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
2018 x-ens-maths__psi

20 maths questions

Q1 Second order differential equations Second-order ODE with initial or boundary value conditions View
Let $\lambda \in \mathbb { R }$. Show that the problem
$$\left\{ \begin{array} { l } - v _ { \lambda } ^ { \prime \prime } ( x ) + c ( x ) v _ { \lambda } ( x ) = f ( x ) , x \in [ 0,1 ] \\ v _ { \lambda } ( 0 ) = 0 \\ v _ { \lambda } ^ { \prime } ( 0 ) = \lambda \end{array} \right.$$
admits a unique solution $v _ { \lambda } \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$.
Q2 Second order differential equations Second-order ODE with initial or boundary value conditions View
Show that for all $\lambda \in \mathbb { R } , v _ { \lambda }$ can be expressed in the form:
$$v _ { \lambda } = \lambda w _ { 1 } + w _ { 2 }$$
with $w _ { 1 } \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$ the unique solution of the system
$$\left\{ \begin{array} { l } - w _ { 1 } ^ { \prime \prime } ( x ) + c ( x ) w _ { 1 } ( x ) = 0 , x \in [ 0,1 ] \\ w _ { 1 } ( 0 ) = 0 \\ w _ { 1 } ^ { \prime } ( 0 ) = 1 \end{array} \right.$$
and $w _ { 2 }$ a function independent of $\lambda$ to be characterized.
Q3 Second order differential equations Second-order ODE with initial or boundary value conditions View
Show that $w _ { 1 } ( 1 ) \neq 0$.
Q4 Second order differential equations Second-order ODE with initial or boundary value conditions View
Deduce that there exists a solution $u \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$ to problem (1): $$\left\{ \begin{array} { l } - u ^ { \prime \prime } ( x ) + c ( x ) u ( x ) = f ( x ) , x \in [ 0,1 ] \\ u ( 0 ) = u ( 1 ) = 0 \end{array} \right.$$ Show that this solution is unique.
Q5 Second order differential equations Qualitative Analysis of DE Solutions View
Show that if $f$ is positive, then $u$ is also positive.
Q6 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
Let $V = {}^{ t } \left( v _ { 1 } , \ldots , v _ { n } \right)$ be an eigenvector of $A _ { n }$ associated with a complex eigenvalue $\lambda$, where $A_n$ is the square matrix of size $n$:
$$A _ { n } = \left( \begin{array} { c c c c c c } 2 & - 1 & 0 & \ldots & \ldots & 0 \\ - 1 & 2 & - 1 & \ddots & & \vdots \\ 0 & - 1 & 2 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 2 & - 1 \\ 0 & \ldots & \ldots & 0 & - 1 & 2 \end{array} \right)$$
Show that $\lambda$ is necessarily real and that the components $v _ { i }$ of $V$ satisfy the relation:
$$v _ { i + 1 } - ( 2 - \lambda ) v _ { i } + v _ { i - 1 } = 0, \quad 1 \leq i \leq n$$
where we set $v _ { 0 } = v _ { n + 1 } = 0$.
Q7 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View
Show that every eigenvalue of $A _ { n }$ is in the interval $]0,4[$.
Q8 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
Let $\lambda$ be an eigenvalue of $A _ { n }$.
(a) Show that the complex roots $r _ { 1 } , r _ { 2 }$ of the polynomial
$$P ( r ) = r ^ { 2 } - ( 2 - \lambda ) r + 1$$
are distinct and conjugate.
(b) We set $r _ { 1 } = \overline { r _ { 2 } } = \rho e ^ { i \theta }$ with $\rho > 0$ and $\theta \in \mathbb { R }$.
Show that we necessarily have $\sin ( ( n + 1 ) \theta ) = 0$ and $\rho = 1$.
Q9 Invariant lines and eigenvalues and vectors Spectral properties of structured or special matrices View
Determine the set of eigenvalues of $A _ { n }$ and a basis of eigenvectors.
Q11 Matrices Linear System and Inverse Existence View
By applying the previous results to $A _ { n } + \varepsilon I _ { n }$ with $\varepsilon > 0$, show that all coefficients of $A _ { n } ^ { - 1 }$ are positive.
