grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2017 x-ens-maths2__mp

16 maths questions

Q1 Proof Direct Proof of an Inequality View
Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits a variance relative to $m$. Show that $fm$ is integrable. As a consequence, the real $$\operatorname { Var } _ { m } ( f ) = \int f ( x ) ^ { 2 } m ( x ) d x - \left( \int f ( x ) m ( x ) d x \right) ^ { 2 }$$ is well defined. Show that $\operatorname { Var } _ { m } ( f ) \geqslant 0$.
Q2 Proof Direct Proof of an Inequality View
Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits an entropy relative to $m$. We consider the function $h : [ 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $h ( 0 ) = 0$ and for $x > 0$, $h ( x ) = x \ln ( x )$.
2a. Show that $f ^ { 2 } m$ is integrable. As a consequence, the real $$\operatorname { Ent } _ { m } ( f ) = \int h \left( f ( x ) ^ { 2 } \right) m ( x ) d x - h \left( \int f ( x ) ^ { 2 } m ( x ) d x \right)$$ is well defined.
2b. Let $a > 0$. Show that $$\forall x \geqslant 0 , \quad h ( x ) \geqslant ( x - a ) h ^ { \prime } ( a ) + h ( a ) ,$$ with strict inequality if $x \neq a$.
2c. Show that $\operatorname { Ent } _ { m } ( f ) \geqslant 0$. You may use the previous question with $a = \int f ( x ) ^ { 2 } m ( x ) d x$.
2d. We assume here that for all $x \in \mathbb { R } , m ( x ) > 0$. Characterize the functions $f$ such that $\operatorname { Ent } _ { m } ( f ) = 0$.
Q3 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We denote $L$ the operator that associates to a function $f : \mathbb { R } \rightarrow \mathbb { R }$ of class $\mathscr { C } ^ { 2 }$, the function $Lf$ defined by $$\forall x \in \mathbb { R } , \quad L f ( x ) = \frac { 1 } { 2 } f ^ { \prime \prime } ( x ) - x f ^ { \prime } ( x )$$ We recall that the measure $\mu$ is defined by $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$.
3a. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be of class $\mathscr { C } ^ { 2 }$. Show that $L f = \frac { 1 } { 2 \mu } \left( \mu f ^ { \prime } \right) ^ { \prime }$.
3b. Let $h _ { 1 } , h _ { 2 }$ be two functions in $\mathscr { C } _ { b } ^ { 2 }$. Show that $$\int h _ { 1 } ( x ) \left( L h _ { 2 } \right) ( x ) \mu ( x ) d x = - \frac { 1 } { 2 } \int h _ { 1 } ^ { \prime } ( x ) h _ { 2 } ^ { \prime } ( x ) \mu ( x ) d x$$ after having justified the existence of each term of the formula.
Q4 Proof Proof That a Map Has a Specific Property View
We consider a function $f \in \mathscr { C } _ { b } ^ { 0 }$. We define for $( t , x ) \in \mathbb { R } ^ { 2 }$ $$\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$$ where $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$. Show that the function $\Phi _ { f } : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$ is well defined and continuous.
Q5 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We consider a function $f \in \mathscr { C } _ { b } ^ { 0 }$. We define for $( t , x ) \in \mathbb { R } ^ { 2 }$ $$\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$$ where $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $Lf(x) = \frac{1}{2} f''(x) - x f'(x)$. We assume that $f \in \mathscr { C } _ { b } ^ { 2 }$.
5a. Show that, on $\mathbb { R } ^ { 2 } , \Phi _ { f }$ is of class $\mathscr { C } ^ { 1 }$ and $\partial _ { x x } \Phi _ { f }$ is well defined, continuous and bounded.
5b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Find a relation between $\partial _ { x } \Phi _ { f } ( t , x )$ and $\Phi _ { f ^ { \prime } } ( t , x )$.
5c. Show that for all $( t , x ) \in \mathbb { R } ^ { 2 }$, we have $\partial _ { t } \Phi _ { f } ( t , x ) \cos t = L \Phi _ { f } ( t , x ) \sin t$.
