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
2010 centrale-maths2__mp

10 maths questions

QII.B.4 Proof Simultaneous diagonalization or commutant structure View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $h^{-1} \circ h'$ has $n$ distinct eigenvalues. Show that there exists a basis of $E$ orthogonal for both $q$ and $q'$.
QII.C.1 Proof Existence Proof View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $x \in E$ such that $q(x) = 0$ and such that $x \neq 0$.
We propose to demonstrate that there exists a plane $\Pi \subset E$ containing $x$ such that $(\Pi, q_{/\Pi})$ is an artinian plane (where $q_{/\Pi}$ denotes the restriction of the application $q$ to the plane $\Pi$).
a) Demonstrate that there exists $z \in E$ such that $\varphi(x,z) = 1$.
b) We set $y = z - \frac{q(z)}{2}x$. Compute $q(y)$.
c) Conclude.
QII.C.2 Proof Proof by Induction or Recursive Construction View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $F$ be a singular vector subspace of $E$. We assume that $(e_1, \ldots, e_s)$ is a basis of $F \cap F^\perp$. We denote by $G$ a supplementary subspace of $F \cap F^\perp$ in $F$.
a) Show that $G$ is non-singular.
b) Demonstrate by induction on the dimension of $F \cap F^\perp$ (starting with $\operatorname{dim}(F \cap F^\perp) = 1$, then $\operatorname{dim}(F \cap F^\perp) > 1$) that there exist $s$ planes $P_1, \ldots, P_s$ of $E$ such that the following three properties are verified:
  1. For all $i \in \{1,\ldots,s\}$, $(P_i, q_{/P_i})$ is an artinian plane containing $e_i$.
  2. For all $(i,j) \in \{1,\ldots,s\}^2$ with $i \neq j$, $P_i$ is orthogonal to $P_j$.
  3. For all $i \in \{1,\ldots,s\}$, $P_i$ is orthogonal to $G$.
QII.C.3 Proof Proof of Set Membership, Containment, or Structural Property View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Show that $\bar{F} = G \oplus P_1 \oplus \ldots \oplus P_s$ is non-singular. We will say that $\bar{F}$ is a non-singular completion of $F$.
QII.C.4 Proof Direct Proof of an Inequality View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Show that if $q_{/F} = 0$, then $\operatorname{dim}(F) \leq \frac{n}{2}$.
QII.C.5 Proof Proof of Equivalence or Logical Relationship Between Conditions View
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
We assume that $n = 2p$. Show that $(E,q)$ is an Artin space if and only if there exists a vector subspace $F$ of $E$ of dimension $p$ such that $q_{/F} = 0$.
QIII.A.1 Proof Proof That a Map Has a Specific Property View
We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself, that is, the set of automorphisms $f$ of $E$ satisfying: for all $x \in E$, $q(f(x)) = q(x)$.
Let $f$ be an endomorphism of $E$.
a) Show that $f \in O(E,q)$ if and only if, for all $(x,y) \in E^2$: $\varphi(f(x),f(y)) = \varphi(x,y)$. Show that if $F$ is a vector subspace of $E$ and if $f \in O(E,q)$, then $f(F^\perp) = (f(F))^\perp$.
b) Let $e$ be a basis of $E$. Compute the matrix of the bilinear form: $$(x,y) \mapsto \varphi(f(x),f(y)) \text{ in terms of } \operatorname{mat}(f,e) \text{ and } \operatorname{mat}(\varphi,e).$$
c) Let us set $M = \operatorname{mat}(f,e)$ and $\Omega = \operatorname{mat}(\varphi,e)$. Show that $f \in O(E,q)$ if and only if $\Omega = {}^t M \Omega M$.
d) Show that if $f \in O(E,q)$, then $\operatorname{det}(f) \in \{1,-1\}$. We will denote: $$O^+(E,q) = \{f \in O(E,q) \mid \operatorname{det}(f) = 1\} \text{ and } O^-(E,q) = \{f \in O(E,q) \mid \operatorname{det}(f) = -1\}.$$
QIII.A.2 Proof Proof That a Map Has a Specific Property View
We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $F$ and $G$ be two vector subspaces of $E$ such that $E = F \oplus G$. We denote by $s$ the symmetry with respect to $F$ parallel to $G$.
a) Show that $s \in O(E,q)$ if and only if $F$ and $G$ are orthogonal (for $\varphi$).
b) Deduce that the symmetries in $O(E,q)$ are the symmetries with respect to $F$ parallel to $F^\perp$, where $F$ is a non-singular subspace of $E$.
c) When $H$ is a non-singular hyperplane, we will call reflection along $H$ the symmetry with respect to $H$ parallel to $H^\perp$. Show that every reflection of $E$ is an element of $O^-(E,q)$.
d) Let $(x,y) \in E^2$ such that $q(x) = q(y)$ and $q(x-y) \neq 0$. We denote by $s$ the reflection along $H = \{x-y\}^\perp$. Show that $s(x) = y$.
QIII.B.1 Proof Proof That a Map Has a Specific Property View
We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Suppose that $E$ is an artinian space of dimension $2p$ and that $F$ is a subspace of $E$ of dimension $p$ such that $q_{/F} = 0$.
If $f \in O(E,q)$ with $f(F) = F$, show that $f \in O^+(E,q)$.
QIII.B.2 Proof Proof That a Map Has a Specific Property View
We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $F$ be a subspace of $E$ such that $\bar{F} = E$ (where $\bar{F}$ is a non-singular completion of $F$). Show that if $f \in O(E,q)$ with $f_{/F} = \operatorname{Id}_F$ (where $\operatorname{Id}_F$ is the identity application from $F$ to $F$), then $f \in O^+(E,q)$.