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__psi

18 maths questions

Q1 Matrices Diagonalizability and Similarity View
Let $A$ be the matrix in $M_{2}(\mathbf{R})$ defined by: $$A = \left(\begin{array}{cc} 1 & 1 \\ -1 & 3 \end{array}\right)$$ Is the matrix $A$ semi-simple?
Q3 Matrices Diagonalizability determination or proof View
Let $M$ be a matrix in $M_{2}(\mathbf{R})$. We assume that $M$ has two complex eigenvalues $\mu = a + ib$ and $\bar{\mu} = a - ib$ with $a \in \mathbf{R}$ and $b \in \mathbf{R}^{*}$. Prove that $M$ is semi-simple and similar in $M_{2}(\mathbf{R})$ to the matrix: $$\left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$$
Q4 Matrices Diagonalizability determination or proof View
Let $M$ be a matrix in $M_{2}(\mathbf{R})$. Prove that $M$ is semi-simple if and only if one of the following conditions is satisfied:
  1. [i)] $M$ is diagonalizable in $M_{2}(\mathbf{R})$;
  2. [ii)] $\chi_{M}$ has two complex conjugate roots with non-zero imaginary part.
Q6 Matrices Diagonalizability and Similarity View
Let $N$ be a matrix in $M_{n}(\mathbf{R})$. Give the factored form of $\chi_{N}$ in $\mathbf{C}[X]$, specifying in the notation the real roots and the complex conjugate roots. Deduce that if $N$ is semi-simple then it is similar in $M_{n}(\mathbf{R})$ to an almost diagonal matrix.
Q7 Matrices Proof of Set Membership, Containment, or Structural Property View
In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$. We assume that $u$ is diagonalizable. We denote by $\mathcal{B} = (v_{1}, v_{2}, \ldots, v_{n})$ a basis of $E$ formed of eigenvectors of $u$. Let $F$ be a vector subspace of $E$, different from $\{0_{E}\}$ and from $E$.
Prove that there exists $k \in \llbracket 1; n \rrbracket$ such that $v_{k} \notin F$ and that then $F$ and the vector line spanned by $v_{k}$ are in direct sum.
Q8 Matrices Linear Transformation and Endomorphism Properties View
In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$. We assume that $u$ is diagonalizable. We denote by $\mathcal{B} = (v_{1}, v_{2}, \ldots, v_{n})$ a basis of $E$ formed of eigenvectors of $u$. Let $F$ be a vector subspace of $E$, different from $\{0_{E}\}$ and from $E$.
We denote $$\mathcal{A} = \left\{H \text{ vector subspace of } E \text{ such that } u(H) \subset H \text{ and } F \cap H = \{0_{E}\}\right\}$$ and $$\mathcal{L} = \left\{p \in \mathbf{N}^{*} \mid \exists H \in \mathcal{A} : p = \operatorname{dim}(H)\right\}$$
Prove that $\mathcal{L}$ has a greatest element which we call $r$.
Q10 Invariant lines and eigenvalues and vectors Diagonalizability and Similarity View
In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$.
Assume that every vector subspace of $E$ has a complement in $E$, stable under $u$. Prove that $u$ is diagonalizable. Deduce a characterization of diagonalizable matrices in $M_{n}(\mathbf{C})$.
Hint: one may reason by contradiction and introduce a vector subspace, whose existence one will justify, of dimension $n-1$ and containing the sum of the eigenspaces of $u$.
Q11 Factor & Remainder Theorem Location and bounds on roots View
Let $\alpha \in \mathbf{R}$. Prove that if $\alpha$ is a root of a polynomial $P$ in $\mathbf{R}[X]$ with strictly positive coefficients, then $\alpha < 0$.
Q12 Factor & Remainder Theorem Algebraic Number Theory and Minimal Polynomials View
Prove that every divisor of a Hurwitz polynomial is a Hurwitz polynomial.
Q13 Factor & Remainder Theorem Coefficient and structural properties of special polynomial families View
Let $P$ be an irreducible Hurwitz polynomial in $\mathbf{R}[X]$ with positive leading coefficient. Prove that all coefficients of $P$ are strictly positive.
Q14 Factor & Remainder Theorem Determine coefficients or parameters from root conditions View
Let $n \in \mathbf{N}^{*}$. Let $(z_{1}, z_{2}, \ldots, z_{n}) \in \mathbf{C}^{n}$. We define the two polynomials $P(X)$ and $Q(X)$ in $\mathbf{C}[X]$ by: $$P(X) = \prod_{k=1}^{n}(X - z_{k}) \quad \text{and} \quad Q(X) = \prod_{(k,l) \in \llbracket 1;n \rrbracket^{2}}(X - z_{k} - z_{l})$$
We assume $n = 2$ and $P \in \mathbf{R}_{2}[X]$. If the coefficients of $Q$ are strictly positive, is $P$ then a Hurwitz polynomial?
Q15 Factor & Remainder Theorem Direct Proof of a Stated Identity or Equality View
Let $A$ and $B$ be two polynomials in $\mathbf{R}[X]$ whose coefficients are all strictly positive. Prove that the coefficients of the product $AB$ are also strictly positive.
Q16 Factor & Remainder Theorem Location and bounds on roots View
Let $n \in \mathbf{N}^{*}$. Let $(z_{1}, z_{2}, \ldots, z_{n}) \in \mathbf{C}^{n}$. We define the two polynomials $P(X)$ and $Q(X)$ in $\mathbf{C}[X]$ by: $$P(X) = \prod_{k=1}^{n}(X - z_{k}) \quad \text{and} \quad Q(X) = \prod_{(k,l) \in \llbracket 1;n \rrbracket^{2}}(X - z_{k} - z_{l})$$
Prove that if $P$ and $Q$ are in $\mathbf{R}[X]$, then we have the equivalence: $P$ is a Hurwitz polynomial if and only if the coefficients of $P$ and $Q$ are strictly positive.
Q17 Second order differential equations Structure of the solution space View
Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
Let $T \in M_{n}(\mathbf{R})$. We assume that $M$ is similar to $T$ in $M_{n}(\mathbf{R})$ and we denote by $(\mathrm{S}^{*})$ the differential system $$(\mathrm{S}^{*}) \quad Y' = TY$$
Prove that the coordinates of a solution $X$ of (S) are linear combinations of the coordinates of a solution $Y$ of $(\mathrm{S}^{*})$.
Q18 Second order differential equations Solving homogeneous second-order linear ODE View
Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We assume $n = 2$, we then denote $X = (x; y)$ where $x$ and $y$ are two functions differentiable from $\mathbf{R}$ to $\mathbf{R}$ and we set $z = x + iy$.
We assume that there exist real numbers $a$ and $b$ such that $M = \left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$.
Prove that $X$ is a solution of (S) if and only if $z$ is a solution of a first-order linear differential equation to be determined. Deduce an expression, as a function of $t$, of the coordinates of the solutions of (S).
Solve the system $X' = BX$ where $B$ is the matrix from question 2).
Q19 Second order differential equations Qualitative and asymptotic analysis of solutions View
Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We assume $n = 2$, we then denote $X = (x; y)$ where $x$ and $y$ are two functions differentiable from $\mathbf{R}$ to $\mathbf{R}$ and we set $z = x + iy$.
Let $M \in M_{2}(\mathbf{R})$ be semi-simple. Give a necessary and sufficient condition, concerning the real and imaginary parts of the eigenvalues of $M$, for every solution of (S) to have each of its coordinates tend to 0 as $+\infty$.
Q20 Second order differential equations Qualitative and asymptotic analysis of solutions View
Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
Let $T \in M_{n}(\mathbf{R})$. We assume that $M$ is similar to $T$ in $M_{n}(\mathbf{R})$ and we denote by $(\mathrm{S}^{*})$ the differential system $$(\mathrm{S}^{*}) \quad Y' = TY$$
We consider the following assertions:
  • [$\mathbf{A}_{1}$] $\chi_{M}$ is a Hurwitz polynomial;
  • [$\mathbf{A}_{2}$] The solutions of (S) tend to $0_{\mathbf{R}^{n}}$ as $t$ tends to $+\infty$;
  • [$\mathbf{A}_{3}$] There exist $\alpha > 0$ and $k > 0$ such that for every solution $\Phi$ of (S), $$\forall t \geq 0 \quad : \quad \|\Phi(t)\| \leq k e^{-\alpha t} \|\Phi(0)\|.$$

