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
2023 mines-ponts-maths1__mp

15 maths questions

Q1 Groups Group Actions and Surjectivity/Injectivity of Maps View
After justifying the existence of the suprema, show that: $$\sup _ { \substack { x \in E \\ x \neq 0 } } \frac { \| u ( x ) \| } { \| x \| } = \sup _ { \substack { x \in E \\ \| x \| = 1 } } \| u ( x ) \| .$$
Q2 Matrices Matrix Norm, Convergence, and Inequality View
We denote $\| u \| = \sup _ { \substack { x \in E \\ x \neq 0 } } \frac { \| u ( x ) \| } { \| x \| }$. Show that $\|.\|$ is a norm on $\mathcal { L } ( E )$.
Q3 Matrices Matrix Norm, Convergence, and Inequality View
Show that it is a sub-multiplicative norm, that is: $$\forall ( u , v ) \in \mathcal { L } ( E ) ^ { 2 } , \quad \| u v \| \leqslant \| u \| \cdot \| v \| ,$$ and deduce a bound on $\left\| u ^ { k } \right\|$, for any natural number $k$, in terms of $\| u \|$ and the integer $k$.
Q4 Invariant lines and eigenvalues and vectors Annihilating or minimal polynomial and spectral deductions View
In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$.
Show that there exist a non-zero natural number $r$, distinct complex numbers $\lambda _ { 1 } , \lambda _ { 2 } , \ldots$, $\lambda _ { r }$, and non-zero natural numbers $m _ { 1 } , m _ { 2 } , \ldots , m _ { r }$, such that: $$\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$$ where for $i \in \llbracket 1 ; r \rrbracket , E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$.
Q7 Matrices Linear Transformation and Endomorphism Properties View
In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$ and inclusions $q_i$ as defined.
Let $( i , j ) \in \llbracket 1 ; r \rrbracket ^ { 2 }$. Express $p _ { i } q _ { j }$ and then $\sum _ { i = 1 } ^ { r } q _ { i } p _ { i }$ in terms of the endomorphisms $id _ { \mathbf { C } ^ { n } }$ and $id _ { E _ { j } }$.
Q8 Matrices Linear Transformation and Endomorphism Properties View
In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$, inclusions $q_i$, and $a_i = p_i a q_i$ the endomorphism of $E_i$.
Show that: $a = \sum _ { i = 1 } ^ { r } q _ { i } a _ { i } p _ { i }$.
Q9 Systems of differential equations View
In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$, inclusions $q_i$, $a_i = p_i a q_i$, and $a = \sum _ { i = 1 } ^ { r } q _ { i } a _ { i } p _ { i }$.
Deduce that: $$\forall t \in \mathbf { R } , \quad e ^ { t a } = \sum _ { i = 1 } ^ { r } q _ { i } e ^ { t a _ { i } } p _ { i }$$
Q10 Systems of differential equations View
In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with $a_i = p_i a q_i$ the endomorphism of $E_i$ and $\|.\|_i$ the norm on $\mathcal{L}(E_i)$.
Show moreover that: $$\forall i \in \llbracket 1 ; r \rrbracket , \quad \forall t \in \mathbf { R } , \quad \left\| e ^ { t a _ { i } } \right\| _ { i } \leqslant \left| e ^ { t \lambda _ { i } } \right| \sum _ { k = 0 } ^ { m _ { i } - 1 } \frac { | t | ^ { k } } { k ! } \left\| a _ { i } - \lambda _ { i } id _ { E _ { i } } \right\| _ { i } ^ { k }$$
Q11 Second order differential equations Qualitative and asymptotic analysis of solutions View
In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with $\|.\|_c$ the norm on $\mathcal{L}(\mathbf{C}^n)$.
Deduce the existence of a polynomial $P$ with real coefficients such that: $$\forall t \in \mathbf { R } , \quad \left\| e ^ { t a } \right\| _ { c } \leqslant P ( | t | ) \sum _ { i = 1 } ^ { r } e ^ { t \operatorname { Re } \left( \lambda _ { i } \right) }$$ where $\operatorname { Re } ( z )$ denotes the real part of a complex number $z$.
