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

2016 centrale-maths1__psi

33 maths questions

QI.A.1 Permutations & Arrangements Combinatorial Structures on Permutation Matrices/Groups View

Justify that $\mathcal{X}_n$ is a finite set and determine its cardinality.

QI.A.2 Matrices Determinant and Rank Computation View

Prove that for all $M \in \mathcal{Y}_n$, $\operatorname{det}(M) \leqslant n!$ and that there is no equality.

QI.A.3 Matrices Structured Matrix Characterization View

Prove that $\mathcal{Y}_n$ is a convex and compact subset of $\mathcal{M}_n(\mathbb{R})$.

QI.A.4 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Let $M \in \mathcal{Y}_n$ and $\lambda$ a complex eigenvalue of $M$. Prove that $|\lambda| \leqslant n$ and give an explicit example where equality holds.

QI.B.1 Matrices Diagonalizability and Similarity View

List the elements of $\mathcal{X}_2' = \mathcal{X}_2 \cap \mathrm{GL}_2(\mathbb{R})$. Specify (by justifying) which ones are diagonalizable over $\mathbb{R}$.

QI.B.2 Matrices Structured Matrix Characterization View

Prove that $\mathcal{X}_2' = \mathcal{X}_2 \cap \mathrm{GL}_2(\mathbb{R})$ generates the vector space $\mathcal{M}_2$. For $n \geqslant 2$, does $\mathcal{X}_n'$ generate the vector space $\mathcal{M}_n(\mathbb{R})$?

QII.A.1 Matrices Matrix Norm, Convergence, and Inequality View

For all $(M, N) \in (\mathcal{M}_n(\mathbb{R}))^2$, we denote $$(M \mid N) = \operatorname{tr}({}^t M N)$$ Prove that this defines an inner product on $\mathcal{M}_n(\mathbb{R})$. Explicitly express $(M \mid N)$ in terms of the coefficients of $M$ and $N$.

QII.A.2 Matrices Matrix Norm, Convergence, and Inequality View

We denote by $\|M\|$ the Euclidean norm associated with the inner product $(M \mid N) = \operatorname{tr}({}^t M N)$. Fix $A \in \mathcal{M}_n(\mathbb{R})$, prove that there exists a matrix $M \in \mathcal{Y}_n$ such that: $$\forall N \in \mathcal{Y}_n \quad \|A - M\| \leqslant \|A - N\|$$

QII.A.3 Matrices Matrix Norm, Convergence, and Inequality View

We denote by $\|M\|$ the Euclidean norm associated with the inner product $(M \mid N) = \operatorname{tr}({}^t M N)$. Justify the uniqueness of the matrix $M \in \mathcal{Y}_n$ minimizing $\|A - M\|$ over $\mathcal{Y}_n$ and explicitly express its coefficients in terms of those of $A$.

QII.B.1 Matrices Determinant and Rank Computation View

Justify that the determinant has a maximum on $\mathcal{X}_n$ (denoted $x_n$) and a maximum on $\mathcal{Y}_n$ (denoted $y_n$).

QII.B.2 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Prove that the sequence $(y_k)_{k \geqslant 2}$ is increasing, where $y_k$ denotes the maximum of the determinant on $\mathcal{Y}_k$.

QII.B.3 Matrices Determinant and Rank Computation View

Let $J \in \mathcal{X}_n$ be the matrix whose coefficients all equal 1. We set $M = J - I_n$.
Calculate $\operatorname{det}(M)$ and deduce that $\lim_{k \to +\infty} y_k = +\infty$.

QII.B.4 Matrices Determinant and Rank Computation View

Let $N = (n_{i,j})_{i,j} \in \mathcal{Y}_n$. Fix $1 \leqslant i, j \leqslant n$ and suppose that $n_{i,j} \in ]0,1[$.
Prove that by replacing $n_{i,j}$ either by 0 or by 1, we can obtain a matrix $N'$ of $\mathcal{Y}_n$ such that $\operatorname{det}(N) \leqslant \operatorname{det}(N')$.
Deduce that $x_n = y_n$.

QIII.A.1 Matrices Matrix Group and Subgroup Structure View

Give two definitions of a vector isometry of $\mathbb{R}^n$ and prove their equivalence.

QIII.A.2 Matrices Determinant and Rank Computation View

Prove that if $M \in \mathrm{O}_n(\mathbb{R})$, then its determinant equals 1 or $-1$. What do you think of the converse?

QIII.A.3 Matrices Matrix Group and Subgroup Structure View

Prove that $\mathcal{P}_n = \mathcal{X}_n \cap \mathrm{O}_n(\mathbb{R})$ and determine its cardinality.

QIII.B.1 Groups Symmetric Group and Permutation Properties View

Let $\sigma$ and $\sigma'$ be two elements of $S_n$.
Prove that $P_\sigma P_{\sigma'} = P_{\sigma \circ \sigma'}$.
Justify that the application $\left\{\begin{array}{l} \mathbb{Z} \rightarrow S_n \\ k \mapsto \sigma^k \end{array}\right.$ is not injective.
Deduce that there exists an integer $N \geqslant 1$ such that $\sigma^N = \operatorname{Id}_{\{1,\ldots,n\}}$, where $\operatorname{Id}_{\{1,\ldots,n\}}$ denotes the identity map on the set $\{1,\ldots,n\}$.

QIII.B.2 Matrices Diagonalizability and Similarity View

Prove that all elements of $\mathcal{P}_n$ are diagonalizable over $\mathbb{C}$.

QIII.B.3 Matrices Eigenvalue and Characteristic Polynomial Analysis View

Determine the eigenvectors common to all elements of $\mathcal{P}_n$ in the cases $n = 2$ and $n = 3$.

