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
2017 centrale-maths1__pc

22 maths questions

QIA Proof Combinatorial Proof or Identity Derivation View
Let $k$ and $n$ be two strictly positive integers. Show that there exists only a finite number of partitions of the set $\llbracket 1 , n \rrbracket$ into $k$ parts.
QIB Proof Combinatorial Proof or Identity Derivation View
Throughout the problem, for every pair $( n , k )$ of strictly positive integers, we denote by $S ( n , k )$ the number of partitions of the set $\llbracket 1 , n \rrbracket$ into $k$ parts. We further set $S ( 0,0 ) = 1$ and, for all $( n , k ) \in \mathbb { N } ^ { * 2 } , S ( n , 0 ) = S ( 0 , k ) = 0$.
Express $S ( n , k )$ as a function of $n$ or of $k$ in the following cases:
I.B.1) $k > n$;
I.B.2) $k = 1$.
QIC Proof Combinatorial Proof or Identity Derivation View
Throughout the problem, for every pair $( n , k )$ of strictly positive integers, we denote by $S ( n , k )$ the number of partitions of the set $\llbracket 1 , n \rrbracket$ into $k$ parts. We further set $S ( 0,0 ) = 1$ and, for all $( n , k ) \in \mathbb { N } ^ { * 2 } , S ( n , 0 ) = S ( 0 , k ) = 0$.
Show that for all strictly positive integers $k$ and $n$, we have $$S ( n , k ) = S ( n - 1 , k - 1 ) + k S ( n - 1 , k )$$ One may distinguish the partitions of $\llbracket 1 , n \rrbracket$ according to whether or not they contain the singleton $\{ n \}$.
QID Proof by induction Combinatorial Proof or Identity Derivation View
Throughout the problem, for every pair $( n , k )$ of strictly positive integers, we denote by $S ( n , k )$ the number of partitions of the set $\llbracket 1 , n \rrbracket$ into $k$ parts. We further set $S ( 0,0 ) = 1$ and, for all $( n , k ) \in \mathbb { N } ^ { * 2 } , S ( n , 0 ) = S ( 0 , k ) = 0$. The recurrence relation $S ( n , k ) = S ( n - 1 , k - 1 ) + k S ( n - 1 , k )$ holds for all strictly positive integers $k$ and $n$.
I.D.1) Write a recursive Python function to compute the number $S ( n , k )$, by direct application of the formula established in question I.C.
I.D.2) Show that, for $n \geqslant 1$, the computation of $S ( n , k )$ by this recursive function requires at least $\binom { n } { k }$ operations (sums or products).
QIIA Proof Combinatorial Proof or Identity Derivation View
We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Show that for $n \geqslant 1$, $B _ { n }$ equals the total number of partitions of the set $\llbracket 1 , n \rrbracket$.
QIIB Proof Functional Equations and Identities via Series View
We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Prove the formula $$\forall n \in \mathbb { N } , \quad B _ { n + 1 } = \sum _ { k = 0 } ^ { n } \binom { n } { k } B _ { k }$$
QIIC Proof Proof of Inequalities Involving Series or Sequence Terms View
We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Show that the sequence $\left( \frac { B _ { n } } { n ! } \right) _ { n \in \mathbb { N } }$ is bounded by 1.
QIID Proof Power Series Expansion and Radius of Convergence View
We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts. The sequence $\left( \frac { B _ { n } } { n ! } \right) _ { n \in \mathbb { N } }$ is bounded by 1.
Deduce a lower bound for the radius of convergence $R$ of the power series $\sum _ { n \geqslant 0 } \frac { B _ { n } } { n ! } z ^ { n }$.
QIIE Proof Properties and Manipulation of Power Series or Formal Series View
We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts. Let $R$ be the radius of convergence of the power series $\sum _ { n \geqslant 0 } \frac { B _ { n } } { n ! } z ^ { n }$, and for $x \in ] - R , R [$, set $f ( x ) = \sum _ { n = 0 } ^ { + \infty } \frac { B _ { n } } { n ! } x ^ { n }$.
Show that for all $x \in ] - R , R [ , f ^ { \prime } ( x ) = \mathrm { e } ^ { x } f ( x )$.
