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
2022 x-ens-maths-a__mp

13 maths questions

Q1 Matrices Projection and Orthogonality View
Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$ (we recall that $p \geqslant 1$).
(a) Show that there exist $u_1 \in V$ and $u_1^{\prime} \in V^{\prime}$ of norm 1 such that $$\left\langle u_1, u_1^{\prime}\right\rangle = \sup\left\{\left\langle a, a^{\prime}\right\rangle \mid\left(a, a^{\prime}\right) \in V \times V^{\prime},\|a\|=\left\|a^{\prime}\right\|=1\right\}.$$
(b) Extend this result by showing that there exist a family $u = (u_1, \ldots, u_p)$ of vectors of $V$ and a family $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ of vectors of $V^{\prime}$ such that $u$ and $u^{\prime}$ are orthonormal and satisfy the following two conditions:
(i) For $k = 1$, we have $$\left\langle u_1, u_1^{\prime}\right\rangle = \sup\left\{\left\langle a, a^{\prime}\right\rangle \mid\left(a, a^{\prime}\right) \in V \times V^{\prime},\|a\|=\left\|a^{\prime}\right\|=1\right\}.$$
(ii) For $k \in \llbracket 2, p \rrbracket$, we have $$\left\langle u_k, u_k^{\prime}\right\rangle = \sup\left\{\left\langle a, a^{\prime}\right\rangle \mid\left(a, a^{\prime}\right) \in V \times V^{\prime},\|a\|=\left\|a^{\prime}\right\|=1,\right.$$ $$\left.\left\langle a, u_l\right\rangle=\left\langle a^{\prime}, u_l^{\prime}\right\rangle=0 \text{ for all } l \in \llbracket 1, k-1 \rrbracket\right\}.$$
(Hint: One may construct the vectors $u_k$ and $u_k^{\prime}$ by induction on the integer $k$.)
Q3 Matrices Projection and Orthogonality View
Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$. Let $u = (u_1, \ldots, u_p)$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ be the orthonormal families constructed in question (1).
(a) Show that $u$ is an orthonormal basis of $V$.
(b) Show that for $k \in \llbracket 1, p-1 \rrbracket$, we have $u_k^{\prime} \in \operatorname{Vect}(u_{k+1}, \ldots, u_p)^{\perp}$. (Hint: One may consider the map $t \mapsto u_k(t) = \frac{u_k + t u_l}{\|u_k + t u_l\|}$ for all $t \in \mathbb{R}$ and $l \in \llbracket k+1, p \rrbracket$ as well as its derivative.)
(c) Show that $u_{k+1} \in \left(\operatorname{Vect}(u_1, \ldots, u_k) + \operatorname{Vect}(u_1^{\prime}, \ldots, u_k^{\prime})\right)^{\perp}$ for all $k$ element of $\llbracket 1, p-1 \rrbracket$.
(d) Deduce that the subspaces $W_k = \operatorname{Vect}(u_k, u_k^{\prime})$ for $k \in \llbracket 1, p \rrbracket$ are pairwise orthogonal.
Q4 Proof Direct Proof of a Stated Identity or Equality View
Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$. Let $u = (u_1, \ldots, u_p)$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ be the orthonormal families constructed in question (1).
(a) Show that there exist $0 \leqslant \theta_1 \leqslant \cdots \leqslant \theta_p \leqslant \pi/2$ such that $\cos(\theta_k) = \langle u_k, u_k^{\prime}\rangle$ for all $k \in \llbracket 1, p \rrbracket$.
(b) Calculate the value of $\operatorname{det}(\operatorname{Gram}(u, u^{\prime}))$ as a function of the $\cos(\theta_k)$.
(c) Deduce that $\operatorname{det}(\operatorname{Gram}(u, u^{\prime})) \leqslant 1$. What can be said about $V$ and $V^{\prime}$ in the case of equality?
Q6 Proof Proof That a Map Has a Specific Property View
For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \operatorname{det}(\operatorname{Gram}(e, u)).$$
(a) Show that for all $e \in E^p$, we have $\Omega_p(e) \in \mathscr{A}_p(E, \mathbb{R})$.
(b) Verify that for all $(e, u) \in E^p \times E^p$, we have $\Omega_p(e)(u) = \Omega_p(u)(e)$.
(c) Show that $\Omega_p \in \mathscr{A}_p(E, \mathscr{A}_p(E, \mathbb{R}))$.
Q7 Proof Proof That a Map Has a Specific Property View
For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \operatorname{det}(\operatorname{Gram}(e, u)).$$
(a) Let $M \in \mathcal{M}_p(\mathbb{R})$, $e = (e_1, \ldots, e_p)$ and $e^{\prime} = (e_1^{\prime}, \ldots, e_p^{\prime})$ in $E^p$ satisfying $e_i^{\prime} = \sum_{j=1}^p M_{ij} e_j$ for all $i \in \llbracket 1, p \rrbracket$. Show that $\Omega_p(e^{\prime}) = \operatorname{det}(M) \Omega_p(e)$.
