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 mines-ponts-maths2__pc

9 maths questions

Q1 Matrices Matrix Norm, Convergence, and Inequality View
Justify that if $M \in \mathcal{M}_{n}(\mathbf{C})$, the map $$X \in \Sigma_{n} \longmapsto \|MX\|$$ attains its maximum, which we denote by $\|M\|_{\text{op}}$. Establish the two properties $$\begin{gathered} \forall M \in \mathcal{M}_{n}(\mathbf{C}), \quad \|M\|_{\mathrm{op}} = \max\left\{\frac{\|MX\|}{\|X\|}; X \in \mathcal{M}_{n,1}(\mathbf{C}) \backslash \{0\}\right\}, \\ \forall (M, M^{\prime}) \in \mathcal{M}_{n}(\mathbf{C})^{2}, \quad \|M^{\prime}M\|_{\mathrm{op}} \leq \|M^{\prime}\|_{\mathrm{op}} \|M\|_{\mathrm{op}}. \end{gathered}$$
Q2 Matrices Matrix Norm, Convergence, and Inequality View
If $U \in \mathcal{M}_{n,1}(\mathbf{C})$, show that $$\max\left\{\left|V^{T}U\right|; V \in \Sigma_{n}\right\} = \|U\|.$$ Deduce that, if $M$ is in $\mathcal{M}_{n}(\mathbf{C})$, then $$\max\left\{\left|X^{T}MY\right|; (X,Y) \in \Sigma_{n} \times \Sigma_{n}\right\} = \|M\|_{\mathrm{op}}.$$
Q3 Matrices Matrix Norm, Convergence, and Inequality View
Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $$b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}.$$
Let $M \in \mathcal{B}_{n}$, $X \in \mathcal{M}_{n,1}(\mathbf{C})$. Show that the sequence $\left(\left\|M^{k}X\right\|\right)_{k \in \mathbf{N}}$ is bounded. If $\lambda \in \sigma(M)$, if $X$ is an eigenvector of $M$ associated with $\lambda$, express for $k \in \mathbf{N}$, the vector $M^{k}X$ in terms of $\lambda$, $k$ and $X$. Deduce that $\sigma(M) \subset \mathbb{D}$.
Q4 Matrices Matrix Norm, Convergence, and Inequality View
Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded.
Assume that $n \geq 2$. Indicate, with justification, a matrix $M$ in $\mathcal{M}_{n}(\mathbf{C})$, upper triangular, such that $\sigma(M) \subset \mathbb{D}$, but not belonging to $\mathcal{B}_{n}$.
Q5 Matrices Diagonalizability and Similarity View
We say that the element $M$ of $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$ if, for every $(i,j)$ in $\{1,\ldots,n\}^{2}$, there exists an element $P_{M,i,j}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad \left(R_{z}(M)\right)_{i,j} = \frac{P_{M,i,j}(z)}{\chi_{M}(z)}$$
Show that the diagonalizable matrices of $\mathcal{M}_{n}(\mathbf{C})$ satisfy $\mathcal{P}$. Begin with the case of diagonal matrices.
Q6 Matrices Deduction or Consequence from Prior Results View
We say that the element $M$ of $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$ if, for every $(i,j)$ in $\{1,\ldots,n\}^{2}$, there exists an element $P_{M,i,j}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad \left(R_{z}(M)\right)_{i,j} = \frac{P_{M,i,j}(z)}{\chi_{M}(z)}$$
We admit that every matrix in $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$. Deduce that, if $M \in \mathcal{M}_{n}(\mathbf{C})$ and $(X,Y) \in \mathcal{M}_{n,1}(\mathbf{C})^{2}$, there exists an element $P_{M,X,Y}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad X^{T}R_{z}(M)Y = \frac{P_{M,X,Y}(z)}{\chi_{M}(z)}.$$
Q7 Matrices Matrix Exponentials and Series of Matrices View
Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$.
Let $M \in \mathcal{B}_{n}$ and $z \in \mathbf{C} \backslash \mathbb{D}$. Show that the series of matrices $\sum \frac{M^{j}}{z^{j+1}}$ converges. We will admit the following fact: let $(E, N)$ be a finite-dimensional normed vector space; if $(v_{j})_{j \in \mathbf{N}}$ is a sequence of elements of $E$ such that the series $\sum N(v_{j})$ converges, then the series $\sum v_{j}$ converges in $E$. If $m \in \mathbf{N}$, give a simplified expression for $\left(zI_{n} - M\right)\sum_{j=0}^{m} \frac{M^{j}}{z^{j+1}}$. Deduce that $$R_{z}(M) = \sum_{j=0}^{+\infty} \frac{M^{j}}{z^{j+1}}$$
Q8 Matrices Matrix Norm, Convergence, and Inequality View
Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$. For $M \in \mathcal{B}_{n}$, we define the function $$\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}.$$
Deduce from the previous question the inequality $$\forall M \in \mathcal{B}_{n}, \quad \forall z \in \mathbf{C} \backslash \mathbb{D}, \quad \varphi_{M}(z) \leq b(M)$$
Q10 Matrices Evaluation of a Finite or Infinite Sum View
Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$.
Let $M \in \mathcal{B}_{n}$, $r \in ]1, +\infty[$ and $(X,Y) \in \mathcal{M}_{n,1}(\mathbf{C})^{2}$. Determine a sequence of complex numbers $(c_{j})_{j \in \mathbf{N}}$ such that the series $\sum c_{j}$ converges absolutely and that $$\forall t \in \mathbf{R}, \quad X^{T}R_{re^{it}}(M)Y = \sum_{j=0}^{+\infty} c_{j} e^{-i(j+1)t}.$$ If $k \in \mathbf{N}$, deduce, using question 9, an integral expression for $X^{T}M^{k}Y$.