QIIF Proof Evaluation of a Finite or Infinite Sum View
We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts. Let $R$ be the radius of convergence of the power series $\sum _ { n \geqslant 0 } \frac { B _ { n } } { n ! } z ^ { n }$, and for $x \in ] - R , R [$, set $f ( x ) = \sum _ { n = 0 } ^ { + \infty } \frac { B _ { n } } { n ! } x ^ { n }$. It has been shown that $f ^ { \prime } ( x ) = \mathrm { e } ^ { x } f ( x )$ for all $x \in ] - R , R [$.
Deduce an expression for the function $f$ on $] - R , R [$.
QIIIA Proof Properties and Manipulation of Power Series or Formal Series View
We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$
Show that the family $( H _ { 0 } , \ldots , H _ { n } )$ is a basis of the space $\mathbb { R } _ { n } [ X ]$.
QIIIB Sequences and Series Functional Equations and Identities via Series View
We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$ and $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
III.B.1) For all $k \in \mathbb { N }$, establish a simplified expression for $H _ { k + 1 } ( X ) + k H _ { k } ( X )$.
III.B.2) Deduce that, for every natural integer $n$ $$X ^ { n } = \sum _ { k = 0 } ^ { n } S ( n , k ) H _ { k } ( X )$$
QIIIC Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$ and $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Let $k \in \mathbb { N }$.
III.C.1) Show that the function $f _ { k } : x \mapsto \sum _ { n = k } ^ { + \infty } S ( n , k ) \frac { x ^ { n } } { n ! }$ is defined on $] - 1,1 [$.
III.C.2) For $k \in \mathbb { N }$, we consider the function $g _ { k } : x \mapsto \frac { \left( \mathrm { e } ^ { x } - 1 \right) ^ { k } } { k ! }$.
Show that the function $g _ { k }$ satisfies the differential equation $$y ^ { \prime } = \frac { \left( \mathrm { e } ^ { x } - 1 \right) ^ { k - 1 } } { ( k - 1 ) ! } + k y$$
III.C.3) Deduce that for all $k \in \mathbb { N }$ and for all $x \in ] - 1,1 [$, $$\frac { \left( \mathrm { e } ^ { x } - 1 \right) ^ { k } } { k ! } = \sum _ { n = k } ^ { + \infty } S ( n , k ) \frac { x ^ { n } } { n ! }$$
QIIID Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We define the sequence of polynomials $\left( H _ { k } \right) _ { k \in \mathbb { N } }$ in $\mathbb { R } [ X ]$ by $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $$H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$$ It has been established that for all $k \in \mathbb { N }$ and $x \in ] - 1,1 [$, $$\frac { \left( \mathrm { e } ^ { x } - 1 \right) ^ { k } } { k ! } = \sum _ { n = k } ^ { + \infty } S ( n , k ) \frac { x ^ { n } } { n ! }$$
III.D.1) For $x \in ] - 1,1 [$ and $\alpha \in \mathbb { R }$, simplify $\sum _ { k = 0 } ^ { + \infty } H _ { k } ( \alpha ) \frac { x ^ { k } } { k ! }$.
III.D.2) Show that for $u < \ln 2$ $$\mathrm { e } ^ { u \alpha } = \sum _ { k = 0 } ^ { + \infty } H _ { k } ( \alpha ) \frac { \left( \mathrm { e } ^ { u } - 1 \right) ^ { k } } { k ! }$$
QIVA Discrete Probability Distributions Radius of convergence and analytic properties of PGF View
We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. Let $m$ be a strictly positive integer. We say that a random variable $Y : \Omega \rightarrow \mathbb { N }$ admits a finite moment of order $m$ if $Y$ admits a finite expectation, that is, if the series $\sum n ^ { m } P ( Y = n )$ converges. We then call the moment of order $m$ of $Y$ the real number $$\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$$
Show that if $Y : \Omega \rightarrow \mathbb { N }$ is a random variable associated with a generating function $G _ { Y }$ of radius strictly greater than 1, then $Y$ admits a finite moment of all orders.
QIVB Discrete Probability Distributions Deriving moments or distribution from a PGF View
We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. We define $H_k(X) = X(X-1)\cdots(X-k+1)$ for $k \in \mathbb{N}^*$ and $H_0(X)=1$. Let $m$ be a strictly positive integer. We say that a random variable $Y : \Omega \rightarrow \mathbb { N }$ admits a finite moment of order $m$ if the series $\sum n ^ { m } P ( Y = n )$ converges, and $\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$.
Let $Y : \Omega \rightarrow \mathbb { N }$ be a random variable admitting a finite moment of all orders.
