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
2013 centrale-maths2__mp

31 maths questions

QI.A.1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $u$ be an endomorphism of $\mathbb{R}^n$. Show that $u$ is self-adjoint positive definite if and only if its matrix in any orthonormal basis belongs to $\mathcal{S}_n^{++}(\mathbb{R})$.
QI.A.2 Matrices Linear System and Inverse Existence View
Show that if $S \in \mathcal{S}_n^{++}(\mathbb{R})$, then $S$ is invertible and $S^{-1} \in \mathcal{S}_n^{++}(\mathbb{R})$.
QI.B.1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
In this question, $u$ denotes an endomorphism of $\mathbb{R}^n$ that is self-adjoint positive definite. We propose to prove that there exists a unique endomorphism $v$ of $\mathbb{R}^n$ that is self-adjoint, positive definite, such that $v^2 = u$.
Let $v$ be an endomorphism of $\mathbb{R}^n$, self-adjoint positive definite and satisfying $v^2 = u$, and let $\lambda$ be an eigenvalue of $u$. Show that $v$ induces an endomorphism of $\operatorname{Ker}(u - \lambda \mathrm{Id})$ which we shall determine.
QI.B.2 Matrices Diagonalizability and Similarity View
In this question, $u$ denotes an endomorphism of $\mathbb{R}^n$ that is self-adjoint positive definite. We propose to prove that there exists a unique endomorphism $v$ of $\mathbb{R}^n$ that is self-adjoint, positive definite, such that $v^2 = u$.
Deduce $v$, then conclude.
QI.B.3 Matrices Eigenvalue and Characteristic Polynomial Analysis View
In this question, $u$ denotes an endomorphism of $\mathbb{R}^n$ that is self-adjoint positive definite. We propose to prove that there exists a unique endomorphism $v$ of $\mathbb{R}^n$ that is self-adjoint, positive definite, such that $v^2 = u$.
Show that there exists a polynomial $Q$ with real coefficients such that $v = Q(u)$.
QI.C.1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in \mathrm{GL}_n(\mathbb{R})$. Show that ${}^t A A \in \mathcal{S}_n^{++}(\mathbb{R})$.
QI.C.2 Matrices Matrix Decomposition and Factorization View
Let $A \in \mathrm{GL}_n(\mathbb{R})$. Deduce that there exists a unique pair $(O, S) \in \mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$ such that $A = OS$.
QI.C.3 Matrices Matrix Decomposition and Factorization View
Let $A \in \mathrm{GL}_n(\mathbb{R})$. Determine the matrices $O$ and $S$ when $A = \left(\begin{array}{ccc} 3 & 0 & -1 \\ \sqrt{2}/2 & 3\sqrt{2} & -3\sqrt{2}/2 \\ -\sqrt{2}/2 & 3\sqrt{2} & 3\sqrt{2}/2 \end{array}\right)$.
QI.D.1 Matrices Matrix Group and Subgroup Structure View
Show that $\mathrm{O}(n)$ is a compact subset of $\mathcal{M}_n(\mathbb{R})$.
QI.D.2 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Show that $\mathcal{S}_n^+(\mathbb{R})$ is a closed subset of $\mathcal{M}_n(\mathbb{R})$.
QI.D.3 Matrices Determinant and Rank Computation View
Show that $\mathrm{GL}_n(\mathbb{R})$ is a dense subset of $\mathcal{M}_n(\mathbb{R})$.
QI.D.4 Matrices Matrix Decomposition and Factorization View
Let $A \in \mathcal{M}_n(\mathbb{R})$. Show that there exists a pair $(O, S) \in \mathrm{O}(n) \times \mathcal{S}_n^+(\mathbb{R})$ such that $A = OS$. Is such a pair unique?
QI.E Matrices Matrix Group and Subgroup Structure View
Let $\varphi$ be the map from $\mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$ to $\mathrm{GL}_n(\mathbb{R})$ defined by $\varphi(O, S) = OS$ for every pair $(O, S)$ of $\mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$.
Show that $\varphi$ is bijective, continuous, and that its inverse is continuous.
QII.A.1 Matrices Matrix Algebra and Product Properties View
In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a square matrix $U$ of size $n$, invertible, with complex coefficients, such that $U {}^t\bar{U} = I_n$ and $A = UBU^{-1}$, where $\bar{U}$ denotes the matrix whose coefficients are the conjugates of those of $U$.
Justify that ${}^t A = U({}^t B)U^{-1}$.
QII.A.2 Matrices Diagonalizability and Similarity View
In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a square matrix $U$ of size $n$, invertible, with complex coefficients, such that $U {}^t\bar{U} = I_n$ and $A = UBU^{-1}$, where $\bar{U}$ denotes the matrix whose coefficients are the conjugates of those of $U$.
We propose to show that there exists a matrix $P \in \mathrm{GL}_n(\mathbb{R})$ such that $A = PBP^{-1}$ and ${}^t A = P {}^t B P^{-1}$. For this, we denote by $X$ and $Y$ the matrices of $\mathcal{M}_n(\mathbb{R})$ such that $U = X + \mathrm{i}Y$.
a) Show that there exists $\mu \in \mathbb{R}$ such that $X + \mu Y \in \mathrm{GL}_n(\mathbb{R})$.
b) Show that $AX = XB$ and $AY = YB$.
c) Conclude.
QII.A.3 Matrices Diagonalizability and Similarity View
In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a matrix $P \in \mathrm{GL}_n(\mathbb{R})$ such that $A = PBP^{-1}$ and ${}^t A = P {}^t B P^{-1}$. We write $P$ in the form $P = OS$, with $O \in \mathrm{O}(n)$ and $S \in \mathcal{S}_n^{++}(\mathbb{R})$.
a) Show that $BS^2 = S^2 B$, then that $BS = SB$.
b) Deduce that there exists $O \in \mathrm{O}(n)$ such that $A = OB {}^t O$.