Q12 Second order differential equations Qualitative and asymptotic analysis of solutions View
For any matrix $A \in \mathscr { M } _ { n } ( \mathbf { R } )$, we denote by $u _ { A }$ the endomorphism canonically associated with $A$ in $\mathbf { R } ^ { n }$ and $v _ { A }$ the endomorphism of $\mathbf { C } ^ { n }$ canonically associated with $A$, viewed as a matrix in $\mathscr { M } _ { n } ( \mathbf { C } )$. We keep the notation $\| . \| _ { c }$ for the norm introduced in part A on $\mathcal { L } \left( \mathbf { C } ^ { n } \right)$ and we use $\| . \| _ { r }$ on $\mathcal { L } \left( \mathbf { R } ^ { n } \right)$. Show that there exists $C > 0$ such that: $$\forall A \in \mathscr { M } _ { n } ( \mathbf { R } ) , \quad \forall t \in \mathbf { R } , \quad \left\| e ^ { t u _ { A } } \right\| _ { r } \leqslant C \left\| e ^ { t v _ { A } } \right\| _ { c }$$
Q13 Second order differential equations Qualitative and asymptotic analysis of solutions View
We consider $u$ an endomorphism of $\mathbf { R } ^ { n }$, and $A \in \mathscr { M } _ { n } ( \mathbf { R } )$ its matrix in the canonical basis. We denote by $S p ( A )$ the complex spectrum of $A$. Let $g _ { x _ { 0 } }$ be the unique $\mathcal { C } ^ { 1 }$ solution on $\mathbf { R } _ { + }$ of: $$\left\{ \begin{aligned} y ^ { \prime } & = u ( y ) \\ y ( 0 ) & = x _ { 0 } \end{aligned} \right.$$
Show that: $$\forall x _ { 0 } \in \mathbf { R } ^ { n } , \quad \lim _ { t \rightarrow + \infty } \left\| g _ { x _ { 0 } } ( t ) \right\| = 0 \Longleftrightarrow S p ( A ) \subset \mathbf { R } _ { - } ^ { * } + i \mathbf { R } .$$
Q14 Second order differential equations Qualitative and asymptotic analysis of solutions View
We consider $u$ an endomorphism of $\mathbf { R } ^ { n }$, and $A \in \mathscr { M } _ { n } ( \mathbf { R } )$ its matrix in the canonical basis. Let $g _ { x _ { 0 } }$ be the unique $\mathcal { C } ^ { 1 }$ solution on $\mathbf { R } _ { + }$ of: $$\left\{ \begin{aligned} y ^ { \prime } & = u ( y ) \\ y ( 0 ) & = x _ { 0 } \end{aligned} \right.$$
In this question we assume that all eigenvalues of $A$ have strictly negative real part. Show then that there exist two strictly positive constants $C _ { 2 }$ and $\alpha$ such that: $$\forall t \in \mathbf { R } _ { + } , \quad \left\| e ^ { t u } \right\| _ { r } \leqslant C _ { 2 } e ^ { - \alpha t }$$ and deduce a bound on $\left\| g _ { x _ { 0 } } ( t ) \right\|$ for $t \in \mathbf { R } _ { + }$.
Q15 Second order differential equations Qualitative and asymptotic analysis of solutions View
In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part.
Show that the function $$b : \left\lvert \, \begin{array} { r l l } \mathbf { R } ^ { n } \times \mathbf { R } ^ { n } & \rightarrow & \mathbf { R } \\ ( x , y ) & \mapsto & \int _ { 0 } ^ { + \infty } \left\langle e ^ { t a } ( x ) \mid e ^ { t a } ( y ) \right\rangle d t \end{array} \right.$$ is well-defined and that it defines an inner product on $\mathbf { R } ^ { n }$.
Q16 Proof Direct Proof of a Stated Identity or Equality View
In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ be the inner product on $\mathbf{R}^n$, and $q$ the associated quadratic form, i.e., $q(x) = b(x,x)$ for all $x \in \mathbf{R}^n$.
Prove then that: $$\forall x \in \mathbf { R } ^ { n } , \quad d q ( x ) ( a ( x ) ) = 2 b ( x , a ( x ) ) = - \| x \| ^ { 2 }$$
Q17 Proof Direct Proof of a Stated Identity or Equality View
In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. For any function $y$ defined on $\mathbf{R}_+$, define: $$\varepsilon ( y ) : \mathbf { R } _ { + } \rightarrow \mathbf { R } ^ { n }, \quad t \mapsto \varphi ( y ( t ) ) - a ( y ( t ) )$$
Verify the equality $$\forall t \in \mathbf { R } _ { + } , \quad q \left( f _ { x _ { 0 } } \right) ^ { \prime } ( t ) = - \left\| f _ { x _ { 0 } } ( t ) \right\| ^ { 2 } + 2 b \left( f _ { x _ { 0 } } ( t ) , \varepsilon \left( f _ { x _ { 0 } } ( t ) \right) \right)$$