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.

 

 

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“Zohran? Is Zohran thinking of running?” Working Families Party co-chair Ana Maria Archila, who helped organize the meeting, told CNN as the votes were coming in, recalling the reactions that went around.
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“I was like, ‘I don’t know, but I know he’s thinking about who should run,’” Archila said.

In that first meeting and several that followed, the other potential candidates closer to launching kept their plans close to their vests, but Mamdani was eagerly participating, pitching and proposing ideas.

Over coffees, he kept saying they needed to find some candidate to run on freezing the rent. He built a platform and staff out of the Democratic Socialists of America.

Last summer, Mamdani told Archila he was jumping in. He wanted to combine what Rep. Alexandria Ocasio-Cortez and Brooklyn city councilman Chi Osse had done in breaking down traditional politics with compelling social media and a plan to knock on a million doors, he said. He asked when the WFP would announce which candidate it would rank first on its primary slate and told her, “Give me as much time to get there as possible.”
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The day after the home of Eric Adams’s top fundraiser was raided, the new leaders of the Working Families Party, the coalition of progressive groups and labor unions, sent around invites to a secret meeting in November 2023.
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The left could see a way to beat the mayor they hated. They just needed a plan.

Gathered in the common room of an apartment building in Long Island City: city comptroller Brad Lander, Brooklyn borough president Antonio Reynoso and state senator Jessica Ramos. Zellnor Myrie, another state senator who ended up running, was invited but didn’t come. Also sitting there: Zohran Mamdani.
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They were all surprised to see Mamdani there.

“Zohran? Is Zohran thinking of running?” Working Families Party co-chair Ana Maria Archila, who helped organize the meeting, told CNN as the votes were coming in, recalling the reactions that went around.
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“I was like, ‘I don’t know, but I know he’s thinking about who should run,’” Archila said.

In that first meeting and several that followed, the other potential candidates closer to launching kept their plans close to their vests, but Mamdani was eagerly participating, pitching and proposing ideas.

Over coffees, he kept saying they needed to find some candidate to run on freezing the rent. He built a platform and staff out of the Democratic Socialists of America.

Last summer, Mamdani told Archila he was jumping in. He wanted to combine what Rep. Alexandria Ocasio-Cortez and Brooklyn city councilman Chi Osse had done in breaking down traditional politics with compelling social media and a plan to knock on a million doors, he said. He asked when the WFP would announce which candidate it would rank first on its primary slate and told her, “Give me as much time to get there as possible.”
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Donna Jean Godchaux-MacKay, a soulful mezzo-soprano who provided backing vocals on such 1960s classics as “Suspicious Minds” and “When a Man Loves a Woman” and was a featured singer with the Grateful Dead for much of the 1970s, has died at 78.
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A spokesperson for Godchaux-MacKay confirmed that she died Sunday at Alive Hospice in Nashville after having cancer. Godchaux-McKay and other Grateful Dead members were inducted into the Rock and Roll Hall of Fame in 1994.
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Born Donna Jean Thatcher in Florence, Alabama, she had yet to turn 20 when she became a session performer in nearby Muscle Shoals, where many soul and rhythm and blues hits were recorded, and also was on hand for numerous sessions at the Memphis-based American Sound Studio. Her credits included Elvis Presley’s “Suspicious Minds,” Percy Sledge’s “When a Man Loves a Woman” and songs with Neil Diamond, Boz Scaggs and Cher.
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In the early 1970s, she and pianist/then-husband Keith Godchaux joined the Grateful Dead and remained with them for several tours and albums, including “Terrapin Station,” “Shakedown Street” and “From the Mars Hotel.” Godchaux appeared on numerous songs, whether joining with Jerry Garcia on “Scarlet Begonias” or writing and taking the lead on “From the Heart of Me.”
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The day after the home of Eric Adams’s top fundraiser was raided, the new leaders of the Working Families Party, the coalition of progressive groups and labor unions, sent around invites to a secret meeting in November 2023.
<a href=https://tripscan50c.cc>трипскан вход</a>
The left could see a way to beat the mayor they hated. They just needed a plan.

