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
2015 x-ens-maths__pc

30 maths questions

Q1a Matrices Structured Matrix Characterization View
Recall why $\mathcal{S}_{n}(\mathbb{R})$ is a real vector space and what is its dimension. Why is the map $s^{\downarrow}$ well-defined on $\mathcal{S}_{n}(\mathbb{R})$?
Q1b Matrices Linear Transformation and Endomorphism Properties View
Is the map $s^{\downarrow}$ linear? Justify your answer.
Q1c Matrices Eigenvalue and Characteristic Polynomial Analysis View
If $M \in \mathcal{S}_{n}(\mathbb{R})$, express $s^{\downarrow}(-M)$ as a function of the coordinates $\left(m_{1}, \ldots, m_{n}\right)$ of $s^{\downarrow}(M)$.
Q1d Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $M = \left(\begin{array}{cc}\lambda & h \\ h & \mu\end{array}\right)$ be a matrix of $\mathcal{S}_{2}(\mathbb{R})$. Calculate $s^{\downarrow}(M)$.
Q2a Matrices Matrix Decomposition and Factorization View
Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum. Show that there exists an orthonormal basis $\left(v_{1}, \ldots, v_{n}\right)$ of $\mathbb{R}^{n}$ such that $$M = \sum_{i=1}^{n} m_{i} v_{i}\, {}^{t}v_{i}$$ Such a decomposition of $M$ will be called in the sequel a spectral resolution of $M$.
Q2b Matrices Matrix Norm, Convergence, and Inequality View
Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum, and $\left(v_{1}, \ldots, v_{n}\right)$ an orthonormal basis from the spectral resolution of $M$. Calculate $$\sup_{\|x\|=1} \langle x, Mx \rangle$$ as a function of the coordinates of $m$. Is this supremum attained? (One may decompose $x$ and $Mx$ on the orthonormal basis $\left(v_{1}, \ldots, v_{n}\right)$ of question 2a).
Q2c Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum, and $\left(v_{1}, \ldots, v_{n}\right)$ an orthonormal basis from the spectral resolution of $M$. Let $j$ be an integer, $1 \leqslant j \leqslant n$. We denote by $\mathcal{V}_{j}$ the vector subspace of $\mathbb{R}^{n}$ spanned by $\left(v_{1}, \ldots, v_{j}\right)$, and by $\mathcal{W}_{j}$ the one spanned by $\left(v_{j}, v_{j+1}, \ldots, v_{n}\right)$. Show the equalities $$\inf_{x \in \mathcal{V}_{j},\, \|x\|=1} \langle x, Mx \rangle = \sup_{x \in \mathcal{W}_{j},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$
Q3a Proof Proof of Set Membership, Containment, or Structural Property View
Let $\mathcal{U}$ and $\mathcal{V}$ be two vector subspaces of $\mathbb{R}^{n}$ such that $$\operatorname{dim} \mathcal{U} + \operatorname{dim} \mathcal{V} > n.$$ Show that $\mathcal{U} \cap \mathcal{V}$ is not reduced to $\{0\}$.
Q3b Proof Direct Proof of an Inequality View
Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$. Let $j$ be an integer, $1 \leqslant j \leqslant n$, and $\mathcal{V}$ be a vector subspace of $\mathbb{R}^{n}$ of dimension $j$. Show that $$\inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle \leqslant m_{j}.$$ (One may use questions $\mathbf{2c}$ and $\mathbf{3a}$, by choosing $\mathcal{U} = \mathcal{W}_{j}$.)
Q3c Proof Deduction or Consequence from Prior Results View
By using the notations of question 3b, deduce that $$\sup_{\mathcal{V} \subset \mathbb{R}^{n},\, \operatorname{dim} \mathcal{V} = j} \inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$ Is this supremum attained?
Q4a Proof Deduction or Consequence from Prior Results View
Let $\ell$ and $m$ be two $n$-tuples of real numbers. We write $$\ell \preccurlyeq m \quad \text{if and only if, for every integer } j,\, 1 \leqslant j \leqslant n, \quad \ell_{j} \leqslant m_{j}.$$ Let $L, M \in \mathcal{S}_{n}(\mathbb{R})$ such that $(0, \ldots, 0) \preccurlyeq s^{\downarrow}(M - L)$. Show that $s^{\downarrow}(L) \preccurlyeq s^{\downarrow}(M)$.