5d. Show that for all $t \in \mathbb { R }$, we have $\int \Phi _ { f } ( t , x ) \mu ( x ) d x = \int f ( x ) \mu ( x ) d x$.
Q6 Proof Computation of a Limit, Value, or Explicit Formula View
Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. Show that $J : \mathbb { R } \rightarrow \mathbb { R }$ is continuous, and calculate $J ( 0 )$ and $J \left( \frac { \pi } { 2 } \right)$.
Q7 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. We assume throughout this question that $f \in \mathscr { C } _ { b } ^ { 2 }$ and that there exists $\delta > 0$ such that $$\forall x \in \mathbb { R } , \quad f ( x ) \geqslant \delta .$$ We denote $g = \left( f ^ { \prime } \right) ^ { 2 } / f$.
7a. Show that $J$ is then of class $\mathscr { C } ^ { 1 }$ on $\mathbb { R }$ and that $$\forall t \in \mathbb { R } , \quad J ^ { \prime } ( t ) \cos t = - \frac { \sin t } { 2 } \int \frac { \left( \partial _ { x } \Phi _ { f } ( t , x ) \right) ^ { 2 } } { \Phi _ { f } ( t , x ) } \mu ( x ) d x$$
7b. Let $( t , x ) \in \mathbb { R } ^ { 2 }$. Show that $$\Phi _ { f ^ { \prime } } ( t , x ) ^ { 2 } \leqslant \Phi _ { f } ( t , x ) \Phi _ { g } ( t , x )$$
7c. Conclude that $$\int h ( f ( x ) ) \mu ( x ) d x - h \left( \int f ( y ) \mu ( y ) d y \right) \leqslant \frac { 1 } { 4 } \int g ( x ) \mu ( x ) d x$$
Q8 Continuous Probability Distributions and Random Variables Entropy, Information, or Log-Sobolev Functional Analysis View
Show that for all $f \in \mathscr { C } _ { b } ^ { 2 }$, $f$ admits an entropy relative to $\mu$ and that $$\operatorname { Ent } _ { \mu } ( f ) \leqslant \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } \mu ( x ) d x$$ You may consider the family of functions defined by $f _ { \delta } = \delta + f ^ { 2 }$ for $\delta > 0$.
Q9 Continuous Probability Distributions and Random Variables Entropy, Information, or Log-Sobolev Functional Analysis View
Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Show that $\int \left( 1 + | x | + x ^ { 2 } \right) m ( x ) d x < + \infty$.
Q10 Continuous Probability Distributions and Random Variables Entropy, Information, or Log-Sobolev Functional Analysis View
Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Let $f \in \mathscr { C } _ { b } ^ { 1 }$. We wish to show that $f$ admits a variance relative to $m$ and that $$\operatorname { Var } _ { m } ( f ) \leqslant \frac { C } { 2 } \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{2}$$
10a. Show that $fm$ and $f ^ { 2 } m$ are integrable, and that it suffices to show (2) in the case where we additionally have $\int f ( x ) m ( x ) d x = 0$ and $\int f ( x ) ^ { 2 } m ( x ) d x = 1$.
10b. Under the hypotheses of the previous question, show (2). You may apply (1) to the family of functions $f _ { \varepsilon } = 1 + \varepsilon f$ for $\varepsilon > 0$.
Q11 Continuous Probability Distributions and Random Variables Entropy, Information, or Log-Sobolev Functional Analysis View
Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Let $f$ be a function in $\mathscr { C } _ { b } ^ { 1 }$, such that for all $x \in \mathbb { R }$, we have $\left| f ^ { \prime } ( x ) \right| \leqslant 1$. We denote, for $\lambda \in \mathbb { R }$, $$H ( \lambda ) = \int e ^ { \lambda f ( x ) } m ( x ) d x$$ We admit that $H$ is of class $\mathscr { C } ^ { 1 }$ and that we obtain an expression of $H ^ { \prime } ( \lambda )$ by differentiating under the integral sign in the usual manner.
