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
2017 x-ens-maths__pc

18 maths questions

Q1 Matrices Matrix Norm, Convergence, and Inequality View
a) For any matrix $M \in M _ { n } ( \mathbb { C } )$ and any real number $C > 0$, show the equivalence $$\| M \| \leqslant C \Longleftrightarrow \forall x \in \mathbb { C } ^ { n } : \| M x \| _ { 1 } \leqslant C \| x \| _ { 1 } .$$ b) Show that the map $M \longmapsto \| M \|$ is a norm on $M _ { n } ( \mathbb { C } )$.
Q2 Matrices Matrix Norm, Convergence, and Inequality View
Show that for $A , B \in M _ { n } ( \mathbb { C } ) , \| A B \| \leqslant \| A \| \| B \|$.
Q3 Matrices Matrix Norm, Convergence, and Inequality View
Let $A \in M _ { n } ( \mathbb { C } )$. We denote by $a _ { i , j }$ the coefficient of $A$ with row index $i$ and column index $j$. Show that $$\| A \| = \max _ { 1 \leqslant j \leqslant n } \left( \sum _ { i = 1 } ^ { n } \left| a _ { i , j } \right| \right)$$
Q4 Matrices Matrix Norm, Convergence, and Inequality View
We say that a sequence $\left( A ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ of matrices in $M _ { n } ( \mathbb { C } )$ converges to a matrix $B \in M _ { n } ( \mathbb { C } )$ when $$\forall i \in \llbracket 1 , n \rrbracket , \forall j \in \llbracket 1 , n \rrbracket , \lim _ { k \rightarrow + \infty } \left( a _ { i , j } \right) ^ { ( k ) } = b _ { i , j }$$ Show that the sequence $( A ^ { ( k ) } )$ converges to $B$ if and only if $\lim _ { k \rightarrow + \infty } \left\| A ^ { ( k ) } - B \right\| = 0$.
Q5 Matrices Matrix Norm, Convergence, and Inequality View
We consider in this question a matrix $A \in M _ { n } ( \mathbb { C } )$ that is upper triangular, $$A = \left( \begin{array} { c c c c c } a _ { 1,1 } & a _ { 1,2 } & \ldots & \ldots & a _ { 1 , n } \\ 0 & a _ { 2,2 } & \ldots & \ldots & a _ { 2 , n } \\ \vdots & \ddots & \ddots & & \vdots \\ \vdots & & \ddots & \ddots & \vdots \\ 0 & \ldots & \ldots & 0 & a _ { n , n } \end{array} \right)$$ We assume that $$\forall i \in \llbracket 1 , n \rrbracket , \left| a _ { i , i } \right| < 1$$ For any real $b > 0$, we set $P _ { b } = \operatorname { diag } \left( 1 , b , b ^ { 2 } , \ldots , b ^ { n - 1 } \right) \in M _ { n } ( \mathbb { R } )$. a) Compute $P _ { b } ^ { - 1 } A P _ { b }$. What happens when $b$ tends to 0? b) Show that there exists $b > 0$ such that $$\left\| P _ { b } ^ { - 1 } A P _ { b } \right\| < 1$$ c) Deduce that the sequence $\left( A ^ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ converges to 0.
Q6 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Determine the spectral radius of the following matrices $$\left( \begin{array} { l l } 0 & 0 \\ 0 & 1 \end{array} \right) , \quad \left( \begin{array} { l l } 0 & 0 \\ 1 & 0 \end{array} \right) , \quad \left( \begin{array} { l l } 1 & 0 \\ 0 & 0 \end{array} \right) , \quad \left( \begin{array} { c c } 0 & - 1 \\ 2 & 0 \end{array} \right) , \quad \left( \begin{array} { l l } 3 & 2 \\ 1 & 2 \end{array} \right)$$
Q7 Matrices True/False or Multiple-Select Conceptual Reasoning View
Say, by briefly justifying the answer, whether the following assertions are correct for all $A , B \in M _ { n } ( \mathbb { C } ) , \mu \in \mathbb { C }$. i) $\rho ( \mu A ) = | \mu | \rho ( A )$. ii) $\rho ( A + B ) \leqslant \rho ( A ) + \rho ( B )$. iii) $\rho ( A B ) \leqslant \rho ( A ) \rho ( B )$. iv) For $P \in M _ { n } ( \mathbb { C } )$ invertible, $\rho \left( P ^ { - 1 } A P \right) = \rho ( A )$. v) $\rho \left( { } ^ { t } A \right) = \rho ( A )$.
Q8 Matrices Matrix Norm, Convergence, and Inequality View
Show that for any matrix $A \in M _ { n } ( \mathbb { C } )$, $$\rho ( A ) \leqslant \| A \| .$$
Q9 Matrices Matrix Norm, Convergence, and Inequality View
Let $A \in M _ { n } ( \mathbb { C } )$. Show that if $\rho ( A ) < 1$, then the sequence $\left( A ^ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ converges to 0.
Q10 Matrices Matrix Norm, Convergence, and Inequality View
Let $A \in M _ { n } ( \mathbb { C } )$. a) Show that, for all $k \in \mathbb { N } ^ { * } , \left\| A ^ { k } \right\| \geqslant \rho ( A ) ^ { k }$. b) We define the subset of $\mathbb { R } _ { + }$ $$E _ { A } = \left\{ \alpha > 0 \left\lvert \, \lim _ { k \rightarrow + \infty } \left( \frac { A } { \alpha } \right) ^ { k } = 0 \right. \right\} .$$ Show that $\left. E _ { A } = \right] \rho ( A ) , + \infty [$.
