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: 
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Initial interviews with the two surviving housemates gave investigators a loose timeline and a general description of the killer – an athletic, White male who wore a mask that covered most of his face – but little else.

Police later found a Ka-Bar knife sheath next to Madison’s body that would prove to be critical in capturing her killer.

One of the surviving housemates told police about a month before the attacks, Kaylee saw “a dark figure staring at her from the tree line when she took her dog Murphy out to pee.”

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That decision further fueled the mystery around his motive for murdering Xana Kernodle, Madison Mogen, Ethan Chapin and Kaylee Goncalves.

“There’s no reason for these crimes that could approach anything resembling rationality,” Idaho District Judge Steven Hippler said during Kohberger’s sentencing. “The more we try to extract a reason, the more power and control we give to him.”

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“We want to squeeze any silver lining that we can out of these tragedies,” said Molly Amman, a retired profiler who spent years leading the FBI’s Behavioral Threat Assessment Center.

“The silver lining is anything we can use to prevent another crime. It starts with learning absolutely, positively everything about the person and the crime that we possibly can.”

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All four victims had been ruthlessly stabbed to death before the attacker vanished through the kitchen’s sliding glass door and into the night.

“The female lying on the left half of the bed … was unrecognizable,” one officer would later write of the attack that killed Kaylee Goncalves. “I was unable to comprehend exactly what I was looking at while trying to discern the nature of the injuries.”

Initial interviews with the two surviving housemates gave investigators a loose timeline and a general description of the killer – an athletic, White male who wore a mask that covered most of his face – but little else.

Police later found a Ka-Bar knife sheath next to Madison’s body that would prove to be critical in capturing her killer.

One of the surviving housemates told police about a month before the attacks, Kaylee saw “a dark figure staring at her from the tree line when she took her dog Murphy out to pee.”

“There has been lighthearted talk and jokes made about a stalker in the past,” the officer noted. “All the girls were slightly nervous about it being a fact, though.”

But after years of investigating the murders, detectives told CNN they were never able to establish a connection between Kohberger and any of the victims, or a motive.

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That decision further fueled the mystery around his motive for murdering Xana Kernodle, Madison Mogen, Ethan Chapin and Kaylee Goncalves.

“There’s no reason for these crimes that could approach anything resembling rationality,” Idaho District Judge Steven Hippler said during Kohberger’s sentencing. “The more we try to extract a reason, the more power and control we give to him.”

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“We want to squeeze any silver lining that we can out of these tragedies,” said Molly Amman, a retired profiler who spent years leading the FBI’s Behavioral Threat Assessment Center.

“The silver lining is anything we can use to prevent another crime. It starts with learning absolutely, positively everything about the person and the crime that we possibly can.”

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Only Kohberger knows
Even seasoned police officers who arrived at 1122 King Road on November 13, 2022, struggled to process the brutality of the crime scene.

All four victims had been ruthlessly stabbed to death before the attacker vanished through the kitchen’s sliding glass door and into the night.

“The female lying on the left half of the bed … was unrecognizable,” one officer would later write of the attack that killed Kaylee Goncalves. “I was unable to comprehend exactly what I was looking at while trying to discern the nature of the injuries.”

Initial interviews with the two surviving housemates gave investigators a loose timeline and a general description of the killer – an athletic, White male who wore a mask that covered most of his face – but little else.

Police later found a Ka-Bar knife sheath next to Madison’s body that would prove to be critical in capturing her killer.

One of the surviving housemates told police about a month before the attacks, Kaylee saw “a dark figure staring at her from the tree line when she took her dog Murphy out to pee.”

“There has been lighthearted talk and jokes made about a stalker in the past,” the officer noted. “All the girls were slightly nervous about it being a fact, though.”

But after years of investigating the murders, detectives told CNN they were never able to establish a connection between Kohberger and any of the victims, or a motive.

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And when he was finally given the opportunity to answer their questions, he said, “I respectfully decline.”

That decision further fueled the mystery around his motive for murdering Xana Kernodle, Madison Mogen, Ethan Chapin and Kaylee Goncalves.

“There’s no reason for these crimes that could approach anything resembling rationality,” Idaho District Judge Steven Hippler said during Kohberger’s sentencing. “The more we try to extract a reason, the more power and control we give to him.”

But, he added, investigators and researchers may wish to study his actions – if only to learn how to prevent similar crimes from occurring in the future.
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Indeed, academics and former FBI profilers told CNN the challenge of unravelling the criminal mind of a man like Bryan Kohberger is enticing. And while his trial may be over, in many ways, the story of what can be learned from his crimes may have only just begun.

“We want to squeeze any silver lining that we can out of these tragedies,” said Molly Amman, a retired profiler who spent years leading the FBI’s Behavioral Threat Assessment Center.

“The silver lining is anything we can use to prevent another crime. It starts with learning absolutely, positively everything about the person and the crime that we possibly can.”

CNN
Only Kohberger knows
Even seasoned police officers who arrived at 1122 King Road on November 13, 2022, struggled to process the brutality of the crime scene.

All four victims had been ruthlessly stabbed to death before the attacker vanished through the kitchen’s sliding glass door and into the night.

“The female lying on the left half of the bed … was unrecognizable,” one officer would later write of the attack that killed Kaylee Goncalves. “I was unable to comprehend exactly what I was looking at while trying to discern the nature of the injuries.”

Initial interviews with the two surviving housemates gave investigators a loose timeline and a general description of the killer – an athletic, White male who wore a mask that covered most of his face – but little else.

Police later found a Ka-Bar knife sheath next to Madison’s body that would prove to be critical in capturing her killer.

One of the surviving housemates told police about a month before the attacks, Kaylee saw “a dark figure staring at her from the tree line when she took her dog Murphy out to pee.”

“There has been lighthearted talk and jokes made about a stalker in the past,” the officer noted. “All the girls were slightly nervous about it being a fact, though.”

But after years of investigating the murders, detectives told CNN they were never able to establish a connection between Kohberger and any of the victims, or a motive.

Kohberger is far from the first killer to deny families and survivors the catharsis that comes with confessing, in detail, to his crimes. But that, former FBI profilers tell CNN, is part of what makes the prospect of studying him infuriating and intriguing.

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