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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Commentaires

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.

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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.”

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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.

Watch out, Orlando, a new world theme park capital is rising in the Arabian desert
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For decades, Orlando has reigned as the global capital of theme parks — a place where Disney, Universal, SeaWorld and countless other attractions have drawn millions of visitors.

But a challenger for the crown has emerged from an unlikely place: the deserts of the Arabian Gulf. In a destination once known more for oil wealth and camel racing than roller coasters, Abu Dhabi is building an adrenaline-charged playground that could give Orlando a run for its money.

And it just landed the ultimate weapon: Disney.
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In May 2025, when Disney announced its first new theme park in 15 years, it chose Abu Dhabi over other key theme park destinations in California, Japan and even Orlando.

There was “no question,” says Josh D’Amaro, chairman of Disney Experiences. The UAE capital, already home to Ferrari World, with the world’s fastest roller coaster; Warner Bros. World (built under license by CNN’s parent company, Warner Brothers Discovery); Yas Waterworld, an epic network of slides and pools; and more recently, SeaWorld Yas Island Abu Dhabi. It’s clear the emirate is emerging as the most serious challenger Orlando has ever faced.

Ferrari World Abu Dhabi is home to the world's fastest rollercoaster and the highest loop ride.
Ferrari World Abu Dhabi is home to the world's fastest rollercoaster and the highest loop ride. Leisa Tyler/LightRocket/Getty Images
Disneyland Abu Dhabi, expected to open on Yas Island in the early 2030s, will be the company’s most technologically advanced park ever. Renderings show a shimmering, futuristic tower at its center — more closely resembling Abu Dhabi’s gleaming skyline than a traditional European castle. It will be the first Disney resort set on an accessible shoreline, located just 20 minutes from downtown Abu Dhabi.

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<p>What began as a shared passion between two friends has grown into the "Abu Dhabi House Movement" — a fast-growing community redefining the city’s music scene. Co-founder Tom Worton takes us inside this grassroots world, where music lovers, DJs, and cultural spaces collide. </p>
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The theme park will be developed, built and operated by Miral, the Abu Dhabi company behind Yas Island’s roster of other attractions. Disney Imagineers will handle creative design and operational oversight, making sure the new park is in keeping with Disney’s brand.

Miral’s CEO, Mohamed Abdalla Al Zaabi, says demand already exists: 2024 saw a 20% rise in theme park attendance on Yas Island. And expansion is already in the works — a Harry Potter–themed land at Warner Bros. World, more record-breaking rides at Ferrari World, new themed hotels, and even two beaches along Yas Bay Waterfront.

‘This isn’t about building another theme park’

disney 3.jpg
Why Disney chose Abu Dhabi for their next theme park location
7:02
Abu Dhabi’s location, a medium-haul flight away from both Europe and Asia, and relatively short hop away from India, means millions of potential visitors are within relatively easy reach.

“This isn’t about building another theme park,” Saleh Mohamed Al Geziry, Abu Dhabi’s director general of tourism, told CNN. “It’s about defining Abu Dhabi as a global destination where culture, entertainment and luxury intersect.”

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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.

“Where else can you start your day under the Louvre’s iconic rain-of-light dome and end it in the immersive, story-driven worlds of Warner Bros. World or Ferrari World?”
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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.

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