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
2022 centrale-maths1__mp

22 maths questions

Q1 Matrices Matrix Algebra and Product Properties View
Let $A$ and $B$ be two matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$ such that
$$\forall ( X , Y ) \in \left( \mathcal { M } _ { n , 1 } ( \mathbb { R } ) \right) ^ { 2 } , \quad X ^ { \top } A Y = X ^ { \top } B Y .$$
Show that $A = B$.
Q2 Matrices Eigenvalue constraints from matrix properties View
Let $M \in \mathrm { GL } _ { n } ( \mathbb { R } )$. Show that the eigenvalues of $M ^ { \top } M$ are all strictly positive.
Deduce that there exists a symmetric matrix $S$ with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$.
Q3 Proof Deduction or Consequence from Prior Results View
Show that, if $\omega$ is a symplectic form on $E$, then for every vector $x$ in $E$, $\omega ( x , x ) = 0$.
Q4 Proof Proof of Set Membership, Containment, or Structural Property View
Let $F$ be a vector subspace of a symplectic space $(E , \omega)$. The $\omega$-orthogonal of $F$ is defined as $$F ^ { \omega } = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}.$$ Justify that $F ^ { \omega }$ is a vector subspace of $E$.
Q5 Proof True/False Justification View
Let $F$ be a vector subspace of a symplectic space $(E , \omega)$, and let $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$. Is the subspace $F ^ { \omega }$ necessarily in direct sum with $F$?
Q6 Proof Proof That a Map Has a Specific Property View
For every $x \in E$, we denote by $\omega ( x , \cdot )$ the linear map from $E$ to $\mathbb { R }$, $y \mapsto \omega ( x , y )$ and we consider
$$d _ { \omega } : \left\lvert \, \begin{array} { c c c } E & \rightarrow & \mathcal { L } ( E , \mathbb { R } ) \\ x & \mapsto & \omega ( x , \cdot ) \end{array} \right.$$
Show that $d _ { \omega }$ is an isomorphism.
Q7 Proof Proof That a Map Has a Specific Property View
For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. Show that the restriction map $r _ { F } : \begin{array} { c c c } \mathcal { L } ( E , \mathbb { R } ) & \rightarrow & \mathcal { L } ( F , \mathbb { R } ) \\ \ell & \mapsto & \left. \ell \right| _ { F } \end{array}$ is surjective.
Q8 Proof Linear Transformation and Endomorphism Properties View
For $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. The restriction map is $r _ { F } : \mathcal { L } ( E , \mathbb { R } ) \rightarrow \mathcal { L } ( F , \mathbb { R } )$, $\ell \mapsto \left. \ell \right| _ { F }$, and $d_{\omega} : E \rightarrow \mathcal{L}(E,\mathbb{R})$, $x \mapsto \omega(x,\cdot)$. The $\omega$-orthogonal is $F ^ { \omega } = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$.
Specify the kernel of $r _ { F } \circ d _ { \omega }$. Deduce that $\operatorname { dim } F ^ { \omega } = \operatorname { dim } E - \operatorname { dim } F$.
Q19 Matrices Structured Matrix Characterization View
We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of the matrix $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that, for every matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } ) \left( = \mathcal { M } _ { 2 m } ( \mathbb { R } ) \right)$,
$$M \in \mathcal { C } _ { J } \quad \Longleftrightarrow \quad \exists ( U , V ) \in \mathcal { M } _ { m } ( \mathbb { R } ) \times \mathcal { M } _ { m } ( \mathbb { R } ) , \quad M = \left( \begin{array} { c c } U & - V \\ V & U \end{array} \right) .$$
Q20 Matrices Block Matrix Multiplication and Determinant Identity View
We denote by $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$ the centralizer of $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$, and every $M \in \mathcal{C}_J$ has the form $M = \left( \begin{array}{cc} U & -V \\ V & U \end{array} \right)$ for some $U, V \in \mathcal{M}_m(\mathbb{R})$. Deduce that, for every matrix $M \in \mathcal { C } _ { J } , \operatorname { det } ( M ) \geqslant 0$.
One may consider the product of block matrices $\left( \begin{array} { c c } I _ { m } & 0 \\ \mathrm { i } I _ { m } & I _ { m } \end{array} \right) \left( \begin{array} { c c } U & - V \\ V & U \end{array} \right) \left( \begin{array} { c c } I _ { m } & 0 \\ - \mathrm { i } I _ { m } & I _ { m } \end{array} \right)$.