IV.B.1) Show that the generating function $G _ { Y }$ is of class $C ^ { \infty }$ on $[ - 1,1 ]$.
IV.B.2) Express $G _ { Y } ^ { ( k ) } ( 1 )$ using the polynomials $H _ { k } ( X )$ and the random variable $Y$.
IV.B.3) Does the generating function $G _ { Y }$ necessarily have a radius of convergence strictly greater than 1? One may use the power series $\sum \mathrm { e } ^ { - \sqrt { n } } x ^ { n }$.
QIVC Poisson distribution Combinatorial PGF for counting problems View
We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. We set for every integer $n \geqslant 0$, $B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$ where $S(n,k)$ is the number of partitions of $\llbracket 1,n \rrbracket$ into $k$ parts. Let $m$ be a strictly positive integer, and $\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$.
We assume in this question that $Y$ follows the Poisson distribution with parameter 1.
IV.C.1) Show that for all $n \in \mathbb { N } , B _ { n } = \mathbb { E } \left( Y ^ { n } \right)$.
IV.C.2) Deduce that for every polynomial $Q ( X )$ with integer coefficients, the series $\sum _ { n = 0 } ^ { + \infty } \frac { Q ( n ) } { n ! }$ is convergent and its sum is of the form $N e$, where $N$ is an integer.
QVA Indefinite & Definite Integrals Asymptotic Equivalents and Growth Estimates for Sequences/Series View
We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$
Using an enclosure by integrals, determine an asymptotic equivalent of $U _ { n } ( p ) = \sum _ { k = 0 } ^ { p } k ^ { n }$, with $n \geqslant 1$ fixed, as $p$ tends to $+ \infty$.
QVB Matrices Matrix Entry and Coefficient Identities View
We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$ We define $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$.
Let $\Delta _ { n }$ be the endomorphism induced by $\Delta$ on the stable subspace $\mathbb { R } _ { n } [ X ]$. Determine the matrix $A$ of $\Delta _ { n }$ in the basis $( H _ { 0 } , \ldots , H _ { n } )$.
QVC Sequences and Series Evaluation of a Finite or Infinite Sum View
We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$ We define $H _ { 0 } ( X ) = 1$ and, for all $k \in \mathbb { N } ^ { * }$, $H _ { k } ( X ) = X ( X - 1 ) \cdots ( X - k + 1 )$, and $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Deduce that $U _ { n } ( p ) = \sum _ { k = 0 } ^ { n } \frac { S ( n , k ) } { k + 1 } H _ { k + 1 } ( p + 1 )$.
QVD Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$ We denote $F = \left\{ P \in \mathbb { R } _ { n } [ X ] \mid P ( 0 ) = 0 \right\}$, then $G = \operatorname { Vect } \left( X ^ { 2 k + 1 } ; 0 \leqslant k \leqslant n - 1 \right)$.
Let $Q ( X )$ be the polynomial such that $\forall p \in \mathbb { N } , Q ( p ) = \sum _ { k = 0 } ^ { p } k$.
V.D.1) Recall the explicit expression of the polynomial $Q ( X )$.
V.D.2) Show that the map: $$\begin{aligned} \Phi : F & \rightarrow G \\ P ( X ) & \mapsto \Delta ( P ( Q ( X - 1 ) ) ) \end{aligned}$$ is an isomorphism.
V.D.3) Deduce that for all $r \in \mathbb { N }$, there exists a unique polynomial $P _ { r } ( X )$ such that $$\forall p \in \mathbb { N } , \quad \sum _ { k = 1 } ^ { p } k ^ { 2 r + 1 } = P _ { r } \left( \frac { p ( p + 1 ) } { 2 } \right)$$
QVE Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We fix $n \in \mathbb { N }$. For all $r \in \mathbb { N }$, there exists a unique polynomial $P _ { r } ( X )$ such that $$\forall p \in \mathbb { N } , \quad \sum _ { k = 1 } ^ { p } k ^ { 2 r + 1 } = P _ { r } \left( \frac { p ( p + 1 ) } { 2 } \right)$$
V.E.1) Determine the leading term in $P _ { r } ( X )$.
V.E.2) Show that for $r \geqslant 1$, $X ^ { 2 }$ divides $P _ { r } ( X )$.
V.E.3) Explicitly give the polynomials $P _ { 1 } ( X )$ and $P _ { 2 } ( X )$.