QII.B.1 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in \mathcal{M}_n(\mathbb{R})$. We propose to give a necessary and sufficient condition for the existence of a solution $X \in \mathrm{GL}_n(\mathbb{R})$ to the system $$(*) : \left\{\begin{array}{l} {}^t A A + {}^t X X = I_n \\ {}^t A X - {}^t X A = 0_n \end{array}\right.$$
Show that if the system $(*)$ admits a solution in $\mathrm{GL}_n(\mathbb{R})$, then the eigenvalues of ${}^t A A$ belong to the interval $[0, 1[$.
QII.B.2 Matrices Matrix Decomposition and Factorization View
Let $A \in \mathcal{M}_n(\mathbb{R})$. Consider the system $$(*) : \left\{\begin{array}{l} {}^t A A + {}^t X X = I_n \\ {}^t A X - {}^t X A = 0_n \end{array}\right.$$
We assume in this question that the eigenvalues of ${}^t A A$ belong to the interval $[0, 1[$.
a) Justify that we can seek the solutions $X$ of $(*)$ in the form $X = UH$, with $U \in \mathrm{O}(n)$ and $H \in \mathcal{S}_n^{++}(\mathbb{R})$.
b) Determine $H$.
c) Show the existence of a solution $X \in \mathrm{GL}_n(\mathbb{R})$ of $(*)$ belonging to $\mathrm{GL}_n(\mathbb{R})$.
QIII.A Matrices Determinant and Rank Computation View
For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$ We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.
Show that for fixed $x \in \mathbb{R}$, the sequence $(P_p(x))_{p \in \mathbb{N}^*}$ satisfies a linear relation of order 2, which we shall specify.
QIII.B Matrices Determinant and Rank Computation View
For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$ We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.
Let $x \in \mathbb{R}$ such that $|2 - x| < 2$. After justifying the existence of a unique $\theta \in ]0, \pi[$ such that $2 - x = 2\cos\theta$, determine $P_p(x)$ as a function of $\sin((p+1)\theta)$ and $\sin(\theta)$.
QIII.C Matrices Eigenvalue and Characteristic Polynomial Analysis View
For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$ We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.
Determine the eigenvalues of $A_p$.
QIII.D Matrices Diagonalizability and Similarity View
For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$
Show that $A_p$ is diagonalizable, and determine a basis of eigenvectors, specifying for each one the associated eigenvalue.
QIV.A Matrices Matrix Entry and Coefficient Identities View
Let $f$ be a linear form on $\mathcal{M}_n(\mathbb{R})$.
Show that there exists a unique matrix $A \in \mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.
QIV.B.1 Matrices Matrix Norm, Convergence, and Inequality View
Let $f$ be a linear form on $\mathcal{M}_n(\mathbb{R})$, and let $A$ be the unique matrix in $\mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.
Justify the existence of $M_n = \sup(\{f(O), O \in \mathrm{O}(n)\})$.
QIV.B.2 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $f$ be a linear form on $\mathcal{M}_n(\mathbb{R})$, and let $A$ be the unique matrix in $\mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.
Justify that ${}^t A A$ admits $n$ positive eigenvalues $\mu_1, \ldots, \mu_n$, counted with multiplicities.
QIV.B.3 Matrices Matrix Decomposition and Factorization View
Let $f$ be a linear form on $\mathcal{M}_n(\mathbb{R})$, and let $A$ be the unique matrix in $\mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$. Let $\mu_1, \ldots, \mu_n$ be the $n$ positive eigenvalues of ${}^t A A$ counted with multiplicities.
Show that $M_n = \sup(\{\operatorname{Tr}(D\Omega), \Omega \in \mathrm{O}(n)\})$, where $D$ is the diagonal matrix whose diagonal elements are $\sqrt{\mu_1}, \ldots, \sqrt{\mu_n}$.
QIV.B.4 Matrices Projection and Orthogonality View
Let $f$ be a linear form on $\mathcal{M}_n(\mathbb{R})$, and let $A$ be the unique matrix in $\mathcal{M}_n(\mathbb{R})$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$. Let $\mu_1, \ldots, \mu_n$ be the $n$ positive eigenvalues of ${}^t A A$ counted with multiplicities, and $D$ the diagonal matrix whose diagonal elements are $\sqrt{\mu_1}, \ldots, \sqrt{\mu_n}$.
Deduce that $M_n = \sum_{k=1}^n \sqrt{\mu_k}$.
QIV.C.1 Matrices Matrix Entry and Coefficient Identities View
In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$.
Determine the matrix $A$ such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.
QIV.C.2 Matrices Linear System and Inverse Existence View
In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $A$ is the matrix such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$.
Show that $$A^{-1} = \left(\begin{array}{rrrrr} 1 & -1 & 0 & \cdots & 0 \\ 0 & 1 & -1 & \ddots & \vdots \\ \vdots & \ddots & 1 & \ddots & 0 \\ \vdots & & \ddots & \ddots & -1 \\ 0 & \cdots & \cdots & 0 & 1 \end{array}\right)$$
QIV.C.3 Matrices Eigenvalue and Characteristic Polynomial Analysis View
In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $A$ is the matrix such that $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \operatorname{Tr}(AM)$, with $A^{-1}$ as given in IV.C.2.
Determine the eigenvalues of $A^{-1} {}^t A^{-1}$.
QIV.C.5 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $M_n = \sum_{k=1}^n \dfrac{1}{2\cos\dfrac{k\pi}{2n+1}}$.
Give an equivalent of $M_n$ as $n$ tends to $+\infty$.