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 x-ens-maths1__mp

24 maths questions

Q1a Sequences and Series 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 Sequences and Series 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 Sequences and Series 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 Sequences and Series 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 Sequences and Series 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 Proof 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 Groups 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 Groups 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 Number Theory 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 Groups 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 Groups 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 Groups 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 $H \in \mathrm{GL}(V)$.
Q14 Matrices Linear Transformation and Endomorphism Properties View
We assume that the conditions of question 12 are satisfied and that $\lambda(0) \neq 0$. Show that $E \circ F = F \circ E + H - H^{-1}$ if and only if for all $i \in \mathbf{Z}$, $$\mu(i) = \mu(i-1) + \lambda(0) q^{-2i} - \lambda(0)^{-1} q^{2i}$$
Q15 Matrices Linear Transformation and Endomorphism Properties View
We assume that the conditions of questions 12 and 14 are satisfied and that $\lambda(0) \neq 0$.
15a. Show that $\lambda$ and $\mu$ are periodic on $\mathbf{Z}$, with periods dividing $\ell$.
15b. Show that the period of $\lambda$ is equal to $\ell$.
15c. Show that the period of $\mu$ is also equal to $\ell$.
Q16 Matrices Eigenvalue and Characteristic Polynomial Analysis View
We assume that the conditions of questions 12 and 14 are satisfied and that $\lambda(0) \neq 0$. Let $C = (q - q^{-1}) E \circ F + q^{-1} H + q H^{-1}$ with $H^{-1}$ the inverse of $H$.
16a. Show that $C = (q - q^{-1}) F \circ E + q H + q^{-1} H^{-1}$.
16b. For $i \in \mathbf{Z}$, show that $v_i$ is an eigenvector of $C$.
16c. Deduce that $C$ is a homothety of $V$ and calculate its ratio $R(\lambda(0), \mu(0), q)$ in terms of $\lambda(0), \mu(0)$ and $q$.
16d. We fix $q$ and $\lambda(0)$. Show that the map $\mu(0) \mapsto R(\lambda(0), \mu(0), q)$ is a bijection from $\mathbf{C}$ to $\mathbf{C}$.
16e. We fix $q$ and $\mu(0)$. Show that the map $\lambda(0) \mapsto R(\lambda(0), \mu(0), q)$ is a surjection from $\mathbf{C}^*$ to $\mathbf{C}$ but not a bijection.
Q17 Matrices Linear Transformation and Endomorphism Properties View
Let $\ell, W_{\ell}, a, P_a$ be as in Part II. We say that an element $\phi$ of $\mathcal{L}(V)$ is compatible with $P_a$ if $P_a \circ \phi \circ P_a = P_a \circ \phi$.
17a. Show that if $\phi \in \mathcal{L}(V)$ commutes with $P_a$, then $\phi$ is compatible with $P_a$.
17b. Show that $H$ and $H^{-1}$ are compatible with $P_a$.
Q18 Matrices Matrix Group and Subgroup Structure View
Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$. Show that $\mathcal{U}_q$ is a subalgebra of $\mathcal{L}(V)$.
Q19 Matrices Linear Transformation and Endomorphism Properties View
Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$. Show that $E \in \mathcal{U}_q$ and $F \in \mathcal{U}_q$.
Q20 Matrices Linear Transformation and Endomorphism Properties View
Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$.
20a. Show that there exists a unique algebra morphism $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ such that $$\forall \phi \in \mathcal{U}_q, \quad \Psi_a(\phi) \circ P_a = P_a \circ \phi$$
20b. Show that $\phi \in \mathcal{U}_q$ is contained in the kernel of $\Psi_a$ if and only if the image of $\phi$ is in the subspace of $V$ spanned by the vectors $v_i - a^p v_r, i \in \mathbf{Z}$, where $i = p\ell + r$ is the Euclidean division of $i$ by $\ell$.
Q21 Matrices Linear Transformation and Endomorphism Properties View
Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. We study $\Psi_a(E)$ in this question.
21a. Determine $\Psi_a(E)(v_0)$.
21b. Deduce $\Psi_a(E^{\ell})$.
21c. Calculate the dimension of the vector subspace $\mathbf{C}[\Psi_a(E)]$.
21d. Calculate the eigenvectors of $\Psi_a(E)$.
Q22 Matrices Linear Transformation and Endomorphism Properties View
Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. Let $W$ be a non-zero subspace of $W_{\ell}$ stable under $\Psi_a(H)$.
22a. Show that $W$ contains at least one of the vectors $v_i$.
22b. What can be said if $W$ is moreover stable under $\Psi_a(E)$?
Q23 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$, and $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ the unique algebra morphism such that $\Psi_a(\phi) \circ P_a = P_a \circ \phi$ for all $\phi \in \mathcal{U}_q$. Give a necessary and sufficient condition on $R(\lambda(0), \mu(0), q)$ for the operator $\Psi_a(F)$ to be nilpotent.