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
2022 mines-ponts-maths1__mp

25 maths questions

Q1 Sequences and series, recurrence and convergence Series convergence and power series analysis View
Let $z \in D$. Show the convergence of the series $\sum _ { n \geq 1 } \frac { z ^ { n } } { n }$. Specify the value of its sum when $z \in ] - 1,1 [$. We denote
$$L ( z ) : = \sum _ { n = 1 } ^ { + \infty } \frac { z ^ { n } } { n }$$
Q3 Sequences and Series Convergence/Divergence Determination of Numerical Series View
Show that $| L ( z ) | \leq - \ln ( 1 - | z | )$ for all $z$ in $D$. Deduce the convergence of the series $\sum _ { n \geq 1 } L \left( z ^ { n } \right)$ for all $z$ in $D$. In what follows, we denote, for $z$ in $D$,
$$P ( z ) : = \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$$
Q4 Sequences and Series Recurrence Relations and Sequence Properties View
For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality.
Let $n \in \mathbf { N }$. Show that $P _ { n , N }$ is finite for all $N \in \mathbf { N } ^ { * }$, that the sequence $\left( p _ { n , N } \right) _ { N \geq 1 }$ is increasing, and that it is constant from rank $\max ( n , 1 )$ onward.
Q6 Sequences and Series Power Series Expansion and Radius of Convergence View
Let $z \in D$. We agree that $p _ { n , 0 } = 0$ for all $n \in \mathbf { N }$. By examining the summability of the family $\left( \left( p _ { n , N + 1 } - p _ { n , N } \right) z ^ { n } \right) _ { ( n , N ) \in \mathbf { N } ^ { 2 } }$, prove that
$$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$
Deduce the radius of convergence of the power series $\sum _ { n } p _ { n } x ^ { n }$.
Q7 Proof Direct Proof of a Stated Identity or Equality View
Let $n \in \mathbf { N }$. Show that for all real $t > 0$,
$$p _ { n } = \frac { e ^ { n t } } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } P \left( e ^ { - t + i \theta } \right) \mathrm { d } \theta$$
so that
$$p _ { n } = \frac { e ^ { n t } P \left( e ^ { - t } \right) } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } \frac { P \left( e ^ { - t + i \theta } \right) } { P \left( e ^ { - t } \right) } \mathrm { d } \theta$$
Q8 Trig Proofs Norm or Modulus Computation Involving Trig/Complex Exponentials View
Let $x \in [ 0,1 [$ and $\theta \in \mathbf { R }$. Using the function $L$, show that
$$\left| \frac { 1 - x } { 1 - x e ^ { i \theta } } \right| \leq \exp ( - ( 1 - \cos \theta ) x )$$
Deduce that for all $x \in [ 0,1 [$ and all real $\theta$,
$$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 1 - x } + \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \right)$$
Q9 Trig Proofs Norm or Modulus Computation Involving Trig/Complex Exponentials View
Let $x \in \left[ \frac { 1 } { 2 } , 1 [ \right.$ and $\theta \in \mathbf { R }$. Show that
$$\frac { 1 } { 1 - x } - \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \geq \frac { x ( 1 - \cos \theta ) } { ( 1 - x ) \left( ( 1 - x ) ^ { 2 } + 2 x ( 1 - \cos \theta ) \right) }$$
Deduce that
$$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 - \cos \theta } { 6 ( 1 - x ) ^ { 3 } } \right) \quad \text { or } \quad \left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 3 ( 1 - x ) } \right)$$
Q10 Sequences and Series Convergence/Divergence Determination of Numerical Series View
We fix a real $\alpha > 0$ and an integer $n \geq 1$. Subject to existence, we set
$$S _ { n , \alpha } ( t ) : = \sum _ { k = 1 } ^ { + \infty } \frac { k ^ { n } e ^ { - k t \alpha } } { \left( 1 - e ^ { - k t } \right) ^ { n } }$$
We also introduce the function
$$\varphi _ { n , \alpha } : x \in \mathbf { R } _ { + } ^ { * } \mapsto \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } }$$
which is obviously of class $\mathcal { C } ^ { \infty }$.
Show that $\varphi _ { n , \alpha }$ and $\varphi _ { n , \alpha } ^ { \prime }$ are integrable on $] 0 , + \infty [$.
