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
2013 x-ens-maths1__mp

14 maths questions

Q1a Proof Recurrence Relations and Sequence Properties View
Show that $V$ is a vector subspace of $\mathbf{C}^{\mathbf{Z}}$. Given $f \in \mathbf{C}^{\mathbf{Z}}$, we define $E(f) \in \mathbf{C}^{\mathbf{Z}}$ by $E(f)(k) = f(k+1), k \in \mathbf{Z}$.
Q1b Proof Recurrence Relations and Sequence Properties View
Show that $E \in \mathcal{L}(\mathbf{C}^{\mathbf{Z}})$ and that $V$ is stable under $E$.
Q2 Sequences and Series Recurrence Relations and Sequence Properties View
Show that $E \in \mathrm{GL}(V)$.
Q3 Matrices Recurrence Relations and Sequence Properties View
For $i \in \mathbf{Z}$, we define $v_i$ in $\mathbf{C}^{\mathbf{Z}}$ by $$v_i(k) = \left\{\begin{array}{l} 1 \text{ if } k = i \\ 0 \text{ if } k \neq i \end{array}\right.$$
3a. Show that the family $\{v_i\}_{i \in \mathbf{Z}}$ is a basis of $V$.
3b. Calculate $E(v_i)$.
Q4 Matrices Recurrence Relations and Sequence Properties View
Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$. We define the linear maps $F, H \in \mathcal{L}(V)$ respectively by $$H(v_i) = \lambda(i) v_i \quad \text{and} \quad F(v_i) = \mu(i) v_{i+1}, \quad i \in \mathbf{Z}.$$ Show that $H \circ E = E \circ H + 2E$ if and only if for all $i \in \mathbf{Z}, \lambda(i) = \lambda(0) - 2i$.
Q5 Matrices Recurrence Relations and Sequence Properties View
We assume that the conditions of question 4 are satisfied (i.e., $\lambda(i) = \lambda(0) - 2i$ for all $i \in \mathbf{Z}$). Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$ and $F, H \in \mathcal{L}(V)$ defined by $H(v_i) = \lambda(i) v_i$ and $F(v_i) = \mu(i) v_{i+1}$. Show that $E \circ F = F \circ E + H$ if and only if for all $i \in \mathbf{Z}$, $$\mu(i) = \mu(0) + i(\lambda(0) - 1) - i^2.$$
Q6 Matrices Deduction or Consequence from Prior Results View
We assume that the conditions of question 4 are satisfied (i.e., $\lambda(i) = \lambda(0) - 2i$ for all $i \in \mathbf{Z}$).
6a. Show that for $f \in V$, the vector space spanned by $H^n(f), n \in \mathbf{N}$, is finite-dimensional.
6b. Deduce that a non-zero subspace of $V$ stable under $H$ contains at least one of the $v_i$.
Q7 Proof Group Homomorphisms and Isomorphisms View
We assume that the conditions of questions 4 and 5 are satisfied and that $\lambda(0) = 0, \mu(0) = 1$.
7a. Show that $F \in \mathrm{GL}(V)$.
7b. Show that $E$ and $F$ are not of finite order in the group $\mathrm{GL}(V)$.
7c. Calculate the kernel of $H$ and show that $H^r \neq \operatorname{Id}_V$ for $r \geq 1$.
Q8 Matrices Algebra and Subalgebra Proofs View
We assume that the conditions of questions 4 and 5 are satisfied and that $\lambda(0) = 0, \mu(0) = 1$. We denote by $\mathbf{C}[X]$ the algebra of polynomials with complex coefficients in one indeterminate $X$.
8a. Show that $\mathbf{C}[E]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.
8b. Show that $\mathbf{C}[F]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.
8c. Show that $\mathbf{C}[H]$ is isomorphic (as an algebra) to $\mathbf{C}[X]$.
Q9 Complex Numbers Arithmetic Algebraic Number Theory and Minimal Polynomials View
Throughout the rest of the problem, we fix an odd integer $\ell \geq 3$ and $q$ a primitive $\ell$-th root of unity. Show that $q^2$ is a primitive $\ell$-th root of unity.
Q10 Matrices Decomposition and Basis Construction View
Let $W_{\ell} = \bigoplus_{0 \leq i < \ell} \mathbf{C} v_i$ and $a \in \mathbf{C}^*$. We consider the element $G_a$ of $\mathcal{L}(W_{\ell})$ whose matrix in the basis $\{v_i\}_{0 \leq i < \ell}$ is: $$\left(\begin{array}{cccccc} 0 & 0 & 0 & \cdots & 0 & a \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & 0 & \ddots & \ddots & \vdots & \vdots \\ 0 & \vdots & \ddots & \ddots & 0 & 0 \\ 0 & 0 & \ldots & 0 & 1 & 0 \end{array}\right)$$
10a. Calculate $G_a^{\ell}$. Show that $G_a$ is diagonalizable.
10b. Let $b$ be an $\ell$-th root of $a$. Calculate the eigenvectors of $G_a$ and the associated eigenvalues in terms of $b, q$ and the $v_i$.
Q11 Matrices Group Actions and Surjectivity/Injectivity of Maps View
Let $W_{\ell} = \bigoplus_{0 \leq i < \ell} \mathbf{C} v_i$ and $a \in \mathbf{C}^*$. We define a linear map $P_a : V \rightarrow V$ by $P_a(v_i) = a^p v_r$ where for $i \in \mathbf{Z}$, we define $r$ and $p$ respectively as the remainder and quotient of the Euclidean division of $i$ by $\ell$; in other words, $i = p\ell + r$ where $0 \leq r < \ell$ and $p \in \mathbf{Z}$. Show that $P_a$ is a projector with image $W_{\ell}$.
Q12 Matrices Group Homomorphisms and Isomorphisms View
Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$ and $H \in \mathcal{L}(V)$ defined by $H(v_i) = \lambda(i) v_i$. Show that $H \circ E = q^2 E \circ H$ if and only if for all $i \in \mathbf{Z}, \lambda(i) = \lambda(0) q^{-2i}$.
Q13 Matrices Linear System and Inverse Existence View
We assume that the conditions of question 12 are satisfied and that $\lambda(0) \neq 0$. Show that $H \in \mathrm{GL}(V)$.