Q4b Proof Direct Proof of an Inequality View
Let $\ell$ and $m$ be two $n$-tuples of real numbers. We write $$\ell \preccurlyeq m \quad \text{if and only if, for every integer } j,\, 1 \leqslant j \leqslant n, \quad \ell_{j} \leqslant m_{j}.$$ Show that for every matrix $M \in \mathcal{S}_{n}(\mathbb{R})$, $(0, \ldots, 0) \preccurlyeq s^{\downarrow}\left(\|M\| I_{n} - M\right)$.
Q4c Proof Direct Proof of an Inequality View
Let $\ell$ and $m$ be two $n$-tuples of real numbers. We write $$\ell \preccurlyeq m \quad \text{if and only if, for every integer } j,\, 1 \leqslant j \leqslant n, \quad \ell_{j} \leqslant m_{j}.$$ Let $L, M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ and $\ell = s^{\downarrow}(L)$. Show that $$\max_{1 \leqslant j \leqslant n} \left|\ell_{j} - m_{j}\right| \leqslant \|L - M\|.$$
Q4d Proof Deduction or Consequence from Prior Results View
Conclude that the function $s^{\downarrow} : \mathcal{S}_{n}(\mathbb{R}) \rightarrow \mathbb{R}^{n}$ is continuous.
Q5a Proof Existence Proof View
We denote by $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ the set of $n \times n$ symmetric matrices whose eigenvalues are all simple. Let $M \in \mathcal{S}_{n}^{\dagger}(\mathbb{R})$. Determine a real $r > 0$ such that the open ball of $\mathcal{S}_{n}(\mathbb{R})$ centered at $M$ with radius $r$ is included in $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$. Deduce that $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ is an open set of $\mathcal{S}_{n}(\mathbb{R})$.
Q5b Proof Proof That a Map Has a Specific Property View
We denote by $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ the set of $n \times n$ symmetric matrices whose eigenvalues are all simple. Show that the first component $s_{1}^{\downarrow}$ of $s^{\downarrow}$ is of class $\mathscr{C}^{1}$ on $\mathcal{S}_{2}^{\dagger}(\mathbb{R})$, but not on $\mathcal{S}_{2}(\mathbb{R})$. (One may use question 1d.)
Q6a Proof Direct Proof of a Stated Identity or Equality View
In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
Show that $$\sum_{i=1}^{n} c_{i} = \sum_{i=1}^{n} a_{i} + \sum_{i=1}^{n} b_{i}.$$
Q6b Matrices Matrix Norm, Convergence, and Inequality View
In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
Show that $a_{1} + b_{1} \geqslant c_{1}$.
Q6c Matrices Matrix Norm, Convergence, and Inequality View
In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
Show that $a_{n} + b_{n} \leqslant c_{n}$.
Q7a Matrices Determinant and Rank Computation View
Let $\mathcal{U}, \mathcal{V}$ and $\mathcal{W}$ be three vector subspaces of $\mathbb{R}^{n}$ such that $$\operatorname{dim} \mathcal{U} + \operatorname{dim} \mathcal{V} + \operatorname{dim} \mathcal{W} > 2n.$$ Show that $\mathcal{U} \cap \mathcal{V} \cap \mathcal{W}$ is not reduced to $\{0\}$.
Q7b Matrices Matrix Norm, Convergence, and Inequality View
In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
By using spectral resolutions of $A$, $B$ and $C$, show that if the strictly positive integers $j$ and $k$ satisfy $j + k \leqslant n + 1$, we have $$c_{j+k-1} \leqslant a_{j} + b_{k}.$$ Deduce that for every integer $j$, $1 \leqslant j \leqslant n$, $$a_{j} + b_{n} \leqslant c_{j}.$$
Q8a Matrices Matrix Norm, Convergence, and Inequality View
We denote by $a_{ii}$ for $1 \leqslant i \leqslant n$ the diagonal elements of $A \in \mathcal{S}_{n}(\mathbb{R})$ with ordered spectrum $a = s^{\downarrow}(A)$. Prove that $a_{11} \leqslant a_{1}$.
