Trump nearly a year into his second term.
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In Virginia, moderate former Rep. Abigail Spanberger turned in the strongest Democratic performance in the state’s recent history as she coasted to victory. And in New Jersey, another moderate, Rep. Mikie Sherrill, busted apart the coalition Trump and her Republican rival, former state lawmaker Jack Ciattarelli, had put together to close the Garden State’s gap in recent elections.
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In New York City, democratic socialist Zohran Mamdani’s win marked the second time this year he’d defeated former Gov. Andrew Cuomo — first in the Democratic primary, and then in the general election, with Cuomo running as an independent backed by Trump.
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The Democratic wins by candidates with sharp ideological differences will do little to settle the party’s long-raging internal debate about its way forward, with a host of competitive midterm primaries just months away and the 2028 presidential primary already looming.
But their campaigns had some things in common. Though their solutions were different, the candidates focused on the issue of affordability. And they were all fiercely critical of Trump’s performance.
“It’s not just a message about Democrats; it’s a message about our entire country. I think Americans are appalled by what they are seeing coming out of this administration,” New York Rep. Alexandria Ocasio-Cortez said on CNN from Mamdani’s victory party.
In California, voters approved by a wide margin a redistricting ballot measure intended to boost Democrats’ chances in next year’s battle for control of the House. And in Pennsylvania, Democratic state Supreme Court justices won their retention votes, allowing Democrats to keep their majority on the high court in a perennial battleground state where legal challenges over voting rules are all but certain.
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