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
2025 mines-ponts-maths2__psi

25 maths questions

Q1 First order differential equations (integrating factor) Second-order ODE with initial or boundary value conditions View
Justify that there exists a unique solution $u$ to the Cauchy problem $\left( C _ { \ell } \right)$, give its expression and draw its variation table.
$$\left( C _ { \ell } \right) : \left\{ \begin{array} { l } u ^ { \prime } ( x ) + u ( x ) + 1 = \frac { 1 } { 2 } ( 1 + u ( x ) ) \\ u ( 0 ) = 0 \end{array} . \right.$$
Q2 First order differential equations (integrating factor) Qualitative and asymptotic analysis of solutions View
Show that there exists a unique constant solution of equation $\left( E _ { \ell } \right)$, denoted $\gamma \in \mathbf { R }$, and verify that the solution $u$ found in question 1 satisfies
$$\lim _ { x \rightarrow + \infty } u ( x ) = \gamma .$$
where $\left( E _ { \ell } \right) : \quad u ^ { \prime } ( x ) + u ( x ) + 1 = \frac { 1 } { 2 } ( 1 + u ( x ) )$.
Q3 First order differential equations (integrating factor) Qualitative Analysis of DE Solutions View
Show that $c$ is a constant solution of $(E)$, then that $(E)$ admits exactly two constant solutions denoted $c _ { 1 }$ and $c _ { 2 }$ such that $c _ { 1 } < 0 < c _ { 2 }$. Deduce the value of $c$ as a function of $c _ { 1 }$ and $c _ { 2 }$.
We admit that $y$ is decreasing on $\mathbf { R } _ { + }$ and $\lim _ { x \rightarrow + \infty } y ( x ) = c$, where $c \in \mathbf { R }$. The equation $(E)$ is: $$( E ) : \quad y ^ { \prime } ( x ) + y ( x ) + 1 = \frac { 1 } { 2 } \mathrm { e } ^ { y ( x ) }.$$
Q4 Sequences and series, recurrence and convergence Convergence/Divergence Determination of Numerical Series View
Show that for all $k \in \mathbf { N }$, the real numbers $b _ { k } = \sum _ { n = 1 } ^ { + \infty } \lambda _ { n } ^ { k } a _ { n }$ are well-defined.
The sequences satisfy: $\left| a _ { n } \right| \leq \frac { M } { 2 ^ { n } }$ for some $M \in \mathbf{R}_+^*$, and $\lambda_n$ is strictly increasing with $\lambda_0 = 0$, $\lim_{n\to+\infty} \lambda_n = +\infty$, and $\lambda_n \underset{n\to+\infty}{=} O(n)$.
Q5 Sequences and series, recurrence and convergence Series convergence and power series analysis View
Show that every Dirichlet series $\sum _ { n \geq 0 } f _ { n }$ converges uniformly on $\mathbf { R } _ { + }$. We then denote its sum by $f$. Justify that $f$ is continuous on $\mathbf { R } _ { + }$.
A Dirichlet series satisfies $f_n(x) = a_n e^{-\lambda_n x}$ with $\left| a _ { n } \right| \leq \frac { M } { 2 ^ { n } }$, $\lambda_0 = 0$, $\lim_{n\to+\infty}\lambda_n = +\infty$, and $\lambda_n = O(n)$.
Q6 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Express $f ( 0 )$ and $\lim _ { x \rightarrow + \infty } f ( x )$ in terms of $a _ { 0 }$ and $b _ { 0 }$.
Here $f = \sum_{n\geq 0} f_n$ is the sum of a Dirichlet series with $f_n(x) = a_n e^{-\lambda_n x}$, $\lambda_0 = 0$, and $b_0 = \sum_{n=1}^{+\infty} a_n$.
Q7 Sequences and series, recurrence and convergence Uniform or Pointwise Convergence of Function Series/Sequences View
Let $k \in \mathbf { N } ^ { * }$. Show that $f \in \mathcal { C } ^ { k } \left( \mathbf { R } _ { + } , \mathbf { R } \right)$ and give an expression for $x \mapsto f ^ { ( k ) } ( x )$. Then express $f ^ { ( k ) } ( 0 )$ in terms of $b _ { k }$.
