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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Disney made a smart choice’
Despite the comparisons, Abu Dhabi isn’t positioning itself as a direct rival to Orlando — it’s aiming to be something more. The emirate sees its theme parks as part of a bigger portfolio of attractions, alongside cultural landmarks, luxury hotels, pristine beaches, and desert adventures.
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A 15-minute drive from Yas Island, Saadiyat Island is home to the Louvre Abu Dhabi, a franchised outpost of the famous Paris art museum, which welcomed 1.4 million visitors last year, 84% from abroad. The Guggenheim Abu Dhabi and Zayed National Museum are both under construction, adding to a cultural district that will be one of the region’s most concentrated hubs of art and heritage.
“Abu Dhabi’s unique appeal lies in the diversity of our tourism offering,” Al Geziry added. “For thrill-seekers, we have record-breaking roller coasters and dune bashing in the desert. For culture lovers, historic sites like Al Ain Oasis and institutions like the Saadiyat museums. And for luxury travelers, world-class dining, private island resorts, and high-end shopping.
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Still, not everyone is convinced that Disney’s expansion into the Middle East is a sure bet.
“The region has seen its share of false starts,” says Dennis Speigel, founder of the International Theme Park Services consultancy, comparing it to neighboring Dubai’s patchy record with theme park expansion ambitions in the mid-2010s. “Several of them struggled for profitability in their first decade.”
Related article
Saadiyat Cultural District in Abu Dhabi is set to become one of the world’s preeminent arts and culture hubs, with one of the highest concentrations of cultural institutions globally. But the area isn’t just for art connoisseurs. Explore what to do in the new district, from iconic museums to luxurious beach days to decadent dining options.
You can walk between the Louvre and the Guggenheim in this new art district
Spiegel believes Abu Dhabi is different. “Disney made a smart choice. The infrastructure, safety, and existing leisure developments create an ideal entry point,” he told CNN earlier this year. “It’s a much more controlled and calculated move.”
Under its Tourism Strategy 2030, Abu Dhabi aims to grow annual visitors from 24 million in 2023 to more than 39 million by the end of the decade. With Disneyland as a centerpiece, those targets may well be surpassed. The city’s population has already grown from 2.7 million in 2014 to more than 4.1 million today, a reflection of its rising profile as a regional hub.
Yas Island alone has been transformed in the space of a decade from a largely undeveloped stretch of sand to a self-contained resort destination, complete with golf courses, marinas, a mall, more than 160 restaurants, and a cluster of high-end hotels.
Orlando’s head start remains formidable — it still offers multiple Disney and Universal parks, has decades of brand loyalty, and an infrastructure built to handle tens of millions of tourists annually.
But Abu Dhabi is catching up fast. Its combination of frictionless travel, year-round comfort, cutting-edge attractions, and a cultural scene that adds depth to the experience gives Abu Dhabi its own unique selling point, potentially offering a model for the next generation of theme park capital.
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A 15-minute drive from Yas Island, Saadiyat Island is home to the Louvre Abu Dhabi, a franchised outpost of the famous Paris art museum, which welcomed 1.4 million visitors last year, 84% from abroad. The Guggenheim Abu Dhabi and Zayed National Museum are both under construction, adding to a cultural district that will be one of the region’s most concentrated hubs of art and heritage.
“Abu Dhabi’s unique appeal lies in the diversity of our tourism offering,” Al Geziry added. “For thrill-seekers, we have record-breaking roller coasters and dune bashing in the desert. For culture lovers, historic sites like Al Ain Oasis and institutions like the Saadiyat museums. And for luxury travelers, world-class dining, private island resorts, and high-end shopping.
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Still, not everyone is convinced that Disney’s expansion into the Middle East is a sure bet.
“The region has seen its share of false starts,” says Dennis Speigel, founder of the International Theme Park Services consultancy, comparing it to neighboring Dubai’s patchy record with theme park expansion ambitions in the mid-2010s. “Several of them struggled for profitability in their first decade.”
Related article
Saadiyat Cultural District in Abu Dhabi is set to become one of the world’s preeminent arts and culture hubs, with one of the highest concentrations of cultural institutions globally. But the area isn’t just for art connoisseurs. Explore what to do in the new district, from iconic museums to luxurious beach days to decadent dining options.
You can walk between the Louvre and the Guggenheim in this new art district
Spiegel believes Abu Dhabi is different. “Disney made a smart choice. The infrastructure, safety, and existing leisure developments create an ideal entry point,” he told CNN earlier this year. “It’s a much more controlled and calculated move.”
Under its Tourism Strategy 2030, Abu Dhabi aims to grow annual visitors from 24 million in 2023 to more than 39 million by the end of the decade. With Disneyland as a centerpiece, those targets may well be surpassed. The city’s population has already grown from 2.7 million in 2014 to more than 4.1 million today, a reflection of its rising profile as a regional hub.
Yas Island alone has been transformed in the space of a decade from a largely undeveloped stretch of sand to a self-contained resort destination, complete with golf courses, marinas, a mall, more than 160 restaurants, and a cluster of high-end hotels.
Orlando’s head start remains formidable — it still offers multiple Disney and Universal parks, has decades of brand loyalty, and an infrastructure built to handle tens of millions of tourists annually.
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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.
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The inside story of Zohran Mamdani’s triumph trip scan
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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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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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.
<a href=https://tripscan50c.cc>tripscan top</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>трипскан</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 сайт</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 сайт</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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