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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Unity, a brand of custom home company Bensonwood, is one of several companies in New England building homes in a factory. It’s a modern spin on the 1900s Sears Roebuck catalogue of mail-order houses, now with energy efficiency front of mind.
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The number of factory-made homes still pales in comparison to traditional stick-built homes in the United States — just 3% of all single-family homes as of 2024, according to the National Association of Home Builders. The trend is much more popular in Europe; in Sweden, most new homes are built in factories.
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Three New England companies specializing in factory-built and energy efficient homes told CNN they are seeing a rise in interest. Prefabricated homes used to be associated with cheap, poorly made housing, but that has changed.
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Unity, a brand of custom home company Bensonwood, is one of several companies in New England building homes in a factory. It’s a modern spin on the 1900s Sears Roebuck catalogue of mail-order houses, now with energy efficiency front of mind.
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Companies like Unity and Maine-based BrightBuilt Home offer several basic designs that owners can customize. At Unity, much of the design is done ahead of time, before the house’s walls, windows and doors are assembled inside the company’s Keene, New Hampshire, factory, then wrapped tightly in reusable plastic and put on a truck bed. Once it’s on site, a panelized house can be built in a matter of days.
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“That’s been relatively flat for a number of years,” said Devin Perry, an assistant vice president at the National Association of Home Builders. “But anecdotally, there seems to be momentum behind that method of construction, and people looking for energy efficient options.”
Three New England companies specializing in factory-built and energy efficient homes told CNN they are seeing a rise in interest. Prefabricated homes used to be associated with cheap, poorly made housing, but that has changed.
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Built in a factory
Unity, a brand of custom home company Bensonwood, is one of several companies in New England building homes in a factory. It’s a modern spin on the 1900s Sears Roebuck catalogue of mail-order houses, now with energy efficiency front of mind.
<a href=https://tripscan.info>трипскан</a>
Companies like Unity and Maine-based BrightBuilt Home offer several basic designs that owners can customize. At Unity, much of the design is done ahead of time, before the house’s walls, windows and doors are assembled inside the company’s Keene, New Hampshire, factory, then wrapped tightly in reusable plastic and put on a truck bed. Once it’s on site, a panelized house can be built in a matter of days.
It is a very different model from traditional “stick-built” home construction, where the structure is built “stick by stick” on site and can take months to finish, all the while leaving wood and materials exposed to rain, snow and wind.
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“There’s a logic to building a structure in a climate-controlled environment. That really resonates with a lot of folks,” said Parlin Meyer, managing principal for BrightBuilt Home.
The number of factory-made homes still pales in comparison to traditional stick-built homes in the United States — just 3% of all single-family homes as of 2024, according to the National Association of Home Builders. The trend is much more popular in Europe; in Sweden, most new homes are built in factories.
“That’s been relatively flat for a number of years,” said Devin Perry, an assistant vice president at the National Association of Home Builders. “But anecdotally, there seems to be momentum behind that method of construction, and people looking for energy efficient options.”
Three New England companies specializing in factory-built and energy efficient homes told CNN they are seeing a rise in interest. Prefabricated homes used to be associated with cheap, poorly made housing, but that has changed.
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The Goodsons didn’t start out intending to build a home. The couple scoured the southern Maine housing market for nearly two years, striking out time after time.
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House-hunting in cities comes with the same problem. When Tim Buntel and Cynthia Graber started looking for homes in Somerville, Massachusetts, they kept finding condos listed for far more than they were worth.
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Free Loot boxes in CS:GO
Free Loot boxes for CS:GO
Finding free case drops in Counter-Strike: Global Offensive can be rewarding and occasionally unexpected. Numerous users seek ways to get cases without paying real funds. This guide explains legitimate options and gives suggestions to help you accumulate no-cost loot boxes responsibly.
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Join casual games on Valve platforms to earn loot boxes. Limited promotions and operations frequently include loot boxes as rewards. Periodically competitions and community giveaways award cases to participants.
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Free Case drops within CSGO
No-cost Loot boxes in CS:GO
Finding no-cost cases in Counter-Strike: Global Offensive may be fun and occasionally surprising. Several players seek methods to receive loot boxes without spending real money. This article describes legitimate paths and gives suggestions to help you gather free cases responsibly.
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Free Loot boxes in Counter-Strike: Global Offensive
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Discovering no-cost case drops in CSGO is fun and periodically surprising. Several players want methods to get loot boxes without spending real funds. This guide explains safe paths and offers tips to enable you gather free cases responsibly.
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Tammydex (non vérifié)
jeu, 08/14/2025 - 17:05
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Exness Trading Platform a350s
Я думаю, что это — неправда.
{online chat, {email|email}, #file_links<>C:\Users\Admin\Desktop\file\gsa+en+35XU797T6G2P2URLBB.txt",1,N], and telephone support are available to {clients|customers|users}. {clients|visitors} can configure automatic {forex strategies|trading strategies|profitable strategies|trading algorithms|trading systems|profitable forex algorithms|profitable trading systems|forex systems|forex algorithms|forex systems|forex trading strategies|forex trading strategies} {through|with the help of|with the help of} advisors.
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