Here $f = \sum_{n\geq 0} f_n$ is the sum of a Dirichlet series with $f_n(x) = a_n e^{-\lambda_n x}$, and $b_k = \sum_{n=1}^{+\infty} \lambda_n^k a_n$.
Q8 Sequences and series, recurrence and convergence Uniform or Pointwise Convergence of Function Series/Sequences View
Show that if $f ( x ) = 0$ for all $x \in \mathbf { R } _ { + }$ then $a _ { n } = 0$ for all $n \in \mathbf { N }$.
Here $f = \sum_{n\geq 0} a_n e^{-\lambda_n x}$ is the sum of a Dirichlet series.
Q9 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Suppose that $y$ is the sum of a Dirichlet series: $$\forall x \in \mathbf { R } _ { + } \quad y ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \mathrm { e } ^ { - \lambda _ { n } x },$$ where $y(0) = 0$ and $\lim_{x\to+\infty} y(x) = c$. Express $a _ { 0 }$ and $b _ { 0 }$ in terms of the constant $c$ introduced in Part I.
Q10 Sequences and series, recurrence and convergence Higher-Order and Special DEs (Proof/Theory) View
Using equation $(E)$ satisfied by $y$, calculate $b _ { 1 }$.
The equation $(E)$ is $y ^ { \prime } ( x ) + y ( x ) + 1 = \frac { 1 } { 2 } \mathrm { e } ^ { y ( x ) }$, and $y$ is the sum of a Dirichlet series $y(x) = \sum_{n=0}^{+\infty} a_n e^{-\lambda_n x}$ with $b_k = \sum_{n=1}^{+\infty} \lambda_n^k a_n$.
Q11 Sequences and series, recurrence and convergence Recursive or implicit derivative computation for series coefficients View
Show that for all $k \in \mathbf { N } ^ { * }$,
$$g ^ { ( k ) } ( 0 ) = ( - 1 ) ^ { k } d _ { k }$$
where the coefficients $d _ { k }$ are defined by
$$d _ { 0 } = 1 , \quad \text { and } \quad \forall k \geq 1 \quad d _ { k } = \sum _ { i = 1 } ^ { k } \binom { k - 1 } { i - 1 } d _ { k - i } b _ { i } ,$$
and $g \in \mathcal { C } ^ { \infty } \left( \mathbf { R } _ { + } , \mathbf { R } \right)$ is defined by $g ( x ) = \mathrm { e } ^ { y ( x ) }$, with $y(x) = \sum_{n=0}^{+\infty} a_n e^{-\lambda_n x}$ and $b_k = \sum_{n=1}^{+\infty} \lambda_n^k a_n$.
Q12 Sequences and series, recurrence and convergence Higher-Order and Special DEs (Proof/Theory) View
Let $k \in \mathbf { N } ^ { * }$. Using equation $(E)$ satisfied by $y$, exhibit a recurrence relation linking $b _ { k + 1 } , b _ { k }$ and $d _ { k }$.
The equation $(E)$ is $y ^ { \prime } ( x ) + y ( x ) + 1 = \frac { 1 } { 2 } \mathrm { e } ^ { y ( x ) }$, with $b_k = \sum_{n=1}^{+\infty} \lambda_n^k a_n$ and $d_k$ as defined in question 11.
Q13 Sequences and series, recurrence and convergence Uniform or Pointwise Convergence of Function Series/Sequences View
Show that
$$\left\| y _ { N } - y \right\| _ { \infty , \mathbf { R } _ { + } } \leq \frac { M } { 2 ^ { N } }$$
and deduce that $y _ { N }$ converges uniformly to $y$ on $\mathbf { R } _ { + }$. Then propose an interval $J \subset \mathbf { R } _ { + }$ where the bound on $\left\| y _ { N } - y \right\| _ { \infty , J }$ would be sharper.
