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
2019 x-ens-maths__psi

9 maths questions

Q17 Proof Matrix Norm, Convergence, and Inequality View
We keep the notations from Parts II and III. We denote $e _ { k } = x _ { k } - \tilde { x }$ and $e _ { 0 } = x _ { 0 } - \tilde { x }$. We recall that $I _ { N }$ is the identity matrix of order $N$, and $\| \cdot \|$ denotes the matrix norm defined in question 2.
Show that $$\left\| e _ { k } \right\| _ { A } \leq \left\| e _ { 0 } \right\| _ { A } \min \left\{ \left\| I _ { N } + A Q ( A ) \right\| \mid Q \in \mathbb { R } [ X ] , \operatorname { deg } ( Q ) \leq k - 1 \right\}$$ (One may use the properties of $A ^ { 1 / 2 }$ demonstrated in question 6.)
Q19 Proof by induction Recurrence Relations and Sequence Properties View
Let $k$ be a non-negative integer. We define the function $f _ { k }$ from the interval $[ - 1,1 ]$ to itself by $f _ { k } ( x ) = \cos ( k \arccos x )$.
a) Expand the expression $f _ { k + 1 } ( x ) + f _ { k - 1 } ( x )$, and deduce the relation $$\forall x \in [ - 1,1 ] \quad f _ { k + 1 } ( x ) = 2 x f _ { k } ( x ) - f _ { k - 1 } ( x )$$
b) Deduce that $f _ { k }$ identifies on $[ - 1,1 ]$ with a polynomial $T _ { k }$, of degree $k$, with the same parity as $k$.
Q20 Hyperbolic functions View
Let $k$ be a non-negative integer. We define the function $f _ { k }$ from the interval $[ - 1,1 ]$ to itself by $f _ { k } ( x ) = \cos ( k \arccos x )$, and $T_k$ is the polynomial of degree $k$ that identifies with $f_k$ on $[-1,1]$. We denote by arcosh the inverse function of the hyperbolic cosine, defined from $[ 1 , + \infty [$ to $[ 0 , + \infty [$.
Show that $$\forall x \in ] - \infty , - 1 ] \quad T _ { k } ( x ) = ( - 1 ) ^ { k } \cosh ( k \operatorname { arcosh } ( - x ) ) .$$
Q21 Proof Proof of Set Membership, Containment, or Structural Property View
Let $k$ be a non-negative integer. We define the function $f _ { k }$ from the interval $[ - 1,1 ]$ to itself by $f _ { k } ( x ) = \cos ( k \arccos x )$, and $T_k$ is the polynomial of degree $k$ that identifies with $f_k$ on $[-1,1]$. We recall that $A \in \mathcal{S}_N^+(\mathbb{R})$ is not proportional to the identity, $\lambda_1$ is the smallest eigenvalue of $A$, $\lambda_N$ is the largest eigenvalue of $A$, and $$\Lambda _ { k } = \{ Q \in \mathbb { R } [ X ] \mid \operatorname { deg } ( Q ) \leq k , Q ( 0 ) = 1 \}$$
We set $\omega _ { k } = \frac { 1 } { T _ { k } \left( - \frac { \lambda _ { N } + \lambda _ { 1 } } { \lambda _ { N } - \lambda _ { 1 } } \right) }$. Show that $\omega _ { k }$ is well defined, that the polynomial $$Q _ { k } ( X ) = \omega _ { k } T _ { k } \left( \frac { 2 X - \lambda _ { 1 } - \lambda _ { N } } { \lambda _ { N } - \lambda _ { 1 } } \right)$$ is an element of $\Lambda _ { k }$, and that the maximum of $| Q _ { k } ( t ) |$ on $\left[ \lambda _ { 1 } , \lambda _ { N } \right]$ is $\left| \omega _ { k } \right|$.
Q22 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
We set $\theta = \operatorname { arcosh } \left( \frac { \lambda _ { N } + \lambda _ { 1 } } { \lambda _ { N } - \lambda _ { 1 } } \right) > 0$ and $\alpha = e ^ { - \theta }$. Show that $\alpha$ is a root of the polynomial $$X ^ { 2 } - 2 \frac { \lambda _ { N } + \lambda _ { 1 } } { \lambda _ { N } - \lambda _ { 1 } } X + 1$$ and deduce the expression of $\alpha$ in terms of the quantity $\beta = \frac { \lambda _ { N } + \lambda _ { 1 } } { \lambda _ { N } - \lambda _ { 1 } }$.
