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
2025 polytechnique-maths__fui

16 maths questions

QP2-1 Matrices Matrix Norm, Convergence, and Inequality View
Problem 2, Part 1: Adapted norms
We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients and we identify $\mathbb { C } ^ { d }$ with the space of column vectors of size $d$.
For a vector $x = \left( x _ { 1 } , \ldots , x _ { d } \right) \in \mathbb { C } ^ { d }$, we define $\| x \| _ { \infty } = \max _ { 1 \leqslant i \leqslant d } \left| x _ { i } \right|$ and $\| x \| _ { 1 } = \sum _ { i = 1 } ^ { d } \left| x _ { i } \right|$.
Let $A \in \mathrm { M } _ { d } ( \mathbb { C } )$. Determine a necessary and sufficient condition on $A$ for the map $x \mapsto \| A x \| _ { \infty }$ to define a norm on $\mathbb { C } ^ { d }$.
QP2-2 Matrices Matrix Norm, Convergence, and Inequality View
Problem 2, Part 1: Adapted norms
We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients and we identify $\mathbb { C } ^ { d }$ with the space of column vectors of size $d$.
For a vector $x = \left( x _ { 1 } , \ldots , x _ { d } \right) \in \mathbb { C } ^ { d }$, we define $\| x \| _ { \infty } = \max _ { 1 \leqslant i \leqslant d } \left| x _ { i } \right|$ and $\| x \| _ { 1 } = \sum _ { i = 1 } ^ { d } \left| x _ { i } \right|$.
Given a matrix $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ we define $$\| A \| = \sup _ { \| x \| _ { \infty } \leqslant 1 } \| A x \| _ { \infty } .$$
a. Show that this defines a norm on $\mathrm { M } _ { d } ( \mathbb { C } )$ and that there exists $x _ { 0 } \in \mathbb { C } ^ { d }$ such that $\left\| x _ { 0 } \right\| _ { \infty } = 1$ and $\left\| A x _ { 0 } \right\| _ { \infty } = \| A \|$. b. Show that for all $( A , B ) \in \mathrm { M } _ { d } ( \mathbb { C } )$ we have $\| A B \| \leqslant \| A \| \cdot \| B \|$.
QP2-3 Matrices Matrix Norm, Convergence, and Inequality View
Problem 2, Part 1: Adapted norms
We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients and we identify $\mathbb { C } ^ { d }$ with the space of column vectors of size $d$.
For a vector $x = \left( x _ { 1 } , \ldots , x _ { d } \right) \in \mathbb { C } ^ { d }$, we define $\| x \| _ { \infty } = \max _ { 1 \leqslant i \leqslant d } \left| x _ { i } \right|$ and $\| x \| _ { 1 } = \sum _ { i = 1 } ^ { d } \left| x _ { i } \right|$.
Given a matrix $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ we define $\| A \| = \sup _ { \| x \| _ { \infty } \leqslant 1 } \| A x \| _ { \infty }$. For $1 \leqslant i \leqslant d$ we define $L _ { i } = \left( a _ { i , j } \right) _ { 1 \leqslant j \leqslant d }$ as the $i ^ { \mathrm { th } }$ row vector of $A$. Show that $$\| A \| = \max _ { 1 \leqslant i \leqslant d } \left\| L _ { i } \right\| _ { 1 } .$$
QP2-4 Matrices Diagonalizability and Similarity View
Problem 2, Part 1: Adapted norms
We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients.
a. Let $u \in \mathcal { L } \left( \mathbb { C } ^ { d } \right)$ be an endomorphism of $\mathbb { C } ^ { d }$ and $M = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant d }$ the matrix of $u$ in a basis $\mathcal { B } = \left( e _ { 1 } , \ldots , e _ { d } \right)$. Express the matrix $M ^ { \prime } = \left( m _ { i , j } ^ { \prime } \right) _ { 1 \leqslant i , j \leqslant d }$ of $u$ in the basis $\mathcal { B } ^ { \prime } = \left( \alpha _ { 1 } e _ { 1 } , \ldots , \alpha _ { d } e _ { d } \right)$, where the $\alpha _ { i }$ are complex numbers. b. Suppose that $M$ is upper triangular. Show that for all $\varepsilon > 0$ we can choose the $\alpha _ { i }$ such that for $j > i$ we have $\left| m _ { i , j } ^ { \prime } \right| < \varepsilon$.