Here $y _ { N } ( x ) = \sum _ { n = 0 } ^ { N } a _ { n } \mathrm { e } ^ { - \lambda _ { n } x }$ and $y ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \mathrm { e } ^ { - \lambda _ { n } x }$ with $\left| a_n \right| \leq \frac{M}{2^n}$.
Q14 Matrices Linear System and Inverse Existence View
Show that $V A = B$ where $V \in \mathcal { M } _ { N } ( \mathbf { R } )$ is a matrix that you will make explicit.
Here $A = \left( a_1, a_2, \ldots, a_N \right)^\top \in \mathbf{R}^N$, $B = \left( \beta_0, \beta_1, \ldots, \beta_{N-1} \right)^\top \in \mathbf{R}^N$, and $\beta_k = \sum_{n=1}^{N} \lambda_n^k a_n$.
Q15 Matrices Linear System and Inverse Existence View
Prove that the system $V A = B$ admits a unique solution $A \in \mathbf { R } ^ { N }$.
Here $V$ is the matrix from question 14, with entries $V_{k,n} = \lambda_n^{k-1}$ for $k = 1,\ldots,N$ and $n = 1,\ldots,N$, and the $\lambda_n$ are strictly increasing real numbers.
Q16 Differential equations Solving Separable DEs with Initial Conditions View
Suppose that $S _ { 0 } = 0$. Give the expression of the solution triplet $( S , I , R )$ of system $( F )$.
The system $(F)$ is: $$( F ) : \left\{ \begin{array} { l } S ^ { \prime } ( x ) = - I ( x ) S ( x ) \\ I ^ { \prime } ( x ) = I ( x ) S ( x ) - I ( x ) \\ R ^ { \prime } ( x ) = I ( x ) \\ S ( 0 ) = S _ { 0 } , \quad I ( 0 ) = I _ { 0 } , \quad R ( 0 ) = R _ { 0 } \end{array} \right.$$ with $S_0 + I_0 + R_0 = 1$ and $S_0, I_0, R_0 \in [0,1]$.
Q17 Differential equations Qualitative Analysis of DE Solutions View
Show that if $S _ { 0 } > 0$ then the function $S$ of the solution triplet $( S , I , R )$ of $( F )$ never vanishes, and deduce that $S$ is strictly positive.
The system $(F)$ is: $$( F ) : \left\{ \begin{array} { l } S ^ { \prime } ( x ) = - I ( x ) S ( x ) \\ I ^ { \prime } ( x ) = I ( x ) S ( x ) - I ( x ) \\ R ^ { \prime } ( x ) = I ( x ) \\ S ( 0 ) = S _ { 0 } , \quad I ( 0 ) = I _ { 0 } , \quad R ( 0 ) = R _ { 0 } \end{array} \right.$$
Q18 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Suppose that $S _ { 0 } > 0$. Show that the function $S$ of the solution triplet $( S , I , R )$ of $(F)$ satisfies the relation
$$\left( - \frac { S ^ { \prime } } { S } \right) ^ { \prime } = - S ^ { \prime } + \frac { S ^ { \prime } } { S }$$
The system $(F)$ is: $$( F ) : \left\{ \begin{array} { l } S ^ { \prime } ( x ) = - I ( x ) S ( x ) \\ I ^ { \prime } ( x ) = I ( x ) S ( x ) - I ( x ) \\ R ^ { \prime } ( x ) = I ( x ) \\ S ( 0 ) = S _ { 0 } , \quad I ( 0 ) = I _ { 0 } , \quad R ( 0 ) = R _ { 0 } \end{array} \right.$$
Q19 Differential equations Solving Separable DEs with Initial Conditions View
With $S _ { 0 } = 1 / 2$, $I _ { 0 } = 1 / 2$, $R _ { 0 } = 0$, and $h : \mathbf { R } _ { + } \rightarrow \mathbf { R }$ defined by
$$\forall x \in \mathbf { R } _ { + } \quad h ( x ) = \ln \left( \frac { S ( x ) } { S _ { 0 } } \right) = \ln ( 2 S ( x ) ),$$
show that $h$ is a solution of the Cauchy problem $(C)$:
$$( C ) : \left\{ \begin{array} { l } y ^ { \prime } ( x ) + y ( x ) + 1 = \frac { 1 } { 2 } \mathrm { e } ^ { y ( x ) } \\ y ( 0 ) = 0 \end{array} \right.$$
using the relation established in question 18.