Q12 Numerical integration Taylor's formula with integral remainder or asymptotic expansion View
Let $n \in \mathbb{N}^*$, $h = \frac{1}{n+1}$, and $x_i = ih$ for all $i \in \{0, \ldots, n+1\}$. Show that for any function $v \in \mathcal { C } ^ { 4 } ( [ 0,1 ] , \mathbb { R } )$, there exists a constant $C \geq 0$, independent of $n$, such that
$$\forall i \in \{ 1 , \ldots , n \} , \left| v ^ { \prime \prime } \left( x _ { i } \right) - \frac { 1 } { h ^ { 2 } } \left( v \left( x _ { i + 1 } \right) + v \left( x _ { i - 1 } \right) - 2 v \left( x _ { i } \right) \right) \right| \leq C h ^ { 2 }$$
Q13 Differential equations Linear System and Inverse Existence View
Let $n \in \mathbb{N}^*$, $h = \frac{1}{n+1}$, and $x_i = ih$ for all $i \in \{0, \ldots, n+1\}$. Show that there exists a unique family of real numbers $\left( u _ { i } \right) _ { 0 \leq i \leq n + 1 }$ satisfying
$$\left\{ \begin{array} { l } - \frac { 1 } { h ^ { 2 } } \left( u _ { i + 1 } + u _ { i - 1 } - 2 u _ { i } \right) + c \left( x _ { i } \right) u _ { i } = f \left( x _ { i } \right) , \text { for } 1 \leq i \leq n \\ u _ { 0 } = u _ { n + 1 } = 0 \end{array} \right.$$
Q14 Differential equations Verification that a Function Satisfies a DE View
We assume (in this question only) that $c ( x ) = 0$ and $f ( x ) = 1$ for all $x \in [ 0,1 ]$. Let $n \in \mathbb{N}^*$, $h = \frac{1}{n+1}$, and $x_i = ih$. We denote by $u$ the exact solution of problem (1): $$\left\{ \begin{array} { l } - u ^ { \prime \prime } ( x ) + c ( x ) u ( x ) = f ( x ) , x \in [ 0,1 ] \\ u ( 0 ) = u ( 1 ) = 0 \end{array} \right.$$ and $(u_i)_{0 \leq i \leq n+1}$ the unique family satisfying $$\left\{ \begin{array} { l } - \frac { 1 } { h ^ { 2 } } \left( u _ { i + 1 } + u _ { i - 1 } - 2 u _ { i } \right) + c \left( x _ { i } \right) u _ { i } = f \left( x _ { i } \right) , \text { for } 1 \leq i \leq n \\ u _ { 0 } = u _ { n + 1 } = 0 \end{array} \right.$$ Show that for all $i \in \{ 0 , \ldots , n + 1 \}$, we have
$$u _ { i } = u \left( x _ { i } \right) = \frac { 1 } { 2 } x _ { i } \left( 1 - x _ { i } \right)$$
Q15 Differential equations Linear System and Inverse Existence View
Show that if $f$ is positive, then $u _ { i } \geq 0$ for all $i \in \{ 0 , \ldots , n + 1 \}$.
Q16 Matrices Matrix Norm, Convergence, and Inequality View
Let $n \in \mathbb { N } ^ { * }$. We define the map $N$ from $\mathcal { M } _ { n } ( \mathbb { R } )$ to $\mathbb { R }$ by the relation:
$$N ( A ) = \sup \left\{ \| A x \| _ { \infty } , \| x \| _ { \infty } \leq 1 \right\}$$
Show that $N$ is a norm on $\mathcal { M } _ { n } ( \mathbb { R } )$ and that if $A = \left[ a _ { i , j } \right] _ { 1 \leq i , j \leq n }$, then
$$N ( A ) = \max _ { i \in \{ 1 , \ldots , n \} } \sum _ { j = 1 } ^ { n } \left| a _ { i , j } \right|$$
Q17 Matrices Matrix Norm, Convergence, and Inequality View
Let $n \in \mathbb { N } ^ { * }$.