Q21 Matrices Symplectic and Orthogonal Group Properties View
We denote by $\operatorname { OSp } _ { n } ( \mathbb { R } ) = \operatorname { Sp } _ { n } ( \mathbb { R } ) \cap \mathrm { O } _ { n } ( \mathbb { R } )$ the set of real symplectic and orthogonal matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$. We equip $\mathcal { M } _ { n } ( \mathbb { R } )$ with its topology as a normed vector space of finite dimension. Show that $\operatorname { OSp } _ { n } ( \mathbb { R } )$ is a compact subgroup of the symplectic group $\operatorname { Sp } _ { n } ( \mathbb { R } )$.
Q22 Matrices Bilinear and Symplectic Form Properties View
We denote by $\operatorname { OSp } _ { n } ( \mathbb { R } ) = \operatorname { Sp } _ { n } ( \mathbb { R } ) \cap \mathrm { O } _ { n } ( \mathbb { R } )$ and $\mathcal { C } _ { J } = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid J M = M J \right\}$. Show that $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$.
Q23 Matrices Determinant and Rank Computation View
We have $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$ and for every $M \in \mathcal{C}_J$, $\det(M) \geq 0$. Deduce that, for every matrix $M$ in $\operatorname { OSp } _ { n } ( \mathbb { R } ) , \operatorname { det } ( M ) = 1$.
Q24 Matrices Bilinear and Symplectic Form Properties View
Let $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$ and let $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ be a symmetric matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$. Show that $S$ is symplectic.
One may consider a basis of eigenvectors of the endomorphism $s$ of $\mathbb { R } ^ { n }$ canonically associated with $S$, and show that $s$ is a symplectic endomorphism of the standard space $(\mathbb { R } ^ { n } , b _ { s })$.
Q25 Matrices Matrix Group and Subgroup Structure View
Let $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$ and let $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ be a symmetric matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$ and $S \in \mathrm{Sp}_n(\mathbb{R})$. Justify that $S$ is invertible then show that the matrix $O$ defined by $O = M S ^ { - 1 }$ belongs to the group $\mathrm { OSp } _ { n } ( \mathbb { R } )$.
Q26 Matrices Determinant and Rank Computation View
Using the polar decomposition $M = OS$ where $O \in \operatorname{OSp}_n(\mathbb{R})$ and $S \in \mathrm{Sp}_n(\mathbb{R})$ is symmetric with strictly positive eigenvalues, conclude that the determinant of the matrix $M \in \mathrm{Sp}_n(\mathbb{R})$ is equal to 1.
Q27 Matrices Bilinear and Symplectic Form Properties View
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda \in \mathbb { R }$ be a real number. Show that the map $\tau _ { a } ^ { \lambda }$ defined by
$$\forall x \in E , \quad \tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$$
is a transvection of $E$ and that it is a symplectic endomorphism of this same space.
Q28 Matrices Symplectic and Orthogonal Group Properties View
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and let $\lambda$ and $\mu$ be real numbers. The symplectic transvections are defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Show that $\tau _ { a } ^ { \mu } \circ \tau _ { a } ^ { \lambda } = \tau _ { a } ^ { \lambda + \mu }$.
Q29 Matrices Symplectic and Orthogonal Group Properties View
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. The symplectic transvection is defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Show that $\operatorname { det } \left( \tau _ { a } ^ { \lambda } \right) > 0$.
Q30 Matrices Symplectic and Orthogonal Group Properties View
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $a \in E$ be a non-zero vector and $\lambda$ be a real number. The symplectic transvection is defined by $\tau _ { a } ^ { \lambda } ( x ) = x + \lambda \omega ( a , x ) a$. Is the inverse $\left( \tau _ { a } ^ { \lambda } \right) ^ { - 1 }$ still a symplectic transvection?
Q31 Matrices Symplectic and Orthogonal Group Properties View
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. We fix $x$ and $y$, non-zero, in $E$. Suppose that $\omega ( x , y ) \neq 0$. Show that there exists $\lambda \in \mathbb { R }$ such that $\tau _ { y - x } ^ { \lambda } ( x ) = y$, where $\tau_a^{\lambda}(x) = x + \lambda \omega(a,x)a$.
Q42 Matrices Symplectic and Orthogonal Group Properties View
Use the results of subsection III.D to prove the inclusion $\mathrm { Sp } _ { n } ( \mathbb { R } ) \subset \mathrm { SL } _ { n } ( \mathbb { R } )$.