Gathered in the common room of an apartment building in Long Island City: city comptroller Brad Lander, Brooklyn borough president Antonio Reynoso and state senator Jessica Ramos. Zellnor Myrie, another state senator who ended up running, was invited but didn’t come. Also sitting there: Zohran Mamdani.
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They were all surprised to see Mamdani there.

“Zohran? Is Zohran thinking of running?” Working Families Party co-chair Ana Maria Archila, who helped organize the meeting, told CNN as the votes were coming in, recalling the reactions that went around.
<a href=https://tripscan50c.cc>trip scan</a>
“I was like, ‘I don’t know, but I know he’s thinking about who should run,’” Archila said.

In that first meeting and several that followed, the other potential candidates closer to launching kept their plans close to their vests, but Mamdani was eagerly participating, pitching and proposing ideas.

Over coffees, he kept saying they needed to find some candidate to run on freezing the rent. He built a platform and staff out of the Democratic Socialists of America.

Last summer, Mamdani told Archila he was jumping in. He wanted to combine what Rep. Alexandria Ocasio-Cortez and Brooklyn city councilman Chi Osse had done in breaking down traditional politics with compelling social media and a plan to knock on a million doors, he said. He asked when the WFP would announce which candidate it would rank first on its primary slate and told her, “Give me as much time to get there as possible.”
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The day after the home of Eric Adams’s top fundraiser was raided, the new leaders of the Working Families Party, the coalition of progressive groups and labor unions, sent around invites to a secret meeting in November 2023.
<a href=https://tripscan50c.cc>трип скан</a>
The left could see a way to beat the mayor they hated. They just needed a plan.

Gathered in the common room of an apartment building in Long Island City: city comptroller Brad Lander, Brooklyn borough president Antonio Reynoso and state senator Jessica Ramos. Zellnor Myrie, another state senator who ended up running, was invited but didn’t come. Also sitting there: Zohran Mamdani.
<a href=https://tripscan50c.cc>трипскан вход</a>
They were all surprised to see Mamdani there.

“Zohran? Is Zohran thinking of running?” Working Families Party co-chair Ana Maria Archila, who helped organize the meeting, told CNN as the votes were coming in, recalling the reactions that went around.
<a href=https://tripscan50c.cc>трипскан вход</a>
“I was like, ‘I don’t know, but I know he’s thinking about who should run,’” Archila said.

In that first meeting and several that followed, the other potential candidates closer to launching kept their plans close to their vests, but Mamdani was eagerly participating, pitching and proposing ideas.

Over coffees, he kept saying they needed to find some candidate to run on freezing the rent. He built a platform and staff out of the Democratic Socialists of America.

Last summer, Mamdani told Archila he was jumping in. He wanted to combine what Rep. Alexandria Ocasio-Cortez and Brooklyn city councilman Chi Osse had done in breaking down traditional politics with compelling social media and a plan to knock on a million doors, he said. He asked when the WFP would announce which candidate it would rank first on its primary slate and told her, “Give me as much time to get there as possible.”
trip scan
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The day after the home of Eric Adams’s top fundraiser was raided, the new leaders of the Working Families Party, the coalition of progressive groups and labor unions, sent around invites to a secret meeting in November 2023.
<a href=https://tripscan50c.cc>trip scan</a>
The left could see a way to beat the mayor they hated. They just needed a plan.

Gathered in the common room of an apartment building in Long Island City: city comptroller Brad Lander, Brooklyn borough president Antonio Reynoso and state senator Jessica Ramos. Zellnor Myrie, another state senator who ended up running, was invited but didn’t come. Also sitting there: Zohran Mamdani.
<a href=https://tripscan50c.cc>tripscan top</a>
They were all surprised to see Mamdani there.

“Zohran? Is Zohran thinking of running?” Working Families Party co-chair Ana Maria Archila, who helped organize the meeting, told CNN as the votes were coming in, recalling the reactions that went around.
<a href=https://tripscan50c.cc>trip scan</a>
“I was like, ‘I don’t know, but I know he’s thinking about who should run,’” Archila said.