Q8b Matrices Matrix Norm, Convergence, and Inequality View
Let $j$ and $k$ be non-negative integers such that $1 \leqslant j < k$ and $s_{1} \geqslant s_{2} \geqslant \cdots \geqslant s_{k}$ be real numbers. We define $\mathcal{D}_{j,k} = \left\{\left(t_{1}, \ldots, t_{k}\right) \in [0,1]^{k} \mid t_{1} + \cdots + t_{k} = j\right\}$ and $f$ the function from $\mathcal{D}_{j,k}$ to $\mathbb{R}$ defined by $$f\left(t_{1}, \ldots, t_{k}\right) = \sum_{i=1}^{k} s_{i} t_{i}.$$ Prove that for every $\left(t_{1}, \ldots, t_{k}\right) \in \mathcal{D}_{j,k}$, $$\sum_{i=1}^{j} s_{i} - f\left(t_{1}, \ldots, t_{k}\right) \geqslant \sum_{i=1}^{j} \left(s_{i} - s_{j}\right)\left(1 - t_{i}\right).$$ Deduce that $$\sup_{\mathcal{D}_{j,k}} f = \sum_{i=1}^{j} s_{i}.$$
Q8c Matrices Matrix Norm, Convergence, and Inequality View
We denote by $a_{ii}$ for $1 \leqslant i \leqslant n$ the diagonal elements of $A \in \mathcal{S}_{n}(\mathbb{R})$ with ordered spectrum $a = s^{\downarrow}(A)$. Show that, more generally than in 8a, we have for every integer $1 \leqslant j \leqslant n$ $$\sum_{i=1}^{j} a_{ii} \leqslant \sum_{i=1}^{j} a_{i}.$$
Q8d Matrices Matrix Norm, Convergence, and Inequality View
We denote by $a_{ii}$ for $1 \leqslant i \leqslant n$ the diagonal elements of $A \in \mathcal{S}_{n}(\mathbb{R})$ with ordered spectrum $a = s^{\downarrow}(A)$. Deduce that for every integer $1 \leqslant j \leqslant n$ $$\sum_{i=1}^{j} a_{i} = \sup_{\left(x_{1}, \ldots, x_{j}\right) \in \mathcal{R}_{j}} \sum_{i=1}^{j} \langle x_{i}, A x_{i} \rangle,$$ where $\mathcal{R}_{j}$ is the set of orthonormal families of cardinality $j$ in $\mathbb{R}^{n}$.
Q8e Matrices Matrix Norm, Convergence, and Inequality View
In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$. We denote by $\mathcal{R}_{j}$ the set of orthonormal families of cardinality $j$ in $\mathbb{R}^{n}$.
Conclude that for every integer $1 \leqslant j \leqslant n$ $$\sum_{i=1}^{j} c_{i} \leqslant \sum_{i=1}^{j} a_{i} + \sum_{i=1}^{j} b_{i}.$$
Q9 Matrices Matrix Decomposition and Factorization View
In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $$a_{1} - a_{2} \geqslant b_{1} - b_{2}.$$ We seek to identify the set $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$
Show that $\Sigma$ is included in a line segment $L$ of length $\sqrt{2}\left(b_{1} - b_{2}\right)$, and determine its endpoints. One may first study the case where $A$ and $B$ are diagonal.
Q10a Matrices Matrix Decomposition and Factorization View
In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $a_{1} - a_{2} \geqslant b_{1} - b_{2}$, and $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$
Show that $$\Sigma = \left\{s^{\downarrow}(A + B) \,\middle|\, A = \left(\begin{array}{cc} a_{1} & 0 \\ 0 & a_{2} \end{array}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$
Q10b Matrices Matrix Decomposition and Factorization View
In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $b_{1} \geqslant b_{2}$.
Determine a continuous function defined on $[-\pi, \pi]$ whose image equals $S\left(b_{1}, b_{2}\right)$.
Q10c Matrices Matrix Decomposition and Factorization View
In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $a_{1} - a_{2} \geqslant b_{1} - b_{2}$, and $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$ Let $L$ be the line segment identified in question 9.
Show that $\Sigma = L$.