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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Leo celebrates Christmas Holy Mass at the Vatican.
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Leo holds an incent burner at St Peter's Basilica.
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“Fragile is the flesh of defenseless populations, tried by so many wars, ongoing or concluded, leaving behind rubble and open wounds,” Leo said. He quoted an Israeli poet, Yehuda Amichai, who called for peace to blossom “like wildflowers.”
Related article
The acting Latin Patriarch of Jerusalem Pierbattista Pizzaballa attends a morning Mass at Saint Catherine's Church, in the Church of the Nativity, in Bethlehem, in the Israeli-occupied West Bank December 25, 2025. REUTERS/Mussa Qawasma
Christmas celebrated once again in Bethlehem but West Bank suffering persists
Later during his Christmas message, he called for compassion towards those “who are fleeing their homeland to seek a future elsewhere, like the many refugees and migrants who cross the Mediterranean or traverse the American continent.” He offered Christmas greetings in different languages including Italian, English, Arabic, Chinese, Polish.
Since his election, Leo has highlighted the plight of those suffering of those in Gaza, and has been outspoken by calling for the better treatment of migrants. In his first major interview in September, the American pope voiced concern over “some things” happening in the country of his birth, highlighting the significance of a letter his predecessor, Pope Francis, had sent to US bishops earlier this year, rebuking the administration’s deportation plans.
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Pope Leo XIV celebrated the first Christmas since his election by denouncing the suffering of people of Gaza – taking shelter in tents from the “rain, wind and cold” – and by calling for the guns to fall silent in Ukraine.
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Leo, who was elected on May 8, said Thursday that Jesus Christ is “our peace” because he “shows us the way to overcome conflicts, whether interpersonal or international. With his grace, we can and must each day our part to reject hatred, violence and opposition, and to practice dialogue, peace and reconciliation.”
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The pontiff began by asking for “justice, peace and stability” for Lebanon, the Palestinian territories, Israel and Syria. Later, he said that, by becoming man, “Jesus took upon himself our fragility,” allowing him to identify “with those who have nothing left and have lost everything, like the inhabitants of Gaza.”
Leo celebrates Christmas Holy Mass at the Vatican.
Leo celebrates Christmas Holy Mass at the Vatican. Yara Nardi/Reuters
Leo holds an incent burner at St Peter's Basilica.
Leo holds an incent burner at St Peter's Basilica. Tiziana Fabi/AFP/Getty Images
The pope’s first Christmas since his election took place in wet and cold conditions, but that failed to deter large crowds from coming out to hear his message.
Earlier during Mass, he asked how, at Christmas, “can we not think of the tents in Gaza, exposed for weeks to rain, wind and cold.” With more than 400,000 homes destroyed during Israel’s war against Hamas, Gazans are being forced to choose this winter between living in tents exposed to the elements or living inside buildings that could collapse any minute.
“Fragile is the flesh of defenseless populations, tried by so many wars, ongoing or concluded, leaving behind rubble and open wounds,” Leo said. He quoted an Israeli poet, Yehuda Amichai, who called for peace to blossom “like wildflowers.”
Related article
The acting Latin Patriarch of Jerusalem Pierbattista Pizzaballa attends a morning Mass at Saint Catherine's Church, in the Church of the Nativity, in Bethlehem, in the Israeli-occupied West Bank December 25, 2025. REUTERS/Mussa Qawasma
Christmas celebrated once again in Bethlehem but West Bank suffering persists
Later during his Christmas message, he called for compassion towards those “who are fleeing their homeland to seek a future elsewhere, like the many refugees and migrants who cross the Mediterranean or traverse the American continent.” He offered Christmas greetings in different languages including Italian, English, Arabic, Chinese, Polish.
Since his election, Leo has highlighted the plight of those suffering of those in Gaza, and has been outspoken by calling for the better treatment of migrants. In his first major interview in September, the American pope voiced concern over “some things” happening in the country of his birth, highlighting the significance of a letter his predecessor, Pope Francis, had sent to US bishops earlier this year, rebuking the administration’s deportation plans.
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Pope Leo XIV celebrated the first Christmas since his election by denouncing the suffering of people of Gaza – taking shelter in tents from the “rain, wind and cold” – and by calling for the guns to fall silent in Ukraine.
