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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Trump announced the appointment of Jeff Landry, the Governor of Louisiana, as ?special envoy to Greenland on Monday in a post on Truth Social. “Jeff understands how essential Greenland is to our national security, and will strongly advance our country’s interests for the safety, security, and survival of our allies, and indeed, the World,” Trump posted on his social media platform.
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Rasmussen said he would summon the US ambassador to Denmark in response to the Trump administration’s move, Reuters reported.
Denmark's Foreign Minister Lars Loekke Rasmussen and Defense Minister Troels Lund Poulsen address journalists in Copenhagen on September 26.
Denmark's Foreign Minister Lars Loekke Rasmussen and Defense Minister Troels Lund Poulsen address journalists in Copenhagen on September 26. Liselotte Sabroe/AFP/Ritzau Scanpix/Getty Images
Trump defended his decision to pick Landry telling reporters on Monday evening that the US needs Greenland “for national security” and that Landry had approached him about the assignment.
“Louisiana, the Louisiana Purchase. He said I’m governor of Louisiana, and he said I would love … I didn’t call him, he called me. He’s very proactive,” Trump told reporters at Mar-a-Lago.
“We need Greenland for national security, not for minerals. We have so many sites for minerals and oil and everything,” Trump said, trying to make the case for annexing Greenland, despite its status as a self-governing territory of Denmark. “If you take a look at Greenland, you look up and down the coast, you have Russian and Chinese ships all over the place. We need it for national security. We have to have it,” he added.
During his Monday remarks, Trump went on to claim that Denmark has “spent no money” on Greenland and has “no military protection.”
While thanking Trump for his appointment, Landry said it was an “honor to serve you in this volunteer position to make Greenland a part of the US.” He also said that “this in no way affects” his position as Louisiana governor.
Trump has repeatedly stated that he wants to annex Greenland – a huge, resource-rich island in the Atlantic and self-governing territory of Denmark – claiming that this is needed for American security purposes.
Both Greenland and Denmark, a NATO ally of the US, are staunchly opposed to the idea.
Danish Prime Minister Mette Frederiksen and Greenland’s Prime Minister Jens-Frederik Nielsen reiterated their opposition Monday to US plans to take over Greenland, stating “you cannot annex another country. Not even with an argument about international security,” according to Reuters.
“Greenland belongs to the Greenlanders and the US shall not take over Greenland,” they said in a joint statement.
Nielsen said earlier Monday that Trump’s announcement “may sound big, but it does not change anything for us. We decide our own future,” Reuters reported.
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Trump announced the appointment of Jeff Landry, the Governor of Louisiana, as ?special envoy to Greenland on Monday in a post on Truth Social. “Jeff understands how essential Greenland is to our national security, and will strongly advance our country’s interests for the safety, security, and survival of our allies, and indeed, the World,” Trump posted on his social media platform.
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“I am deeply upset by this appointment of a special envoy. And I ?am particularly upset by his statements, which we find completely unacceptable,” Lars Lokke Rasmussen told Denmark’s national broadcaster TV 2, according to Reuters news agency.
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Rasmussen said he would summon the US ambassador to Denmark in response to the Trump administration’s move, Reuters reported.
Denmark's Foreign Minister Lars Loekke Rasmussen and Defense Minister Troels Lund Poulsen address journalists in Copenhagen on September 26.
Denmark's Foreign Minister Lars Loekke Rasmussen and Defense Minister Troels Lund Poulsen address journalists in Copenhagen on September 26. Liselotte Sabroe/AFP/Ritzau Scanpix/Getty Images
Trump defended his decision to pick Landry telling reporters on Monday evening that the US needs Greenland “for national security” and that Landry had approached him about the assignment.
“Louisiana, the Louisiana Purchase. He said I’m governor of Louisiana, and he said I would love … I didn’t call him, he called me. He’s very proactive,” Trump told reporters at Mar-a-Lago.
“We need Greenland for national security, not for minerals. We have so many sites for minerals and oil and everything,” Trump said, trying to make the case for annexing Greenland, despite its status as a self-governing territory of Denmark. “If you take a look at Greenland, you look up and down the coast, you have Russian and Chinese ships all over the place. We need it for national security. We have to have it,” he added.
During his Monday remarks, Trump went on to claim that Denmark has “spent no money” on Greenland and has “no military protection.”
While thanking Trump for his appointment, Landry said it was an “honor to serve you in this volunteer position to make Greenland a part of the US.” He also said that “this in no way affects” his position as Louisiana governor.