(b) Let $e \in E^p$. Show that $\Omega_p(e) \neq 0$ if and only if $e$ is a free family.
(c) Verify that $\Omega_p(e)(e) \geqslant 0$ for all families $e \in E^p$.
Q8 Matrices Determinant and Rank Computation View
For all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = \left(\operatorname{det}(\operatorname{Gram}(e, e))\right)^{1/2}.$$
(a) Calculate $\operatorname{vol}_p(b)$ when $b = (b_1, \ldots, b_p)$ is an orthonormal family of vectors of $E$.
(b) Suppose here that $p \geqslant 2$. Let $e = (e_1, \ldots, e_p) \in E^p$. We denote by $\operatorname{pr}$ the orthogonal projection onto the orthogonal of the space spanned by the family $e_2^p = (e_2, \ldots, e_p)$. Show that $\operatorname{vol}_p(e) = \|\operatorname{pr}(e_1)\| \operatorname{vol}_{p-1}(e_2^p)$.
(c) For all free families $e = (e_1, \ldots, e_p) \in E^p$, show that $\operatorname{vol}_p(e) \leqslant \prod_{i=1}^p \|e_i\|$ with equality if and only if $e$ is a family of pairwise orthogonal vectors.
Q9 Matrices Determinant and Rank Computation View
For all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = \left(\operatorname{det}(\operatorname{Gram}(e, e))\right)^{1/2}.$$
(a) Show that if $e \in E^p$ is a free family and if $b \in E^p$ is an orthonormal basis of $\operatorname{Vect}(e)$, then $\operatorname{vol}_p(e) = \left|\operatorname{det}\left(P_b^e\right)\right|$ where $P_b^e$ is the change of basis matrix from $b$ to $e$, i.e. $e_j = \sum_{i=1}^p \left(P_b^e\right)_{ij} b_i$ for all $j \in \llbracket 1, p \rrbracket$.
(b) Show that for all $e, e^{\prime} \in E^p$, we have $\left|\Omega_p(e)(e^{\prime})\right| \leqslant \operatorname{vol}_p(e) \operatorname{vol}_p(e^{\prime})$.
Q10 Groups Decomposition and Basis Construction View
Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
(a) Show that for all $\omega \in \mathscr{A}_p(E, \mathbb{R})$, we have $\omega = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \Omega_p(e_{\alpha})$.
(b) Deduce that $(\omega, \omega^{\prime}) \mapsto \langle\omega, \omega^{\prime}\rangle$ is an inner product on $\mathscr{A}_p(E, \mathbb{R})$ for which $(\Omega_p(e_{\alpha}))_{\alpha \in \mathcal{I}_p}$ is an orthonormal basis of $\mathscr{A}_p(E, \mathbb{R})$ and give the dimension of $\mathscr{A}_p(E, \mathbb{R})$.
(c) Construct in the case $p = d-1$ an isometry between $\mathscr{A}_p(E, \mathbb{R})$ and $E$.
Q11 Vector Product and Surfaces View
Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
We consider $u, v \in E^p$. Show that $$\Omega_p(u)(v) = \langle\Omega_p(u), \Omega_p(v)\rangle.$$
Q12 Matrices Projection and Orthogonality View
Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
Show that the inner product $(\omega, \omega^{\prime}) \mapsto \langle\omega, \omega^{\prime}\rangle$ defined above depends only on the inner product on $E$ and not on the choice of the orthonormal basis $e$.
Q13 Vector Product and Surfaces View
Let $p \in \llbracket 1, d \rrbracket$. We define a relation on the set of bases of a subspace $V$ of dimension $p$ of $E$ by: $e$ and $e^{\prime}$ are in relation if $\operatorname{det}_e(e^{\prime}) > 0$ where $\operatorname{det}_e(e^{\prime})$ is the determinant of $e^{\prime}$ in the basis $e$. We admit that this relation is an equivalence relation on the set of bases of $V$ for which there exist exactly two equivalence classes called orientations of $V$. An oriented subspace is a pair $(V, C)$ where $C$ is an orientation of $V$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$.
(a) Show that if $e$ and $e^{\prime}$ are two free families of cardinal $p$ of $E$ then $\Omega_p(e)$ and $\Omega_p(e^{\prime})$ are collinear if and only if $\operatorname{Vect}(e) = \operatorname{Vect}(e^{\prime})$.
(b) Show that for all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, there exists a unique $\Psi(V, C) \in \mathscr{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.
Q14 Vector Product and Surfaces View
Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathscr{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$. We equip $\mathscr{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
(a) Verify that $\Psi : (V, C) \mapsto \Psi(V, C)$ is an injection from $\widetilde{\operatorname{Gr}}(p, E)$ into the unit sphere of $\mathscr{A}_p(E, \mathbb{R})$.
(b) Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a compact subset of $\mathscr{A}_p(E, \mathbb{R})$.
Q15 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathscr{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$. We equip $\mathscr{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a path-connected subset of $\mathscr{A}_p(E, \mathbb{R})$ if and only if $p \leqslant d-1$. (Hint: One may use question 3d.)