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On Christmas Day, the first US-born pope, offered the traditional “Urbi et Orbi” blessing (“To the City and to the World”) from the balcony of St Peter’s, surveying a world speckled with conflicts from Yemen to Myanmar, and calling for compassion towards those who have fled their homelands to seek a future in Europe and America.
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The pontiff began by asking for “justice, peace and stability” for Lebanon, the Palestinian territories, Israel and Syria. Later, he said that, by becoming man, “Jesus took upon himself our fragility,” allowing him to identify “with those who have nothing left and have lost everything, like the inhabitants of Gaza.”
Leo celebrates Christmas Holy Mass at the Vatican.
Leo celebrates Christmas Holy Mass at the Vatican. Yara Nardi/Reuters
Leo holds an incent burner at St Peter's Basilica.
Leo holds an incent burner at St Peter's Basilica. Tiziana Fabi/AFP/Getty Images
The pope’s first Christmas since his election took place in wet and cold conditions, but that failed to deter large crowds from coming out to hear his message.
Earlier during Mass, he asked how, at Christmas, “can we not think of the tents in Gaza, exposed for weeks to rain, wind and cold.” With more than 400,000 homes destroyed during Israel’s war against Hamas, Gazans are being forced to choose this winter between living in tents exposed to the elements or living inside buildings that could collapse any minute.
“Fragile is the flesh of defenseless populations, tried by so many wars, ongoing or concluded, leaving behind rubble and open wounds,” Leo said. He quoted an Israeli poet, Yehuda Amichai, who called for peace to blossom “like wildflowers.”
Related article
The acting Latin Patriarch of Jerusalem Pierbattista Pizzaballa attends a morning Mass at Saint Catherine's Church, in the Church of the Nativity, in Bethlehem, in the Israeli-occupied West Bank December 25, 2025. REUTERS/Mussa Qawasma
Christmas celebrated once again in Bethlehem but West Bank suffering persists
Later during his Christmas message, he called for compassion towards those “who are fleeing their homeland to seek a future elsewhere, like the many refugees and migrants who cross the Mediterranean or traverse the American continent.” He offered Christmas greetings in different languages including Italian, English, Arabic, Chinese, Polish.
Since his election, Leo has highlighted the plight of those suffering of those in Gaza, and has been outspoken by calling for the better treatment of migrants. In his first major interview in September, the American pope voiced concern over “some things” happening in the country of his birth, highlighting the significance of a letter his predecessor, Pope Francis, had sent to US bishops earlier this year, rebuking the administration’s deportation plans.
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On Christmas Day, the first US-born pope, offered the traditional “Urbi et Orbi” blessing (“To the City and to the World”) from the balcony of St Peter’s, surveying a world speckled with conflicts from Yemen to Myanmar, and calling for compassion towards those who have fled their homelands to seek a future in Europe and America.
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Leo, who was elected on May 8, said Thursday that Jesus Christ is “our peace” because he “shows us the way to overcome conflicts, whether interpersonal or international. With his grace, we can and must each day our part to reject hatred, violence and opposition, and to practice dialogue, peace and reconciliation.”
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Leo celebrates Christmas Holy Mass at the Vatican.
Leo celebrates Christmas Holy Mass at the Vatican. Yara Nardi/Reuters
Leo holds an incent burner at St Peter's Basilica.
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Earlier during Mass, he asked how, at Christmas, “can we not think of the tents in Gaza, exposed for weeks to rain, wind and cold.” With more than 400,000 homes destroyed during Israel’s war against Hamas, Gazans are being forced to choose this winter between living in tents exposed to the elements or living inside buildings that could collapse any minute.
“Fragile is the flesh of defenseless populations, tried by so many wars, ongoing or concluded, leaving behind rubble and open wounds,” Leo said. He quoted an Israeli poet, Yehuda Amichai, who called for peace to blossom “like wildflowers.”
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Later during his Christmas message, he called for compassion towards those “who are fleeing their homeland to seek a future elsewhere, like the many refugees and migrants who cross the Mediterranean or traverse the American continent.” He offered Christmas greetings in different languages including Italian, English, Arabic, Chinese, Polish.
Since his election, Leo has highlighted the plight of those suffering of those in Gaza, and has been outspoken by calling for the better treatment of migrants. In his first major interview in September, the American pope voiced concern over “some things” happening in the country of his birth, highlighting the significance of a letter his predecessor, Pope Francis, had sent to US bishops earlier this year, rebuking the administration’s deportation plans.
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