11a. Show that for all $\lambda \in \mathbb { R }$, $$\lambda H ^ { \prime } ( \lambda ) - H ( \lambda ) \ln H ( \lambda ) \leqslant \frac { C \lambda ^ { 2 } } { 4 } H ( \lambda )$$
11b. Deduce that for $\lambda \geqslant 0$, $$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right) \tag{3}$$ You may study the function $\lambda \mapsto \frac { 1 } { \lambda } \ln H ( \lambda )$.
Q12 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that inequality (3) $$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right)$$ applies to all $f \in \mathscr{C}_b^1$ with $|f'(x)| \leq 1$. Show that inequality (3) applies to the function defined by $f ( x ) = x$. You may use the sequence of functions defined by $f _ { n } ( x ) = n \arctan \left( \frac { x } { n } \right)$.
Q13 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that for $\lambda \geq 0$, $$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right) \tag{3}$$ applies (in particular to $f(x) = x$).
13a. Let $M = \int x m ( x ) d x$ and $a \geqslant M$. Show that $$\int _ { a } ^ { + \infty } m ( x ) d x \leqslant \exp \left( - \frac { ( a - M ) ^ { 2 } } { C } \right)$$
13b. Conclude that for all $\alpha < \frac { 1 } { C }$, the function $x \mapsto e ^ { \alpha x ^ { 2 } } m ( x )$ is integrable on $\mathbb { R }$.
Q14 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View
Let $p , q , r : \mathbb { R } \rightarrow \mathbb { R } _ { * } ^ { + }$ be three continuous functions, with strictly positive values and integrable on $\mathbb { R }$.
14a. Show that there exists a function $u : ] 0,1 [ \rightarrow \mathbb { R }$ of class $\mathscr { C } ^ { 1 }$ bijective such that $$\forall t \in ] 0,1 [ , \quad u ^ { \prime } ( t ) p ( u ( t ) ) = \int p ( x ) d x$$ Similarly, there exists an analogous function $v : ] 0,1 [ \rightarrow \mathbb { R }$ for $q$.
14b. We assume that $$\forall x , y \in \mathbb { R } , \quad p ( x ) q ( y ) \leqslant \left( r \left( \frac { x + y } { 2 } \right) \right) ^ { 2 } . \tag{4}$$ Show that $$\left( \int p ( x ) d x \right) \left( \int q ( x ) d x \right) \leqslant \left( \int r ( x ) d x \right) ^ { 2 } \tag{5}$$ You may use, after having justified its validity, the change of variable defined by $x = \frac { u ( t ) + v ( t ) } { 2 }$ in the right-hand side of inequality (5).
Q15 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
We recall that $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$ is a measure, and that for $A \in \operatorname{Int}$, $\mu(A) = \int \mathbb{1}_A(x) \mu(x) dx$. We denote $d(x, A) = \inf\{|x - y| : y \in A\}$. Let $A \subset \mathbb { R }$.
15a. Show that for all $x , y \in \mathbb { R }$, we have $$\exp \left( \frac { 1 } { 2 } d ( x , A ) ^ { 2 } - x ^ { 2 } \right) \mathbb { 1 } _ { A } ( y ) \exp \left( - y ^ { 2 } \right) \leqslant \exp \left( - \frac { ( x + y ) ^ { 2 } } { 2 } \right)$$
15b. We assume that $A \in \operatorname { Int }$ and that $\mu ( A ) > 0$. Deduce that $$\int \exp \left( \frac { 1 } { 2 } d ( x , A ) ^ { 2 } \right) \mu ( x ) d x \leqslant \frac { 1 } { \mu ( A ) }$$
Q16 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
We recall that $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$ is a measure, and that for $A \in \operatorname{Int}$, $\mu(A) = \int \mathbb{1}_A(x) \mu(x) dx$. Let $A \in \operatorname{Int}$. For $t \geqslant 0$, we define the set $A _ { t } = \{ x \in \mathbb { R } : d ( x , A ) \leqslant t \}$.
16a. Show that $A _ { t } \in \operatorname { Int }$ for all $t \geqslant 0$.
16b. We further assume that $\mu ( A ) > 0$. Show that for all $t \geqslant 0$, we have $$1 - \mu \left( A _ { t } \right) \leqslant \frac { e ^ { - t ^ { 2 } / 2 } } { \mu ( A ) }$$