QP2-5 Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View
Problem 2, Part 1: Adapted norms
We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients and we identify $\mathbb { C } ^ { d }$ with the space of column vectors of size $d$. For a vector $x = \left( x _ { 1 } , \ldots , x _ { d } \right) \in \mathbb { C } ^ { d }$, we define $\| x \| _ { \infty } = \max _ { 1 \leqslant i \leqslant d } \left| x _ { i } \right|$. If $A$ is a matrix in $\mathrm { M } _ { d } ( \mathbb { C } )$ we denote by $\operatorname { Sp } ( A )$ the spectrum of $A$ and we define the spectral radius $\sigma ( A ) = \max \{ | \lambda | , \lambda \in \operatorname { Sp } ( A ) \}$.
Let $T = \left( t _ { i , j } \right) _ { 1 \leqslant i , j \leqslant d }$ be an upper triangular matrix. Show that for all $\varepsilon > 0$, there exists a norm $\| \cdot \| ^ { \prime }$ on $\mathbb { C } ^ { d }$ such that for all $x \in \mathbb { C } ^ { d }$ we have $$\| T x \| ^ { \prime } \leqslant ( \sigma ( T ) + \varepsilon ) \| x \| ^ { \prime }$$ (one may choose $\| \cdot \| ^ { \prime }$ in the form $\| x \| ^ { \prime } = \| P x \| _ { \infty }$ for a suitably chosen matrix $P$).
QP2-6 Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View
Problem 2, Part 1: Adapted norms
We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients. If $A$ is a matrix in $\mathrm { M } _ { d } ( \mathbb { C } )$ we denote by $\operatorname { Sp } ( A )$ the spectrum of $A$ and we define the spectral radius $\sigma ( A ) = \max \{ | \lambda | , \lambda \in \operatorname { Sp } ( A ) \}$. We define $\| A \| = \sup _ { \| x \| _ { \infty } \leqslant 1 } \| A x \| _ { \infty }$.
Application: norm and spectral radius. a. Let $T \in \mathrm { M } _ { d } ( \mathbb { C } )$ be an upper triangular matrix. Show that for all $\varepsilon > 0$, there exists a constant $C$ such that for all $n$ we have $\left\| T ^ { n } \right\| \leqslant C ( \sigma ( T ) + \varepsilon ) ^ { n }$. b. Show that $\lim _ { n \rightarrow \infty } \left\| T ^ { n } \right\| ^ { 1 / n } = \sigma ( T )$. c. Let now $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ be an arbitrary matrix. Show that $\lim _ { n \rightarrow \infty } \left\| A ^ { n } \right\| ^ { 1 / n } = \sigma ( A )$. d. Show the equivalence $$A ^ { n } \underset { n \rightarrow \infty } { \longrightarrow } 0 \Leftrightarrow \sigma ( A ) < 1 .$$
Q1 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View
Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Study of $f$. a. Show that $f ( x )$ is well defined for all $x \geqslant 0$. b. Show with precision that the function $f$ is of class $C ^ { 2 }$ on $] 0 , \infty [$, and also continuous at 0. c. Calculate $f + f ^ { \prime \prime }$ and deduce that $f$ is a solution of a linear second-order differential equation.
Q2 Second order differential equations Solving homogeneous second-order linear ODE View
Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Study of $g$. a. Show that $g ( x )$ is well defined for all $x \geqslant 0$. b. Show that $g$ is of class $C ^ { 2 }$ on $] 0 , \infty [$.
