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
2024 mines-ponts-maths1__mp

25 maths questions

Q1 Sequences and Series Convergence/Divergence Determination of Numerical Series View
Show that for all $\theta \in ] - \pi ; \pi [$, the function $f$ defined by
$$\begin{aligned} f : ] 0 ; + \infty [ & \longrightarrow \mathbf { C } \\ t & \longmapsto \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \end{aligned}$$
is defined and integrable on $] 0 ; + \infty [$, where $x$ is a fixed element of $]0;1[$.
Q2 Integration by Substitution Substitution within a Multi-Part Proof or Derivation View
Let $r$ be the function defined by
$$\begin{aligned} r : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show that the function $r$ is of class $\mathcal { C } ^ { 1 }$ on $] - \pi ; \pi [$ and that:
$$\forall \theta \in ] - \pi ; \pi \left[ , \quad r ^ { \prime } ( \theta ) = - \mathrm { i } e ^ { \mathrm { i } \theta } \cdot \int _ { 0 } ^ { + \infty } \frac { t ^ { x } } { \left( 1 + t \mathrm { e } ^ { \mathrm { i } \theta } \right) ^ { 2 } } \mathrm {~d} t . \right.$$
Hint: let $\beta \in ] 0 ; \pi [$, show that for all $\theta \in [ - \beta ; \beta ]$ and $t \in [ 0 , + \infty [$, $\left| 1 + t e ^ { i \theta } \right| ^ { 2 } \geq \left| 1 + t e ^ { i \beta } \right| ^ { 2 } = ( t + \cos ( \beta ) ) ^ { 2 } + ( \sin ( \beta ) ) ^ { 2 }$.
Q3 Reduction Formulae Prove Regularity or Structural Properties of an Integral-Defined Function View
Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show that the function $g$ is of class $\mathcal { C } ^ { 1 }$ on $] - \pi ; \pi [$ and that for all $\theta \in ] - \pi ; \pi [$,
$$g ^ { \prime } ( \theta ) = \mathrm { i } e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } h ^ { \prime } ( t ) \mathrm { d } t$$
where $h$ is the function defined by
$$\begin{aligned} h : ] 0 ; + \infty [ & \longrightarrow \mathbf { C } \\ t & \longmapsto \frac { t ^ { x } } { 1 + t e ^ { \mathrm { i } \theta } } . \end{aligned}$$
Calculate $h ( 0 )$ and
$$\lim _ { t \rightarrow + \infty } h ( t ) .$$
Deduce that the function $g$ is constant on $] - \pi ; \pi [$.
Q4 Reduction Formulae Establish an Integral Identity or Representation View
Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show that for all $\theta \in ] 0 ; \pi [$,
$$g ( \theta ) \sin ( x \theta ) = \frac { 1 } { 2 \mathrm { i } } \left( g ( - \theta ) e ^ { \mathrm { i } x \theta } - g ( \theta ) e ^ { - \mathrm { i } x \theta } \right) = \sin ( \theta ) \int _ { 0 } ^ { + \infty } \frac { t ^ { x } } { t ^ { 2 } + 2 t \cos ( \theta ) + 1 } \mathrm {~d} t$$
Q5 Reduction Formulae Perform a Change of Variable or Transformation on a Parametric Integral View
Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Deduce that:
$$\forall \theta \in ] 0 ; \pi \left[ , \quad g ( \theta ) \sin ( \theta x ) = \int _ { \cot ( \theta ) } ^ { + \infty } \frac { ( u \sin ( \theta ) - \cos ( \theta ) ) ^ { x } } { 1 + u ^ { 2 } } \mathrm {~d} u , \right.$$
where $\cot ( \theta ) = \frac { \cos ( \theta ) } { \sin ( \theta ) }$.
