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
2024 mines-ponts-maths2__pc

19 maths questions

Q1 Matrices Structured Matrix Characterization View
Give examples of Hadamard matrices of order 1 and 2.
Q2 Matrices Structured Matrix Characterization View
Show that if $H$ is a Hadamard matrix then any matrix obtained by multiplying a row or column of $H$ by $-1$ or by exchanging two rows or two columns of $H$ is still a Hadamard matrix.
Q3 Matrices Structured Matrix Characterization View
Show that if $H$ is a Hadamard matrix of order $n$ then there exists a Hadamard matrix of order $n$ whose coefficients of the first row are all equal to 1. Deduce that if $n \geq 2$ then $n$ is even.
Q4 Proof Structured Matrix Characterization View
Show that if $H$ is a Hadamard matrix of order $n$ greater than or equal to 4, then $n$ is a multiple of 4. One may begin by showing that we can assume the first row of $H$ is composed only of 1's and its second row is composed of $n/2$ coefficients equal to 1 then $n/2$ coefficients equal to $-1$.
Q5 Invariant lines and eigenvalues and vectors Diagonalizability and Similarity View
Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$. We denote by $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ the eigenvalues ordered in increasing order of $f$.
Justify the existence of an orthonormal basis $(e_1, \ldots, e_n)$ of $\mathbf{R}^n$ formed of eigenvectors of $f$, the vector $e_i$ being associated with $\lambda_i$ for all $i \in \{1, \ldots, n\}$. We keep this basis henceforth.
Q6 Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$ with eigenvalues $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ and associated orthonormal eigenbasis $(e_1, \ldots, e_n)$.
Let $k \in \llbracket 1, n \rrbracket$ and $S_k$ a vector subspace of $\mathbf{R}^n$ of dimension $k$. We set $T_k = \operatorname{Vect}(e_k, \ldots, e_n)$.
Justify that $S_k \cap T_k \neq \{0\}$.
Q7 Invariant lines and eigenvalues and vectors Eigenvalue interlacing and spectral inequalities View
Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$ with eigenvalues $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ and associated orthonormal eigenbasis $(e_1, \ldots, e_n)$. Let $k \in \llbracket 1, n \rrbracket$, $S_k$ a vector subspace of $\mathbf{R}^n$ of dimension $k$, and $T_k = \operatorname{Vect}(e_k, \ldots, e_n)$.
By considering $x \in S_k \cap T_k$, justify that: $$\max_{x \in S_k, \|x\|=1} (x, f(x)) \geq \lambda_k.$$
Q8 Invariant lines and eigenvalues and vectors Eigenvalue interlacing and spectral inequalities View
Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$ with eigenvalues $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ and associated orthonormal eigenbasis $(e_1, \ldots, e_n)$. For $k \in \llbracket 1, n \rrbracket$, let $\pi_k$ denote the set of vector subspaces of $\mathbf{R}^n$ of dimension $k$.
Let $k \in \llbracket 1, n \rrbracket$. Using $S = \operatorname{Vect}(e_1, \ldots, e_k) \in \pi_k$, show the equality: $$\lambda_k = \min_{S \in \pi_k} \left( \max_{x \in S, \|x\|=1} (x, f(x)) \right)$$ This is the Courant-Fischer theorem.
Q9 Invariant lines and eigenvalues and vectors Matrix Decomposition and Factorization View
Let $M$ be a symmetric matrix of $\mathcal{M}_n(\mathbf{R})$. Show that if $M$ is positive, then there exists $B \in \mathcal{M}_n(\mathbf{R})$ such that $M = B^T \cdot B$. Deduce that if $M$ is no longer assumed to be positive, but admits a unique strictly positive eigenvalue $\lambda$ with eigenspace of dimension 1 and unit eigenvector $u$, then there exists $B \in \mathcal{M}_n(\mathbf{R})$ such that $M = \lambda u \cdot u^T - B^T \cdot B$.
Q10 Invariant lines and eigenvalues and vectors Projection and Orthogonality View
We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, and $P$ the matrix of order $n$ defined by $$P = I_n - \frac{1}{n} \mathbf{e} \cdot \mathbf{e}^T.$$
Show that $P$ is symmetric and that the endomorphism of $\mathbf{R}^n$ canonically associated is an orthogonal projection onto $\operatorname{Vect}(\mathbf{e})^\perp$.
Q12 Invariant lines and eigenvalues and vectors Structured Matrix Characterization View
We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1. We denote by $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. We denote by $K$ the application from $\Omega_n$ to $\mathcal{M}_n(\mathbb{R})$ which associates to a matrix $A$ $$K(A) = \mathbf{e} \cdot \mathbf{a}^T + \mathbf{a} \cdot \mathbf{e}^T - 2A$$ where $\mathbf{a}$ is the column matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are the diagonal coefficients of $A$.
Show that for every matrix $A$ of $\Omega_n$ we have $K(A) \in \Delta_n$.
Q13 Invariant lines and eigenvalues and vectors Matrix Algebra and Product Properties View
We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, $\Delta_n$ the set of EDM of order $n$, and $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. The application $T: \Delta_n \to \mathcal{M}_n(\mathbb{R})$ associates to $D$ the matrix $T(D) = -\frac{1}{2}PDP$, and the application $K: \Omega_n \to \mathcal{M}_n(\mathbb{R})$ associates to $A$ the matrix $K(A) = \mathbf{e}\cdot\mathbf{a}^T + \mathbf{a}\cdot\mathbf{e}^T - 2A$ where $\mathbf{a}$ is the column of diagonal coefficients of $A$.
Show that the applications $T: \Delta_n \rightarrow \Omega_n$ and $K: \Omega_n \rightarrow \Delta_n$ satisfy: $$T \circ K = \operatorname{Id}_{\Omega_n}.$$
Q14 Invariant lines and eigenvalues and vectors Proof of Inequalities Involving Series or Sequence Terms View
We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Show that a symmetric matrix $D$ of order $n$ with non-negative coefficients and zero diagonal is EDM if and only if $-\frac{1}{2}PDP$ is positive.
Q15 Invariant lines and eigenvalues and vectors Proof of Inequalities Involving Series or Sequence Terms View
We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Show that every non-zero symmetric matrix with non-negative coefficients and zero diagonal, having a unique strictly positive eigenvalue with eigenspace of dimension 1 and eigenvector $\mathbf{e}$, is EDM.
Q16 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Specify the sum $\displaystyle\sum_{i=1}^{n} \lambda_i$ of the eigenvalues of an EDM of order $n$.
Q17 Invariant lines and eigenvalues and vectors Bilinear and Symplectic Form Properties View
We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Let $D$ be a non-zero EDM of order $n$. Show that for all $x \in \operatorname{Vect}(\mathbf{e})^\perp$, we have $$x^T D x \leqslant 0.$$
Q18 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View
We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Let $D$ be a non-zero EDM of order $n$. Let $\lambda_1, \ldots, \lambda_n$ be its eigenvalues, ordered in increasing order. Show $$\lambda_{n-1} \leqslant 0$$ and deduce that $D$ has exactly one strictly positive eigenvalue.
Q19 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that $$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = 0.$$ We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$.
Show that $D$ is symmetric, with non-negative coefficients and zero diagonal, and has eigenvalues $\lambda_1, \ldots, \lambda_n$, with $\lambda_1$ having eigenspace of dimension 1.
Q20 Invariant lines and eigenvalues and vectors Structured Matrix Characterization View
Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that $$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = 0.$$ We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$.
Show that $D$ is EDM.