Q20 Differential equations Convergence and Approximation of DE Solutions View
Show that $S _ { N }$ converges uniformly to $S$ on $\mathbf { R } _ { + }$ when $N \rightarrow + \infty$ and that
$$\left\| S _ { N } - S \right\| _ { \infty , \mathbf { R } _ { + } } \leq \frac { M \mathrm { e } ^ { 2 M } } { 2 ^ { N + 1 } }$$
where $S _ { N } ( x ) = S _ { 0 } \mathrm { e } ^ { y _ { N } ( x ) } = \frac { 1 } { 2 } \exp \left( \sum _ { n = 0 } ^ { N } a _ { n } \mathrm { e } ^ { - \lambda _ { n } x } \right)$, $S(x) = S_0 e^{y(x)} = \frac{1}{2} e^{y(x)}$, and $\left\| y_N - y \right\|_{\infty, \mathbf{R}_+} \leq \frac{M}{2^N}$.
Q21 Discrete Probability Distributions Conditional Probability with Discrete Random Variable View
Let $( s , i , r ) \in E$ where $E = \{ ( s , i , r ) \in \mathbf{N}^3,\, s + i + r = M \}$. Conditional on the event $\left( \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right)$, what is the probability, denoted $p ( i )$, for a susceptible person to be infected during this day?
Each of the $s$ healthy persons meets, independently of the others, $K$ persons chosen at random from the $M$ persons in the total population. As soon as at least one of the meetings is with an infected person, the healthy person becomes infected the next morning.
Q22 Measures of Location and Spread Proof of a General Conditional Expectation or Independence Property View
Let $Z$ be a random variable taking values in $\{ 0 , \ldots , M \}$. Show that:
$$\mathbf { E } [ Z ] = \sum _ { ( s , i , r ) \in E } \left( \sum _ { k = 0 } ^ { M } k \mathbf { P } \left( Z = k \mid \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right) \right) \mathbf { P } \left( \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right).$$
Q23 Measures of Location and Spread Existence of Expectation or Moments View
Justify that for all $n \geq 0$, the random variables $\tilde { S } _ { n } , \tilde { I } _ { n }$ and $\tilde { R } _ { n }$ as well as the random variables $\Delta \tilde { S } _ { n } , \Delta \tilde { I } _ { n }$ and $\Delta \tilde { R } _ { n }$, have finite expectation.
Here $\Delta U_n = U_{n+1} - U_n$ and the random variables take values in $\{0, \ldots, M\}$.
Q24 Discrete Probability Distributions Conditional Expectation and Total Expectation Applications View
Establish the following identity:
$$\mathbf { E } \left[ \Delta \tilde { R } _ { n } \right] = \rho \mathbf { E } \left[ \tilde { I } _ { n } \right]$$
where each infected person recovers at the end of the day with probability $\rho \in ]0,1[$, independently of others, and $\Delta \tilde{R}_n = \tilde{R}_{n+1} - \tilde{R}_n$.
Q25 Discrete Probability Distributions Derive or Prove a Binomial Distribution Identity View
Establish the following identity: for $( s , i , r ) \in E$, for all $k \in \{ 0 , \cdots , s \}$,
$$\mathbf { P } \left( \Delta \tilde { S } _ { n } = - k \mid \left( \tilde { S } _ { n } , \tilde { I } _ { n } , \tilde { R } _ { n } \right) = ( s , i , r ) \right) = \binom { s } { k } ( p ( i ) ) ^ { k } ( 1 - p ( i ) ) ^ { s - k }$$
where $p(i)$ is the probability for a susceptible person to be infected during the day (as found in question 21), and the $s$ susceptible persons act independently.