Let $T \in M_{n}(\mathbf{R})$. We assume that $T$ satisfies the following condition: $$(\mathrm{C}) \quad \exists \beta \in \mathbf{R}_{+}^{*}, \forall X \in \mathbf{R}^{n} : \langle TX, X \rangle \leq -\beta \|X\|^{2}.$$
Prove that $\mathrm{A}_{3}$ is true with $k = 1$ for every solution $\Phi$ of $(\mathrm{S}^{*})$.
Hint: one may introduce the function $t \mapsto e^{2\beta t} \|\Phi(t)\|^{2}$.
Q21 Second order differential equations Qualitative and asymptotic analysis of solutions View
Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We consider the following assertions:
  • [$\mathbf{A}_{1}$] $\chi_{M}$ is a Hurwitz polynomial;
  • [$\mathbf{A}_{2}$] The solutions of (S) tend to $0_{\mathbf{R}^{n}}$ as $t$ tends to $+\infty$;
  • [$\mathbf{A}_{3}$] There exist $\alpha > 0$ and $k > 0$ such that for every solution $\Phi$ of (S), $$\forall t \geq 0 \quad : \quad \|\Phi(t)\| \leq k e^{-\alpha t} \|\Phi(0)\|.$$

Assume that $M \in M_{n}(\mathbf{R})$ is semi-simple. Prove that the assertions $\mathrm{A}_{1}$, $\mathrm{A}_{2}$ and $\mathrm{A}_{3}$ are equivalent.
Hint: one may start with $A_{3}$ implies $A_{2}$.