In that first meeting and several that followed, the other potential candidates closer to launching kept their plans close to their vests, but Mamdani was eagerly participating, pitching and proposing ideas.

Over coffees, he kept saying they needed to find some candidate to run on freezing the rent. He built a platform and staff out of the Democratic Socialists of America.

Last summer, Mamdani told Archila he was jumping in. He wanted to combine what Rep. Alexandria Ocasio-Cortez and Brooklyn city councilman Chi Osse had done in breaking down traditional politics with compelling social media and a plan to knock on a million doors, he said. He asked when the WFP would announce which candidate it would rank first on its primary slate and told her, “Give me as much time to get there as possible.”
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The day after the home of Eric Adams’s top fundraiser was raided, the new leaders of the Working Families Party, the coalition of progressive groups and labor unions, sent around invites to a secret meeting in November 2023.
<a href=https://tripscan50c.cc>trip scan</a>
The left could see a way to beat the mayor they hated. They just needed a plan.

Gathered in the common room of an apartment building in Long Island City: city comptroller Brad Lander, Brooklyn borough president Antonio Reynoso and state senator Jessica Ramos. Zellnor Myrie, another state senator who ended up running, was invited but didn’t come. Also sitting there: Zohran Mamdani.
<a href=https://tripscan50c.cc>трипскан вход</a>
They were all surprised to see Mamdani there.

“Zohran? Is Zohran thinking of running?” Working Families Party co-chair Ana Maria Archila, who helped organize the meeting, told CNN as the votes were coming in, recalling the reactions that went around.
<a href=https://tripscan50c.cc>трипскан вход</a>
“I was like, ‘I don’t know, but I know he’s thinking about who should run,’” Archila said.

In that first meeting and several that followed, the other potential candidates closer to launching kept their plans close to their vests, but Mamdani was eagerly participating, pitching and proposing ideas.

Over coffees, he kept saying they needed to find some candidate to run on freezing the rent. He built a platform and staff out of the Democratic Socialists of America.

Last summer, Mamdani told Archila he was jumping in. He wanted to combine what Rep. Alexandria Ocasio-Cortez and Brooklyn city councilman Chi Osse had done in breaking down traditional politics with compelling social media and a plan to knock on a million doors, he said. He asked when the WFP would announce which candidate it would rank first on its primary slate and told her, “Give me as much time to get there as possible.”
трипскан
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The day after the home of Eric Adams’s top fundraiser was raided, the new leaders of the Working Families Party, the coalition of progressive groups and labor unions, sent around invites to a secret meeting in November 2023.
<a href=https://tripscan50c.cc>tripscan top</a>
The left could see a way to beat the mayor they hated. They just needed a plan.

Gathered in the common room of an apartment building in Long Island City: city comptroller Brad Lander, Brooklyn borough president Antonio Reynoso and state senator Jessica Ramos. Zellnor Myrie, another state senator who ended up running, was invited but didn’t come. Also sitting there: Zohran Mamdani.
<a href=https://tripscan50c.cc>трипскан</a>
They were all surprised to see Mamdani there.

“Zohran? Is Zohran thinking of running?” Working Families Party co-chair Ana Maria Archila, who helped organize the meeting, told CNN as the votes were coming in, recalling the reactions that went around.
<a href=https://tripscan50c.cc>tripscan</a>
“I was like, ‘I don’t know, but I know he’s thinking about who should run,’” Archila said.

In that first meeting and several that followed, the other potential candidates closer to launching kept their plans close to their vests, but Mamdani was eagerly participating, pitching and proposing ideas.

Over coffees, he kept saying they needed to find some candidate to run on freezing the rent. He built a platform and staff out of the Democratic Socialists of America.

Last summer, Mamdani told Archila he was jumping in. He wanted to combine what Rep. Alexandria Ocasio-Cortez and Brooklyn city councilman Chi Osse had done in breaking down traditional politics with compelling social media and a plan to knock on a million doors, he said. He asked when the WFP would announce which candidate it would rank first on its primary slate and told her, “Give me as much time to get there as possible.”
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