Trump has repeatedly stated that he wants to annex Greenland – a huge, resource-rich island in the Atlantic and self-governing territory of Denmark – claiming that this is needed for American security purposes.
Both Greenland and Denmark, a NATO ally of the US, are staunchly opposed to the idea.
Danish Prime Minister Mette Frederiksen and Greenland’s Prime Minister Jens-Frederik Nielsen reiterated their opposition Monday to US plans to take over Greenland, stating “you cannot annex another country. Not even with an argument about international security,” according to Reuters.
“Greenland belongs to the Greenlanders and the US shall not take over Greenland,” they said in a joint statement.
Nielsen said earlier Monday that Trump’s announcement “may sound big, but it does not change anything for us. We decide our own future,” Reuters reported.
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Related article
The Free Press' Honestly with Bari Weiss (pictured) hosts Senator Ted Cruz presented by Uber and X on January 18, 2025 in Washington, DC.
Inside the Bari Weiss decision that led to a ‘60 Minutes’ crisis
Progressive Substack writers and commentators blasted out the clips and urged people to share them. “This could wind up being the most-watched newsmagazine segment in television history,” the high-profile Trump antagonist George Conway commented on X.
A CBS News spokesperson had no immediate comment on the astonishing turn of events.
Alfonsi’s report was weeks in the making. Weiss screened it for the first time last Thursday night. The story was finalized on Friday, according to CBS sources, and was announced in a press release that same day.
On Saturday morning, Weiss began to change her mind about the story and raised concerns about its content, including the lack of responses from the relevant Trump administration officials.
But networks like CBS sometimes deliver taped programming to affiliates like Global TV ahead of time. That appears to be what happened in this case: The Friday version of the “60 Minutes” episode is what streamed to Canadian viewers.
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People close to Weiss have argued that the piece was imbalanced, however, because it did not include interviews with Trump officials.
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<a href=https://minexchange.net>mine.exchange</a>
On Monday, some Canadian viewers noticed that the pre-planned “60 Minutes” episode was published on a streaming platform owned by Global TV, the network that has the rights to “60 Minutes” in Canada.
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The preplanned episode led with correspondent Sharyn Alfonsi’s story — the one that Weiss stopped from airing in the US because she said it was “not ready.”
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Several Canadian viewers shared clips and summaries of the story on social media, and within hours, the videos went viral on platforms like Reddit and Bluesky.
“Watch fast,” one of the Canadian viewers wrote on Bluesky, predicting that CBS would try to have the videos taken offline.
Related article
The Free Press' Honestly with Bari Weiss (pictured) hosts Senator Ted Cruz presented by Uber and X on January 18, 2025 in Washington, DC.
Inside the Bari Weiss decision that led to a ‘60 Minutes’ crisis
Progressive Substack writers and commentators blasted out the clips and urged people to share them. “This could wind up being the most-watched newsmagazine segment in television history,” the high-profile Trump antagonist George Conway commented on X.
A CBS News spokesperson had no immediate comment on the astonishing turn of events.
Alfonsi’s report was weeks in the making. Weiss screened it for the first time last Thursday night. The story was finalized on Friday, according to CBS sources, and was announced in a press release that same day.
On Saturday morning, Weiss began to change her mind about the story and raised concerns about its content, including the lack of responses from the relevant Trump administration officials.
But networks like CBS sometimes deliver taped programming to affiliates like Global TV ahead of time. That appears to be what happened in this case: The Friday version of the “60 Minutes” episode is what streamed to Canadian viewers.
The inadvertent Canadian stream is “the best thing that could have happened,” a CBS source told CNN on Monday evening, arguing that the Alfonsi piece is “excellent” and should have been televised as intended.
People close to Weiss have argued that the piece was imbalanced, however, because it did not include interviews with Trump officials.
Weiss told staffers on Monday, “We need to be able to get the principals on the record and on camera.” However, in an earlier memo to colleagues, Alfonsi asserted that her team tried, and their “refusal to be interviewed” was “a tactical maneuver designed to kill the story.”
At the end of the segment that streamed on Global TV’s platform, Alfonsi said Homeland Security “declined our request for an interview and referred all questions about CECOT to El Salvador. The government there did not respond to our request.”
The segment included sound bites from President Trump and Homeland Security Secretary Kristi Noem. But it was clearly meant to be a story about Venezuelan men deported to El Salvador, not about the officials who implemented Trump’s mass deportation policy.
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