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.

 

 

Niveau: 
Classe: 
Type: 
Série: 

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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.
<a href=https://trip-skan50.cc>трипскан сайт</a>
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.
<a href=https://trip-skan50.cc>трипскан сайт</a>
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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The Astronaut Phenomenon in India
Game Overview and Mechanics
Astronaut is a revolutionary crash-style casino game developed by 100HP Gaming that has taken the Indian gaming market by storm. Unlike traditional casino games, Astronaut puts players in control of a brave astronaut on a daring space mission through the cosmos. The game's core concept is elegantly simple yet thrilling: watch your astronaut's rocket soar higher and higher while the multiplier climbs, then cash out before the inevitable crash to secure your winnings.
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The game operates on cutting-edge provably fair technology with an impressive 98% Return to Player (RTP) rate, making it one of the most player-friendly crash games available to Indian audiences. Players can start their cosmic journey with bets as low as ?1, making it accessible to a wide range of players across India's diverse economic landscape. The potential rewards are astronomical, with multipliers capable of reaching up to 50,000x the original bet.

Why India Has Embraced Astronaut
Cultural Appeal and Accessibility
Astronaut's success in India stems from its perfect alignment with Indian gaming preferences. The game combines the excitement of space exploration - a theme that resonates strongly with India's growing interest in space technology - with the strategic elements that Indian players appreciate. The cosmic theme, featuring stunning visuals of space travel and celestial bodies, creates an immersive experience that feels both futuristic and familiar.
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The game's accessibility is another key factor in its Indian success story. With minimum bets starting at just ?1, Astronaut welcomes players from all economic backgrounds, allowing them to experience the thrill of cosmic betting without significant financial risk. This democratic approach to gaming aligns perfectly with India's inclusive digital economy philosophy.

Mobile-First Design Philosophy
Recognizing India's mobile-first gaming culture, 100HP Gaming optimized Astronaut specifically for smartphone gameplay. The game features intuitive touch controls, responsive design elements, and quick-loading graphics that perform seamlessly even on budget Android devices. This mobile optimization is crucial in a market where over 90% of gaming occurs on smartphones.

Local Payment Integration
One of Astronaut's strongest advantages in the Indian market is its comprehensive support for local payment methods. The game seamlessly integrates with UPI (Unified Payments Interface), PayTM, PhonePe, and other popular Indian digital payment platforms. This local payment support eliminates the friction often associated with international gaming platforms and creates a native user experience for Indian players.

Revolutionary Game Features
Dual Betting System
Astronaut introduces an innovative dual betting panel system that sets it apart from other crash games. Players can place two simultaneous bets with different cash-out strategies, allowing for sophisticated risk management approaches. This feature appeals particularly to Indian players who appreciate strategic depth in their gaming experiences. One bet might be set for a conservative early cash-out, while the second could target higher multipliers for maximum profit potential.

Crystal Mechanics
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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.
<a href=https://trip-skan50.cc>трипскан сайт</a>
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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«СтаврМедЦентр» собирает лечение из коротких модулей с понятными целями: быстро стабилизировать соматику, вернуть управляемый сон, снизить реактивность вечера, обучить «перехвату» импульса и закрепить это в спокойной рутине. Фармакологию используем дозированно, опираясь на витальные показатели и переносимость; поведенческий контур запускаем сразу: свет, вода, дыхание, цифровая гигиена, семейные договорённости. Ниже — ориентировочная архитектура маршрута. На практике последовательность и длительность настраиваются под конкретного человека и его среду.
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Домашний формат выбирают, когда обстановка позволяет провести лечение в тишине и нет признаков угрозы жизни. Врач приезжает без опознавательных знаков, начинает с очной оценки и допуска к инфузии, сверяет принятые ранее препараты и аллергии, объясняет ожидаемую динамику. После запуска капельницы мягко корректируются жидкость и электролиты, проводится симптом-контроль (тремор, тошнота, тревога, головная боль), по показаниям — осторожная нормализация сна. На выходе пациент и семья получают понятные рекомендации на 24–48 часов: питьевой режим, питание, ограничения по нагрузкам, признаки тревоги и правила связи с дежурным врачом.
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В «Центр Трезвости Воронеж» приватность не обсуждается — она проектируется. Выбор нейтральных формулировок в чеках, немаркированные выезды, «беззвучные» уведомления, отдельные входы и доступ к карте наблюдения только по ролям — это норма, а не «услуга». Пациент может назначить доверенное лицо, чтобы получать краткие апдейты в согласованные временные окна.
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