Q11 Matrices Matrix Norm, Convergence, and Inequality View
Let $A \in M _ { n } ( \mathbb { C } )$. Show the formula $$\lim _ { k \rightarrow + \infty } \left\| A ^ { k } \right\| ^ { 1 / k } = \rho ( A )$$
Q12 Matrices Matrix Norm, Convergence, and Inequality View
For $A \in M _ { n } ( \mathbb { C } )$ with coefficients $a _ { i , j }$, we set $A _ { + } = \left( b _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$, where $b _ { i , j } = \left| a _ { i , j } \right|$. Show the inequality $$\rho ( A ) \leqslant \rho \left( A _ { + } \right)$$
Q13 Complex Numbers Arithmetic Modulus Inequalities and Bounds (Proof-Based) View
Let $z _ { 1 } , \ldots , z _ { n }$ be complex numbers. Show that if $$\left| z _ { 1 } + \cdots + z _ { n } \right| = \left| z _ { 1 } \right| + \cdots + \left| z _ { n } \right|$$ then the vector $\left( \begin{array} { c } z _ { 1 } \\ \vdots \\ z _ { n } \end{array} \right)$ is collinear with the vector $\left( \begin{array} { c } \left| z _ { 1 } \right| \\ \vdots \\ \left| z _ { n } \right| \end{array} \right)$.
Q14 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $x , y \in \mathbb { C } ^ { n } , \lambda , \mu \in \mathbb { C }$. Show that if $\lambda \neq \mu$, then the following implication holds $$\left( A x = \lambda x \quad \text { and } \quad { } ^ { t } A y = \mu y \right) \Longrightarrow { } ^ { t } x y = 0 .$$
Q15 Matrices Matrix Norm, Convergence, and Inequality View
Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. Suppose that there exist a non-negative real $\mu$ and a positive non-zero vector $w$ such that $A w \geqslant \mu w$. a) Show that for all natural integer $k , A ^ { k } w \geqslant \mu ^ { k } w$. Deduce that $\rho ( A ) \geqslant \mu$. b) Show that if $A w > \mu w$, then $\rho ( A ) > \mu$. c) We now suppose that in the system of inequalities $A w \geqslant \mu w$, the $k$-th inequality is strict, that is $$\sum _ { j = 1 } ^ { n } a _ { k j } w _ { j } > \mu w _ { k } .$$ Show that there exists $\epsilon > 0$ such that, by setting $w _ { j } ^ { \prime } = w _ { j }$ if $j \neq k$ and $w _ { k } ^ { \prime } = w _ { k } + \epsilon$, we have $A w ^ { \prime } > \mu w ^ { \prime }$. Deduce that $\rho ( A ) > \mu$.
Q16 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. Let $\lambda$ be an eigenvalue of $A$ with modulus $\rho ( A )$ and let $x \in \mathbb { C } ^ { n } \backslash \{ 0 \}$ be an eigenvector of $A$ associated with $\lambda$. We define the positive non-zero vector $v _ { 0 }$ by $\left( v _ { 0 } \right) _ { i } = \left| x _ { i } \right|$ for $1 \leqslant i \leqslant n$. a) Show that $A v _ { 0 } \geqslant \rho ( A ) v _ { 0 }$, then that $$A v _ { 0 } = \rho ( A ) v _ { 0 }$$ b) Deduce that $\rho ( A ) > 0$ and $$\forall i \in \llbracket 1 , n \rrbracket , \left( v _ { 0 } \right) _ { i } > 0 .$$ c) Show that $x$ is collinear with $v _ { 0 }$. Deduce that $\lambda = \rho ( A )$.
Q17 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. By applying the previous results to the matrix ${ } ^ { t } A$, we obtain the existence of $w _ { 0 } \in \mathbb { R } ^ { n }$, whose all components are strictly positive, such that ${ } ^ { t } A w _ { 0 } = \rho ( A ) w _ { 0 }$. We set $$F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$$ a) Show that $F$ is a vector subspace of $\mathbb { C } ^ { n }$ stable by $\varphi _ { A }$, and that $$\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$$ b) Show that if $v$ is an eigenvector of $A$ associated with an eigenvalue $\mu \neq \rho ( A )$, then $v \in F$. Deduce property (iii): if $v$ is an eigenvector of $A$ whose all components are positive, then $v \in \operatorname { ker } ( A - \rho ( A ) I _ { n } )$.
Q18 Matrices Matrix Power Computation and Application View
Throughout this part, $A$ is a strictly positive matrix in $M _ { n } ( \mathbb { R } )$. We use the notation from question 17: $w_0$, $v_0$, $F = \left\{ x \in \mathbb { C } ^ { n } \mid { } ^ { t } x w _ { 0 } = 0 \right\}$, and $\mathbb { C } ^ { n } = F \oplus \mathbb { C } v _ { 0 }$. a) We denote by $\psi$ the endomorphism of $F$ defined as the restriction of $\varphi _ { A }$ to $F$. Show that all eigenvalues of $\psi$ have modulus strictly less than $\rho ( A )$. Deduce that $\rho ( A )$ is a simple root of the characteristic polynomial of $A$ and that $$\operatorname { ker } \left( A - \rho ( A ) I _ { n } \right) = \mathbb { C } v _ { 0 }$$ b) Show that if $x \in F , \lim _ { k \rightarrow + \infty } \frac { A ^ { k } x } { \rho ( A ) ^ { k } } = 0$. c) Let $x$ be a positive non-zero vector. Determine the limit of $\frac { A ^ { k } x } { \rho ( A ) ^ { k } }$ when $k$ tends to $+ \infty$.