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
2014 x-ens-maths1__mp

9 maths questions

Q1 Groups Binary Operation Properties View
Prove that $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ is indeed a quadratic form on $\mathbb { K } ^ { n }$, where $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ denotes the quadratic form $q$ defined on $\mathbb { K } ^ { n }$ by the formula $$q \left( x _ { 1 } , \ldots , x _ { n } \right) = a _ { 1 } x _ { 1 } ^ { 2 } + \cdots + a _ { n } x _ { n } ^ { 2 }$$
Q2 Matrices Bilinear and Symplectic Form Properties View
Prove that the map $q \mapsto \widetilde { q }$ is a bijection from the set of quadratic forms on $V$ to the set of symmetric bilinear forms on $V$, where $\widetilde { q } : V \times V \rightarrow \mathbb { K }$ is defined by $( x , y ) \mapsto \widetilde { q } ( x , y ) = \frac { 1 } { 2 } ( q ( x + y ) - q ( x ) - q ( y ) )$.
Q3 Matrices Determinant and Rank Computation View
Let $\mathcal { B } : = \left( e _ { 1 } , \ldots , e _ { n } \right)$ be a basis of $V$. We associate to every symmetric bilinear form $b$ on $V$ a symmetric matrix $\Phi _ { \mathcal { B } } ( b ) : = \left( b \left( e _ { i } , e _ { j } \right) \right) _ { i , j = 1 \ldots n }$ called the matrix of $b$ in the basis $\mathcal { B }$. We recall that $b \mapsto \Phi _ { \mathcal { B } } ( b )$ is an isomorphism between the vector space of symmetric bilinear forms on $V$ and that of square symmetric matrices of size $n$.
(a) Prove that a quadratic form $q$ on $V$ is non-degenerate if and only if the determinant $\operatorname { det } \left( \Phi _ { \mathcal { B } } ( \tilde { q } ) \right)$ is non-zero.
(b) What is the matrix of $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ in the canonical basis of $\mathbb { K } ^ { n }$ ? Deduce that $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle \in \mathcal { Q } \left( \mathbb { K } ^ { n } \right)$.
Q4 Matrices Linear Transformation and Endomorphism Properties View
Let $q \in \mathcal { Q } ( V )$ be a non-degenerate quadratic form on $V$.
(a) Let $V ^ { \prime }$ be a $\mathbb { K }$-vector space of finite dimension and $q ^ { \prime }$ a quadratic form on $V ^ { \prime }$. Prove that if $q$ and $q ^ { \prime }$ are isometric, then $q ^ { \prime }$ is in $\mathcal { Q } \left( V ^ { \prime } \right)$, that is, non-degenerate.
(b) For $x \neq 0$, we denote $\{ x \} ^ { \perp } : = \{ y \in V \mid \widetilde { q } ( x , y ) = 0 \}$. Show that $\{ x \} ^ { \perp }$ is a vector subspace of $V$ of dimension $n - 1$.
(c) Under what condition on $x$ is the subspace $\{ x \} ^ { \perp }$ a complement of the line $\mathbb { K } x$ in $V$ ?
Q5 Groups Symplectic and Orthogonal Group Properties View
Let $q \in \mathcal { Q } ( V )$ and $q ^ { \prime } \in \mathcal { Q } \left( V ^ { \prime } \right)$ where $V ^ { \prime }$ is a $\mathbb { K }$-vector space of finite dimension. Prove that $O ( q )$ is a subgroup of $\mathrm { GL } ( V )$ and that if $q \cong q ^ { \prime }$, then $O ( q )$ and $O \left( q ^ { \prime } \right)$ are two isomorphic groups.
Q17 Proof Proof of Set Membership, Containment, or Structural Property View
We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. If $V$ and $V ^ { \prime }$ are two $\mathbb { K }$-vector spaces of finite dimension, $q \in \mathcal { Q } ( V )$ and $q ^ { \prime } \in \mathcal { Q } \left( V ^ { \prime } \right)$ are two non-degenerate quadratic forms, the orthogonal sum $q \perp q ^ { \prime }$ of $q$ and $q ^ { \prime }$ is the quadratic form on $V \times V ^ { \prime }$ defined by $$q \perp q ^ { \prime } \left( x , x ^ { \prime } \right) = q ( x ) + q ^ { \prime } \left( x ^ { \prime } \right)$$ for all $x \in V$ and all $x ^ { \prime } \in V ^ { \prime }$.