For this you may use the change of variable $u = t + x$ and express $g$ in terms of the functions $C : x \mapsto \int _ { x } ^ { \infty } \frac { \cos u } { u } \mathrm {~d} u$ and $S : x \mapsto \int _ { x } ^ { \infty } \frac { \sin u } { u } \mathrm {~d} u$. c. Determine a linear second-order differential equation satisfied by $f$.
Q3 Differential equations Finding a DE from a Limit or Implicit Condition View
Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Show that on $] 0 , \infty [$ we have $f = g$. For this you may use a differential equation satisfied by $( f - g )$ and use the behavior of $f$ and $g$ at $+ \infty$.
Q5 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Deduce the value of $\int _ { 0 } ^ { \infty } \frac { \sin t } { t } \mathrm {~d} t$.
Q7 Matrices Matrix Power Computation and Application View
Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1.
For $n \geqslant 0$ we define the vector $U _ { n } \in \mathbb { C } ^ { d }$ by $U _ { n } = \left( u _ { n } , \ldots , u _ { n + d - 1 } \right)$ (recall that $U _ { n }$ is identified with a column vector). Show that the sequence $(U _ { n })$ satisfies a recurrence relation of the form $U _ { n + 1 } = A U _ { n } + B$, with $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ and $B \in \mathbb { C } ^ { d }$ are elements that we shall specify.
Q8 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1. The matrix $A \in \mathrm{M}_d(\mathbb{C})$ is as defined in question 7.
Calculate the characteristic polynomial of the matrix $A$ (one may reason by induction on $d$).
Q9 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1.
Suppose in this question that $b = 0$. Show that $(u _ { n })$ tends to 0.
Q10 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1.
In the general case, show that $(u _ { n })$ converges and specify its limit.
Q11 Sequences and series, recurrence and convergence Coefficient and growth rate estimation View
Problem 2, Part 3: Linear recurrence sequences with variable coefficients
We consider a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ satisfying a recurrence of the form $$v _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } b _ { i } ( n ) v _ { n + i }$$ where $v _ { 0 } , \ldots , v _ { d - 1 }$ are given and for all $i \in \{ 0 , \ldots , d - 1 \} , \left( b _ { i } ( n ) \right) _ { n \geqslant 0 }$ is a sequence with complex values converging to $a _ { i }$. We also define for all $n \geqslant 0 , V _ { n } = \left( v _ { n } , \ldots , v _ { n + d - 1 } \right)$. We always assume hypothesis (*) is satisfied (all complex roots of $P(X) = X^d - \sum_{i=0}^{d-1} a_i X^i$ have modulus strictly less than 1), and $A$ is the matrix from question 7.
Let $\varepsilon > 0$ be fixed. Show that there exists an integer $q \geqslant 1$ and an integer $n _ { 0 }$ such that for all $n \geqslant n _ { 0 }$, $$\left\| V _ { n + q } \right\| _ { \infty } \leqslant ( \sigma ( A ) + \varepsilon ) ^ { q } \left\| V _ { n } \right\| _ { \infty }$$ where $A$ is the matrix from question 7.
Q12 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Problem 2, Part 3: Linear recurrence sequences with variable coefficients
We consider a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ satisfying a recurrence of the form $$v _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } b _ { i } ( n ) v _ { n + i }$$ where $v _ { 0 } , \ldots , v _ { d - 1 }$ are given and for all $i \in \{ 0 , \ldots , d - 1 \} , \left( b _ { i } ( n ) \right) _ { n \geqslant 0 }$ is a sequence with complex values converging to $a _ { i }$. We also define for all $n \geqslant 0 , V _ { n } = \left( v _ { n } , \ldots , v _ { n + d - 1 } \right)$. We always assume hypothesis (*) is satisfied.
Deduce that $v _ { n }$ tends to 0.