Q11 Sequences and Series Evaluation of a Finite or Infinite Sum View
We fix a real $\alpha > 0$ and an integer $n \geq 1$. Subject to existence, we set
$$S _ { n , \alpha } ( t ) : = \sum _ { k = 1 } ^ { + \infty } \frac { k ^ { n } e ^ { - k t \alpha } } { \left( 1 - e ^ { - k t } \right) ^ { n } }$$
We also introduce the function
$$\varphi _ { n , \alpha } : x \in \mathbf { R } _ { + } ^ { * } \mapsto \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } }$$
Show, for all real $t > 0$, the existence of $S _ { n , \alpha } ( t )$, its strict positivity, and the identity
$$\int _ { 0 } ^ { + \infty } \varphi _ { n , \alpha } ( x ) \mathrm { d } x = t ^ { n + 1 } S _ { n , \alpha } ( t ) - \sum _ { k = 0 } ^ { + \infty } \int _ { k t } ^ { ( k + 1 ) t } ( x - k t ) \varphi _ { n , \alpha } ^ { \prime } ( x ) \mathrm { d } x$$
Deduce that
$$S _ { n , \alpha } ( t ) = \frac { 1 } { t ^ { n + 1 } } \int _ { 0 } ^ { + \infty } \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } } \mathrm {~d} x + O \left( \frac { 1 } { t ^ { n } } \right) \quad \text { as } t \rightarrow 0 ^ { + }$$
Q12 Reduction Formulae Connect a Discrete Sum to an Integral via Reduction Formulae View
Prove, without using what precedes, that
$$\int _ { 0 } ^ { + \infty } \frac { x e ^ { - x } } { 1 - e ^ { - x } } \mathrm {~d} x = \frac { \pi ^ { 2 } } { 6 }$$
Q13 Moment generating functions Existence and domain of the MGF View
Let $X$ be a real random variable. Show that $\left| \Phi _ { X } ( \theta ) \right| \leq 1$ for all real $\theta$.
Q14 Moment generating functions Compute MGF or characteristic function for a named distribution View
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that for all $( a , b ) \in \mathbf { R } ^ { 2 }$ and all real $\theta$,
$$\Phi _ { a X + b } ( \theta ) = \frac { p e ^ { i ( a + b ) \theta } } { 1 - q e ^ { i a \theta } }$$
Q15 Proof Proof That a Map Has a Specific Property View
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that for all $k \in \mathbf { N }$, the random variable $X ^ { k }$ has finite expectation. Show that $\Phi _ { X }$ is of class $\mathcal { C } ^ { \infty }$ on $\mathbf { R }$ and that $\Phi _ { X } ^ { ( k ) } ( 0 ) = i ^ { k } \mathbf { E } \left( X ^ { k } \right)$ for all $k \in \mathbf { N }$.
Q16 Probability Generating Functions Deriving moments or distribution from a PGF View
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that there exists a sequence $\left( P _ { k } \right) _ { k \in \mathbf { N } }$ of polynomials with coefficients in $\mathbf { C }$, independent of $p$, such that
$$\forall \theta \in \mathbf { R } , \forall k \in \mathbf { N } , \Phi _ { X } ^ { ( k ) } ( \theta ) = p i ^ { k } e ^ { i \theta } \frac { P _ { k } \left( q e ^ { i \theta } \right) } { \left( 1 - q e ^ { i \theta } \right) ^ { k + 1 } } \quad \text { and } \quad P _ { k } ( 0 ) = 1$$
Q17 Probability Generating Functions Bounding probabilities or tail estimates via PGF View
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Deduce that there exists a sequence $\left( C _ { k } \right) _ { k \in \mathbf { N } }$ of strictly positive reals, independent of $p$, such that
$$\forall k \in \mathbf { N } , \left| \mathbf { E } \left( X ^ { k } \right) - \frac { 1 } { p ^ { k } } \right| \leq \frac { C _ { k } q } { p ^ { k } }$$
Q18 Geometric Distribution View
In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Deduce that there exists a real $K > 0$ independent of $p$ such that
$$\mathbf { E } \left( ( X - \mathbf { E } ( X ) ) ^ { 4 } \right) \leq \frac { K q } { p ^ { 4 } }$$
Q19 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Show successively that $Y ^ { 2 }$ and $| Y | ^ { 3 }$ have finite expectation, and that
$$\mathrm { E } \left( Y ^ { 2 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 1 / 2 } \quad \text { then } \quad \mathrm { E } \left( | Y | ^ { 3 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$
Q20 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View
We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Show, for all real $u$, the inequality
$$\left| e ^ { i u } - 1 - i u + \frac { u ^ { 2 } } { 2 } \right| \leq \frac { | u | ^ { 3 } } { 6 }$$
Deduce that for all real $\theta$,
$$\left| \Phi _ { Y } ( \theta ) - 1 + \frac { \mathbf { E } \left( Y ^ { 2 } \right) \theta ^ { 2 } } { 2 } \right| \leq \frac { | \theta | ^ { 3 } } { 3 } \left( \mathbf { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$