QIII.B.4 Matrices Linear Transformation and Endomorphism Properties View

We propose to prove that the only vector subspaces of $\mathbb{R}^n$ stable under all $u_\sigma$, $\sigma \in S_n$ are $\{0_{\mathbb{R}^n}\}$, $\mathbb{R}^n$, the line $D$ generated by $e_1 + e_2 + \cdots + e_n$ and the hyperplane $H$ orthogonal to $D$.
a) Verify that these four vector subspaces are stable under all $u_\sigma$.
b) Let $V$ be a vector subspace of $\mathbb{R}^n$, not contained in $D$ and stable under all $u_\sigma$. Prove that there exists a pair $(i,j) \in \{1,\ldots,n\}^2$ with $i \neq j$ such that $e_i - e_j \in V$, then that the $n-1$ vectors $e_k - e_j$ ($k \in \{1,\ldots,n\}$, $k \neq j$) belong to $V$.
c) Conclude.

QIII.C Matrices Matrix Group and Subgroup Structure View

We are given a matrix $M$ of $\mathrm{GL}_n(\mathbb{R})$ whose coefficients are all natural integers and such that the set formed by all coefficients of all successive powers of $M$ is finite.
Prove that $M^{-1}$ has coefficients in $\mathbb{N}$ and deduce that $M$ is a permutation matrix. What can be said of the converse?

QIV.A.1 Discrete Probability Distributions Proof of Distributional Properties or Symmetry View

QIV.A.2 Binomial Distribution Derive or Prove a Binomial Distribution Identity View

QIV.A.3 Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables, defined on a probability space $(\Omega, \mathcal{A}, P)$ and following the same Bernoulli distribution with parameter $p$.
Let $i$ and $j$ be in $\{1, \ldots, n\}$. Give the distribution of the random variable $X_{i,j} = X_i \times X_j$.

QIV.A.4 Matrices Projection and Orthogonality View

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables, defined on a probability space $(\Omega, \mathcal{A}, P)$ and following the same Bernoulli distribution with parameter $p$. Let $S = X_1 + \ldots + X_n$.
If $\omega \in \Omega$, we introduce the column matrix $$U(\omega) = \begin{pmatrix} X_1(\omega) \\ \vdots \\ X_n(\omega) \end{pmatrix}$$ and the matrix $M(\omega) = U(\omega)\, {}^t(U(\omega))$. The application $M : \left\{\begin{array}{l} \Omega \rightarrow \mathcal{M}_n(\mathbb{R}) \\ \omega \mapsto M(\omega) \end{array}\right.$ is thus a random variable.
a) If $\omega \in \Omega$, justify that $M(\omega) \in \mathcal{X}_n$.
b) If $\omega \in \Omega$, justify that $\operatorname{tr}(M(\omega)) \in \{0, \ldots, n\}$, that $M(\omega)$ is diagonalizable over $\mathbb{R}$ and that $\operatorname{rg}(M(\omega)) \leqslant 1$.
c) If $\omega \in \Omega$, justify that $M(\omega)$ is an orthogonal projection matrix if and only if $S(\omega) \in \{0,1\}$.

QIV.A.5 Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables following the same Bernoulli distribution with parameter $p$. Let $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
Give the distribution, expectation and variance of the random variables $\operatorname{tr}(M)$ and $\operatorname{rg}(M)$.

QIV.A.6 Matrices Matrix Power Computation and Application View

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables following the same Bernoulli distribution with parameter $p$. Let $S = X_1 + \ldots + X_n$, $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
Express $M^k$ in terms of $S$ and $M$.
What is the probability that the sequence of matrices $(M^k)_{k \in \mathbb{N}}$ is convergent?
Show that, in this case, the limit is a projection matrix.

QIV.A.7 Discrete Probability Distributions Proof of Distributional Properties or Symmetry View

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables following the same Bernoulli distribution with parameter $p$. Let $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
What is the probability that $M$ has two distinct eigenvalues?

QIV.B.2 Binomial Distribution Justify Binomial Model and State Parameters View

Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$.
Give the distribution of $N_1$, then the conditional distribution of $N_2$ given $(N_1 = i)$ for $i$ in a set to be specified. Are $N_1$ and $N_2$ independent?

QIV.B.3 Geometric Distribution View

Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently. We denote $q = 1-p$ and $m = n^2$.
Let $i$ and $j$ be in $\{1, \ldots, n\}$. The smallest integer $k \geqslant 1$ such that the coefficient at row $i$, column $j$ of $M_k$ equals 1 is denoted $T_{i,j}$. Give the distribution of $T_{i,j}$.

QIV.B.4 Geometric Distribution View

Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. The smallest integer $k \geqslant 1$ such that the coefficient at row $i$, column $j$ of $M_k$ equals 1 is denoted $T_{i,j}$.
For an integer $k \geqslant 1$, give the value of $P(T_{i,j} \geqslant k)$.

QIV.B.5 Binomial Distribution Justify Binomial Model and State Parameters View

Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. For $k \geqslant 1$, the number of modifications made during the $k$-th pass is denoted $N_k$.
Let $r \geqslant 1$ be an integer and $S_r = N_1 + \cdots + N_r$. What does $S_r$ represent? Give its distribution (you may use the previous question).

QIV.B.6 Discrete Random Variables Expectation and Variance via Combinatorial Counting View

Let $p \in ]0,1[$, $q = 1-p$, $m = n^2$. We denote by $N$ the smallest index $k$ for which the matrix $M_k$ is completely filled.
a) Propose an approach to approximate the expectation of $N$ using a computer simulation with the functions above.
b) Give an expression for the exact value of this expectation involving $q$ and $m$.