Q23 Solving quadratics and applications Matrix Norm, Convergence, and Inequality View
We denote by $\kappa = \lambda _ { N } / \lambda _ { 1 }$. Show that the real number $\alpha$ from question 22 equals $\alpha = \frac { \sqrt { \kappa } - 1 } { \sqrt { \kappa } + 1 }$ and deduce that $$\left\| e _ { k } \right\| _ { A } \leq 2 \left\| e _ { 0 } \right\| _ { A } \left( \frac { \sqrt { \kappa } - 1 } { \sqrt { \kappa } + 1 } \right) ^ { k }$$
Q24 Proof Projection and Orthogonality View
We keep the notations from the previous parts. In particular, we still denote by $x _ { k }$ the minimizer of $J$ on $x _ { 0 } + H _ { k }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$
Show that there exists a family $\left( p _ { 0 } , \ldots , p _ { m - 1 } \right)$ of vectors in $\mathbb { R } ^ { N }$ such that
(i) For all $k \in \{ 1 , \ldots , m \}$, the family $( p _ { 0 } , \ldots , p _ { k - 1 } )$ is a basis of $H _ { k }$.
(ii) The family is orthogonal with respect to the inner product associated with $A$, that is $$\forall i , j \in \{ 0 , \ldots , m - 1 \} \quad i \neq j \Rightarrow \left\langle A p _ { i } , p _ { j } \right\rangle = 0$$
Q25 Proof Projection and Orthogonality View
We keep the notations from the previous parts. In particular, we still denote by $x _ { k }$ the minimizer of $J$ on $x _ { 0 } + H _ { k }$.
Assume that a family $\left( p _ { 0 } , \ldots , p _ { m - 1 } \right)$ of vectors satisfying the properties of question 24 is known. Show that $x _ { k + 1 } - x _ { k }$ is then collinear with $p _ { k }$ for all integer $k \in \{ 0 , \ldots , m - 1 \}$.
Q26 Proof Direct Proof of a Stated Identity or Equality View
We are given $x _ { 0 } \in \mathbb { R } ^ { N }$. We consider the finite real sequences $\left( \alpha _ { k } \right)$ and $\left( \beta _ { k } \right)$, as well as the finite sequences $\left( \tilde { x } _ { k } \right) , \left( \tilde { r } _ { k } \right)$ and $\left( \tilde { p } _ { k } \right)$ of elements of $\mathbb { R } ^ { N }$, constructed according to the following recurrence relations, for $k \in \{ 0 , \ldots , m - 1 \}$, $$\begin{aligned} \alpha _ { k } & = \frac { \left\| \tilde { r } _ { k } \right\| ^ { 2 } } { \left\langle A \tilde { p } _ { k } , \tilde { p } _ { k } \right\rangle } \\ \tilde { x } _ { k + 1 } & = \tilde { x } _ { k } + \alpha _ { k } \tilde { p } _ { k } \\ \tilde { r } _ { k + 1 } & = \tilde { r } _ { k } - \alpha _ { k } A \tilde { p } _ { k } \\ \beta _ { k } & = \frac { \left\| \tilde { r } _ { k + 1 } \right\| ^ { 2 } } { \left\| \tilde { r } _ { k } \right\| ^ { 2 } } \\ \tilde { p } _ { k + 1 } & = \tilde { r } _ { k + 1 } + \beta _ { k } \tilde { p } _ { k } \end{aligned}$$ with $\tilde { x } _ { 0 } = x _ { 0 } , \tilde { r } _ { 0 } = b - A x _ { 0 }$ and $\tilde { p } _ { 0 } = \tilde { r } _ { 0 }$.
Show that the following properties are satisfied:
(i) For all $k \in \{ 0 , \ldots , m - 1 \}$, for all $i \in \{ 0 , \ldots , k - 1 \}$, we have $$\left\langle \tilde { r } _ { i } , \tilde { r } _ { k } \right\rangle = 0 , \left\langle \tilde { p } _ { i } , \tilde { r } _ { k } \right\rangle = 0 , \left\langle \tilde { p } _ { i } , A \tilde { p } _ { k } \right\rangle = 0$$
(ii) For all $k \in \{ 0 , \ldots , m \} , \tilde { x } _ { k }$ is identified with $x _ { k }$, the minimizer of $J$ on $x _ { 0 } + H _ { k }$ defined in question 13.
(iii) For all $k \in \{ 0 , \ldots , m \} , \tilde { r } _ { k }$ is identified with $r _ { k } = b - A x _ { k }$.
(iv) The family $\left( \tilde { p } _ { 0 } , \ldots , \tilde { p } _ { k } \right)$ is a basis of $H _ { k + 1 }$, for all $k \in \{ 0 , \ldots , m - 1 \}$.