(a) Using the results of questions 14 and 15, show that for the matrix $A _ { n }$ defined at the beginning of part 2, we have:
$$N \left( \left( ( n + 1 ) ^ { 2 } A _ { n } \right) ^ { - 1 } \right) \leq \frac { 1 } { 8 }$$
(b) Deduce that for any diagonal matrix $D _ { n } = \left[ d _ { i , j } \right] _ { 1 \leq i , j \leq n }$ such that $d _ { i , i } \geq 0$ for all $i \in \{ 1 , \ldots , n \}$, we also have
$$N \left( \left( ( n + 1 ) ^ { 2 } A _ { n } + D _ { n } \right) ^ { - 1 } \right) \leq \frac { 1 } { 8 }$$
Q18 Differential equations Matrix Norm, Convergence, and Inequality View
Let $u$ be the unique solution of problem (1) and $\left( u _ { i } \right) _ { 0 \leq i \leq n + 1 }$ the family defined by relation (2) for $n \in \mathbb { N } ^ { * }$. Show that there exists a constant $\tilde { C } > 0$, independent of $n$, such that
$$\max _ { 0 \leq i \leq n + 1 } \left| u \left( x _ { i } \right) - u _ { i } \right| \leq \frac { \tilde { C } } { n ^ { 2 } }$$
Hint: one may introduce the vector $X = {}^{ t } \left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { n } \right)$ where we set $\varepsilon _ { i } = u \left( x _ { i } \right) - u _ { i }$ and compute $A _ { n } X$.
Q19 Taylor series Expectation and Moment Inequality Proof View
We assume that $f \in \mathcal { C } ( [ 0,1 ] , \mathbb { R } )$ satisfies: $$\exists \alpha \in ]0,1] , \exists K \geq 0 , \forall ( y , z ) \in [ 0,1 ] ^ { 2 } , | f ( y ) - f ( z ) | \leq K | y - z | ^ { \alpha }$$ and that $c(x) = 0$ for all $x \in [0,1]$. For all $n \in \mathbb{N}^*$, we define: $$B _ { n } f ( X ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } X ^ { k } ( 1 - X ) ^ { n - k }$$
Let $x \in ]0,1[$ and $n \in \mathbb { N } ^ { * }$. We consider $X _ { 1 } , \ldots , X _ { n }$ mutually independent random variables all following the same Bernoulli distribution with parameter $x$. We set
$$S _ { n } = \frac { X _ { 1 } + \cdots + X _ { n } } { n }$$
(a) Express $\mathbb { E } \left( S _ { n } \right) , \mathbb { V } \left( S _ { n } \right)$ and $\mathbb { E } \left( f \left( S _ { n } \right) \right)$ in terms of $x , n$ and the polynomial $B _ { n } f$.
(b) Deduce the inequalities:
$$\sum _ { k = 0 } ^ { n } \left| x - \frac { k } { n } \right| \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } \leq \mathbb { V } \left( S _ { n } \right) ^ { \frac { 1 } { 2 } } \leq \frac { 1 } { 2 \sqrt { n } }$$
Q20 Small angle approximation Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) View
Show that $\lambda ^ { \alpha } \leq 1 + \lambda$ for all real $\lambda > 0$ and deduce the inequality:
$$\left| x - \frac { k } { n } \right| ^ { \alpha } \leq n ^ { - \alpha / 2 } \left( 1 + \sqrt { n } \left| x - \frac { k } { n } \right| \right)$$
for all $x \in ]0,1[ , n \in \mathbb { N } ^ { * }$ and $k \in \{ 0 , \ldots , n \}$.
Q21 Taylor series Expectation and Moment Inequality Proof View
We assume that $f \in \mathcal { C } ( [ 0,1 ] , \mathbb { R } )$ satisfies: $$\exists \alpha \in ]0,1] , \exists K \geq 0 , \forall ( y , z ) \in [ 0,1 ] ^ { 2 } , | f ( y ) - f ( z ) | \leq K | y - z | ^ { \alpha }$$ For all $n \in \mathbb{N}^*$, define $B _ { n } f ( X ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } X ^ { k } ( 1 - X ) ^ { n - k }$.
Let $n \in \mathbb { N } ^ { * }$. Show that
$$\left\| f - B _ { n } f \right\| _ { \infty } \leq \frac { 3 K } { 2 } \frac { 1 } { n ^ { \alpha / 2 } }$$
Hint: One may first express $f ( x ) - B _ { n } f ( x )$ in terms of $\mathbb { E } ( f ( x ) - f \left( S _ { n } \right) )$.