Let $V , V ^ { \prime }$ and $V ^ { \prime \prime }$ be three $\mathbb { K }$-vector spaces of finite dimension and $\left( q , q ^ { \prime } , q ^ { \prime \prime } \right) \in \mathcal { Q } ( V ) \times \mathcal { Q } \left( V ^ { \prime } \right) \times \mathcal { Q } \left( V ^ { \prime \prime } \right)$.
(a) Show that $q \perp q ^ { \prime } \in \mathcal { Q } \left( V \times V ^ { \prime } \right)$ and then that $\left( q \perp q ^ { \prime } \right) \perp q ^ { \prime \prime } \cong q \perp \left( q ^ { \prime } \perp q ^ { \prime \prime } \right)$.
(b) Show that if $q ^ { \prime } \cong q ^ { \prime \prime }$ then $q \perp q ^ { \prime } \cong q \perp q ^ { \prime \prime }$.
(c) Prove that if $V = V ^ { \prime } \oplus V ^ { \prime \prime }$ and $\tilde { q } ( x , y ) = 0$ for all $x \in V ^ { \prime }$ and all $y \in V ^ { \prime \prime }$, then $q \cong q ^ { \prime } \perp q ^ { \prime \prime }$ where $q ^ { \prime }$ is the restriction of $q$ to $V ^ { \prime }$ and $q ^ { \prime \prime }$ is the restriction of $q$ to $V ^ { \prime \prime }$.
Q18 Groups Symplectic and Orthogonal Group Properties View
We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $V$ be a $\mathbb { K }$-vector space of finite dimension, $q \in \mathcal { Q } ( V )$ and $v , w \in V$ be two distinct vectors of $V$ such that $q ( v ) = q ( w ) \neq 0$.
We want to show in this question that there then exists an isometry $h \in O ( q )$ such that $h ( v ) = w$.
(a) Let $x \in V$ such that $q ( x ) \neq 0$. We denote by $s _ { x }$ the endomorphism of $V$ defined by $y \mapsto s _ { x } ( y ) = y - 2 \frac { \widetilde { q } ( x , y ) } { q ( x ) } x$. Show that $s _ { x }$ and $- s _ { x }$ belong to $O ( q )$.
(b) Suppose here that $q ( w - v ) \neq 0$. Show that the map $s _ { w - v }$ is an isometry such that $s _ { w - v } ( v ) = w$.
(c) Suppose here that $q ( w - v ) = 0$. Show that there exists an isometry $g \in O ( q )$ such that $g ( v ) = w$ and conclude.
Q19 Groups Group Homomorphisms and Isomorphisms View
We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $\left( V _ { i } \right) _ { 1 \leq i \leq 3 }$ be three $\mathbb { K }$-vector spaces of finite dimension and $q _ { i } \in \mathcal { Q } \left( V _ { i } \right)$ for $1 \leq i \leq 3$ satisfying $q _ { 1 } \perp q _ { 3 } \cong q _ { 2 } \perp q _ { 3 }$. Show that $q _ { 1 } \cong q _ { 2 }$.
Hint: one may reason by induction and use questions 17 and 18.
Q20 Groups Group Order and Structure Theorems View
We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $V$ be a $\mathbb { K }$-vector space of finite dimension and $q \in \mathcal { Q } ( V )$. Show that there exists a unique non-negative integer $m$ and an anisotropic quadratic form $q _ { \text {an} }$, unique up to isometry, such that $q \cong q _ { an } \perp m \cdot h$ where $m \cdot h = h \perp \cdots \perp h$ is the orthogonal sum of $m$ copies of $h$ and $h$ is the quadratic form defined by $h \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ (introduced in question 6b).
Hint: one may use question 6b and the previous question.