grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2025 mines-ponts-maths1__mp

17 maths questions

Q1 Proof Direct Proof of an Inequality View
Show that $$\forall x , y \in \mathbf { R } _ { + } , \quad x y \leq \frac { x ^ { p } } { p } + \frac { y ^ { q } } { q }$$ where $p , q \in ] 1 , + \infty [$ such that $\frac { 1 } { p } + \frac { 1 } { q } = 1$.
Q2 Proof Expectation and Moment Inequality Proof View
Let $p , q \in ] 1 , + \infty [$ such that $\frac { 1 } { p } + \frac { 1 } { q } = 1$. Let $X , Y \in L ^ { 0 } ( \Omega )$ which we assume are both non-negative. Deduce the following inequality (Hölder's inequality): $$\mathbf { E } ( X Y ) \leq \left( \mathrm { E } \left( X ^ { p } \right) \right) ^ { 1 / p } \left( \mathrm { E } \left( Y ^ { q } \right) \right) ^ { 1 / q } .$$ You may begin by treating the case where $\mathbf { E } \left( X ^ { p } \right) = \mathbf { E } \left( Y ^ { q } \right) = 1$.
Q3 Proof Direct Proof of an Inequality View
What inequality do we recover when $p = q = 2$ ? Give a direct proof of it.
Q5 Proof Upper bound on MGF (sub-Gaussian or exponential inequalities) View
Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Show that: for all $t \geq 0$, for all $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$, $$\mathbf { E } \left( \exp \left( t \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) \right) \leq \exp \left( \frac { t ^ { 2 } } { 2 } \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } \right)$$
Q6 Proof Probability Inequality and Tail Bound Proof View
Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Deduce that: for all $t \geq 0$, for all $x \geq 0$ and for all $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$, $$\mathbf { P } \left( \exp \left( x \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| \right) > \mathrm { e } ^ { t x } \right) \leq 2 \mathrm { e } ^ { - t x } \exp \left( \frac { x ^ { 2 } \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } } { 2 } \right) .$$ You may use Markov's inequality.
Q7 Proof Probability Inequality and Tail Bound Proof View
Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Show that: for all $t \geq 0$ and for all non-zero $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$, $$\mathbf { P } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| > t \right) \leq 2 \exp \left( - \frac { t ^ { 2 } } { 2 \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } } \right) .$$
Q8 Proof Change of Variable and Integral Evaluation View
Let $p \in \left[ 1 , + \infty \right[$. Let $X$ be a positive and finite real random variable. Let $F _ { X }$ be the function defined for all $t \geq 0$ by $$F _ { X } ( t ) = \mathbf { P } ( X > t ) .$$ Show that the integral $\int _ { 0 } ^ { + \infty } t ^ { p - 1 } F _ { X } ( t ) \mathrm { d } t$ converges, then that $$\mathbf { E } \left( X ^ { p } \right) = p \int _ { 0 } ^ { + \infty } t ^ { p - 1 } F _ { X } ( t ) \mathrm { d } t$$
Q9 Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View
Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Suppose in this question that $\sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } = 1$. Show that the integral $\int _ { 0 } ^ { + \infty } t ^ { 3 } \mathrm { e } ^ { - t ^ { 2 } / 2 } \mathrm {~d} t$ converges, then that $$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 4 } \right) \leq 8 \int _ { 0 } ^ { + \infty } t ^ { 3 } \mathrm { e } ^ { - t ^ { 2 } / 2 } \mathrm {~d} t$$
Q10 Discrete Probability Distributions Expectation and Variance of Sums of Independent Variables View
Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Show that $$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) = \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } .$$
Q11 Discrete Probability Distributions Expectation and Moment Inequality Proof View
Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Deduce that there exists a real $\beta _ { p } > 0$ such that $$\mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 1 / p } \leq \beta _ { p } \mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) ^ { 1 / 2 } .$$
Q12 Discrete Probability Distributions Proof of Probabilistic Inequalities or Bounds View
Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Suppose $p \geq 2$. Show that $$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) ^ { 1 / 2 } \leq \mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 1 / p }$$
Q13 Discrete Probability Distributions Expectation and Moment Inequality Proof View
Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Assume $1 \leq p < 2$. Justify that there exists $\theta \in ] 0,1 [$ such that $\frac { 1 } { 2 } = \frac { \theta } { p } + \frac { 1 - \theta } { 4 }$.
Q14 Discrete Probability Distributions Proof of Probabilistic Inequalities or Bounds View
Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Assume $1 \leq p < 2$. Show that $$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) \leq \mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 2 \theta / p } \mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { 4 } \right) ^ { ( 1 - \theta ) / 2 } .$$
Q15 Discrete Probability Distributions Proof of Probabilistic Inequalities or Bounds View
Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Assume $1 \leq p < 2$. Show that there exists $\tilde { \alpha } _ { p } > 0$ such that $$\tilde { \alpha } _ { p } \mathrm { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) ^ { 1 / 2 } \leq \mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 1 / p } .$$
Q16 Discrete Probability Distributions Proof of Probabilistic Inequalities or Bounds View
Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Deduce that there exists a real $\alpha _ { p }$ such that $$\alpha _ { p } \mathrm { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) ^ { 1 / 2 } \leq \mathrm { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 1 / p } .$$
Q17 Discrete Probability Distributions Verification of Probability Measure or Inner Product Properties View
Let $\left( X _ { i } \right) _ { i \in \mathbf { N } }$ be a sequence of independent random variables all following a Rademacher distribution. Show that the map $\varphi$ defined on $\left( L ^ { 0 } ( \Omega ) \right) ^ { 2 }$ by $$\forall X , Y \in L ^ { 0 } ( \Omega ) , \quad \varphi ( X , Y ) = \mathbf { E } ( X Y )$$ is an inner product on $L ^ { 0 } ( \Omega )$.
Q21 Matrices Matrix Norm, Convergence, and Inequality View
In this part, we assume that $n$ is a power of 2: we write $n = 2 ^ { k }$ with $k \in \mathbf { N } ^ { \star }$. Deduce that there exists a vector subspace $F$ of dimension $k$ of $\mathbf { R } ^ { n }$ such that: $$\forall x \in F , \quad \alpha _ { 1 } \sqrt { n } \| x \| _ { 2 } ^ { \mathbf { R } ^ { n } } \leq \| x \| _ { 1 } ^ { \mathbf { R } ^ { n } } \leq \beta _ { 1 } \sqrt { n } \| x \| _ { 2 } ^ { \mathbf { R } ^ { n } } .$$ By ordering the $n$ elements of $\{ - 1,1 \} ^ { k }$ arbitrarily, you may use the map $T$ defined on $\mathbf { R } ^ { k }$ by $T \left( a _ { 1 } , \ldots , a _ { k } \right) = \left( \sum _ { i = 1 } ^ { k } a _ { i } \varepsilon _ { i } \right) _ { \left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { k } \right) \in \{ - 1,1 \} ^ { k } }$.