Q21 Moment generating functions Approximation or bound on characteristic function difference View
We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Conclude that for all real $\theta$,
$$\left| \Phi _ { Y } ( \theta ) - \exp \left( - \frac { \mathbf { E } \left( Y ^ { 2 } \right) \theta ^ { 2 } } { 2 } \right) \right| \leq \frac { | \theta | ^ { 3 } } { 3 } \left( \mathbf { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 } + \frac { \theta ^ { 4 } } { 8 } \mathbf { E } \left( Y ^ { 4 } \right)$$
Q22 Complex Numbers Arithmetic Modulus Inequalities and Bounds (Proof-Based) View
Let $n \in \mathbf { N } ^ { * }$ as well as complex numbers $z _ { 1 } , \ldots , z _ { n } , u _ { 1 } , \ldots , u _ { n }$ all of modulus at most 1. Show that
$$\left| \prod _ { k = 1 } ^ { n } z _ { k } - \prod _ { k = 1 } ^ { n } u _ { k } \right| \leq \sum _ { k = 1 } ^ { n } \left| z _ { k } - u _ { k } \right|$$
Q23 Discrete Random Variables Convergence of Expectations or Moments View
Given a real $t > 0$, we set, following the notations of part $\mathbf{C}$,
$$m _ { t } : = S _ { 1,1 } ( t ) \quad \text { and } \quad \sigma _ { t } : = \sqrt { S _ { 2,1 } ( t ) }$$
Given reals $t > 0$ and $\theta$, we set
$$h ( t , \theta ) = e ^ { - i m _ { t } \theta } \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) }$$
Let $\theta \in \mathbf { R }$ and $t \in \mathbf { R } _ { + } ^ { * }$. We consider, for all $k \in \mathbf { N } ^ { * }$, a random variable $Z _ { k }$ following the distribution $\mathcal { G } \left( 1 - e ^ { - k t } \right)$, and we set $Y _ { k } = k \left( Z _ { k } - \mathrm { E } \left( Z _ { k } \right) \right)$. Prove that
$$h ( t , \theta ) = \lim _ { n \rightarrow + \infty } \prod _ { k = 1 } ^ { n } \Phi _ { Y _ { k } } ( \theta )$$
Deduce, using in particular question $21 \triangleright$, the inequality
$$\left| h ( t , \theta ) - e ^ { - \frac { \sigma _ { t } ^ { 2 } \theta ^ { 2 } } { 2 } } \right| \leq K ^ { 3 / 4 } | \theta | ^ { 3 } S _ { 3,3 / 4 } ( t ) + K \theta ^ { 4 } S _ { 4,1 } ( t )$$
Q24 Central limit theorem View
Given a real $t > 0$, we set
$$m _ { t } : = S _ { 1,1 } ( t ) \quad \text { and } \quad \sigma _ { t } : = \sqrt { S _ { 2,1 } ( t ) }$$
Given reals $t > 0$ and $u$, we set
$$\zeta ( t , u ) = \exp \left( i \frac { u } { \sigma _ { t } } \left( m _ { t } - \frac { \pi ^ { 2 } } { 6 t ^ { 2 } } \right) \right) \quad \text { and } \quad j ( t , u ) = \zeta ( t , u ) h \left( t , \frac { u } { \sigma _ { t } } \right)$$
Show that $\sigma _ { t } \sim \frac { \pi } { \sqrt { 3 } t ^ { 3 / 2 } }$ as $t$ tends to $0 ^ { + }$. Deduce from this that, for all real $u$,
$$j ( t , u ) \underset { t \rightarrow 0 ^ { + } } { \longrightarrow } e ^ { - u ^ { 2 } / 2 }$$
Q25 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View
Show that there exists a real $\alpha > 0$ such that
$$\forall \theta \in [ - \pi , \pi ] , 1 - \cos \theta \geq \alpha \theta ^ { 2 }$$
Using question $9 \triangleright$, deduce from this that there exist three real numbers $t _ { 0 } > 0 , \beta > 0$ and $\gamma > 0$ such that, for all $\left. t \in ] 0 , t _ { 0 } \right]$ and all $\theta \in [ - \pi , \pi ]$,
$$| h ( t , \theta ) | \leq e ^ { - \beta \left( \sigma _ { t } \theta \right) ^ { 2 } } \quad \text { or } \quad | h ( t , \theta ) | \leq e ^ { - \gamma \left( \sigma _ { t } | \theta | \right) ^ { 2 / 3 } }$$
Q26 Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View
Given reals $t > 0$ and $u$, we set
$$\zeta ( t , u ) = \exp \left( i \frac { u } { \sigma _ { t } } \left( m _ { t } - \frac { \pi ^ { 2 } } { 6 t ^ { 2 } } \right) \right) \quad \text { and } \quad j ( t , u ) = \zeta ( t , u ) h \left( t , \frac { u } { \sigma _ { t } } \right)$$
Conclude that
$$\int _ { - \pi \sigma _ { t } } ^ { \pi \sigma _ { t } } j ( t , u ) \mathrm { d } u \underset { t \rightarrow 0 ^ { + } } { \longrightarrow } \sqrt { 2 \pi }$$
Q27 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
We admit that $P \left( e ^ { - t } \right) \sim \sqrt { \frac { t } { 2 \pi } } \exp \left( \frac { \pi ^ { 2 } } { 6 t } \right)$ as $t$ tends to $0 ^ { + }$.
By applying formula (1) to $t = \frac { \pi } { \sqrt { 6 n } }$, prove that
$$p _ { n } \sim \frac { \exp \left( \pi \sqrt { \frac { 2 n } { 3 } } \right) } { 4 \sqrt { 3 } n } \quad \text { as } n \rightarrow + \infty$$