Q6 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show, using the dominated convergence theorem, that:
$$\lim _ { \theta \rightarrow \pi ^ { - } } g ( \theta ) \sin ( x \theta ) = \int _ { - \infty } ^ { + \infty } \frac { \mathrm { d } u } { 1 + u ^ { 2 } }$$
Q7 Laws of Logarithms Prove a Logarithmic Identity View
Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Deduce that
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \frac { \pi } { \sin ( \pi x ) }$$
Q8 Laws of Logarithms Prove a Logarithmic Identity View
Recall that $x$ is a fixed element of $]0;1[$. Show that:
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \int _ { 0 } ^ { 1 } \left( \frac { t ^ { x - 1 } } { 1 + t } + \frac { t ^ { - x } } { 1 + t } \right) \mathrm { d } t$$
Q9 Taylor series Derive series via differentiation or integration of a known series View
Recall that $x$ is a fixed element of $]0;1[$. Show that:
$$\int _ { 0 } ^ { 1 } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \sum _ { k = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { k } } { k + x }$$
Q10 Integration with Partial Fractions View
Recall that $x$ is a fixed element of $]0;1[$. Establish the identity
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } } { n + x } + \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } } { n + 1 - x }$$
Q11 Sequences and Series Evaluation of a Finite or Infinite Sum View
Recall that $x$ is a fixed element of $]0;1[$. Deduce that:
$$\frac { \pi } { \sin ( \pi x ) } = \frac { 1 } { x } - \sum _ { n = 1 } ^ { + \infty } \frac { 2 ( - 1 ) ^ { n } x } { n ^ { 2 } - x ^ { 2 } }$$
Q12 Taylor series Construct series for a composite or related function View
Recall that $x$ is a fixed element of $]0;1[$. Finally deduce that:
$$\forall y \in ] 0 ; \pi \left[ , \quad \sum _ { n = 1 } ^ { + \infty } \frac { 2 ( - 1 ) ^ { n } y \sin ( y ) } { y ^ { 2 } - n ^ { 2 } \pi ^ { 2 } } = 1 - \frac { \sin ( y ) } { y } . \right.$$
Q13 Standard Integrals and Reverse Chain Rule Convergence and Estimation of Improper Integrals View
Show that the integral
$$\int _ { 0 } ^ { + \infty } \frac { 1 - ( \cos ( t ) ) ^ { 2 p + 1 } } { t ^ { 2 } } \mathrm {~d} t$$
converges and that:
$$\int _ { 0 } ^ { + \infty } \frac { 1 - ( \cos ( t ) ) ^ { 2 p + 1 } } { t ^ { 2 } } \mathrm {~d} t = ( 2 p + 1 ) \int _ { 0 } ^ { + \infty } ( \cos ( t ) ) ^ { 2 p } \frac { \sin ( t ) } { t } \mathrm {~d} t$$
Q14 Sequences and Series Evaluation of a Finite or Infinite Sum View
Show that for all $n \in \mathbf { N } ^ { * }$:
$$\int _ { \frac { \pi } { 2 } + ( n - 1 ) \pi } ^ { \frac { \pi } { 2 } + n \pi } ( \cos ( t ) ) ^ { 2 p } \frac { \sin ( t ) } { t } \mathrm {~d} t = \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \cos ( t ) ) ^ { 2 p } \frac { 2 ( - 1 ) ^ { n } t \sin ( t ) } { t ^ { 2 } - n ^ { 2 } \pi ^ { 2 } } \mathrm {~d} t$$
Q15 Sequences and Series Evaluation of a Finite or Infinite Sum View
Deduce that:
$$\int _ { \frac { \pi } { 2 } } ^ { + \infty } ( \cos ( t ) ) ^ { 2 p } \frac { \sin ( t ) } { t } \mathrm {~d} t = \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \cos ( t ) ) ^ { 2 p } \left( \sum _ { n = 1 } ^ { + \infty } \frac { 2 ( - 1 ) ^ { n } t \sin ( t ) } { t ^ { 2 } - n ^ { 2 } \pi ^ { 2 } } \right) \mathrm { d } t$$
Q16 Integration by Parts Prove an Integral Identity or Equality View
Deduce that:
$$\int _ { 0 } ^ { + \infty } ( \cos ( t ) ) ^ { 2 p } \frac { \sin ( t ) } { t } \mathrm {~d} t = \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \cos ( t ) ) ^ { 2 p } \mathrm {~d} t$$
In the case $p = 0$, this integral is commonly called the ``Dirichlet Integral''.
Q17 Complex numbers 2 Proof of General Complex Number Properties View
Show that:
$$( \cos ( t ) ) ^ { 2 p } = \frac { 1 } { 2 ^ { 2 p } } \left( \binom { 2 p } { p } + 2 \sum _ { k = 0 } ^ { p - 1 } \binom { 2 p } { k } \cos ( 2 ( p - k ) t ) \right)$$
Hint: One may develop $\left( \frac { e ^ { \mathrm{i} t } + e ^ { - \mathrm{i} t } } { 2 } \right) ^ { 2 p }$.
Q18 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View
Deduce that:
$$\int _ { 0 } ^ { + \infty } \frac { 1 - ( \cos ( t ) ) ^ { 2 p + 1 } } { t ^ { 2 } } \mathrm {~d} t = \frac { \pi } { 2 } \frac { ( 2 p + 1 ) ! } { 2 ^ { 2 p } \cdot ( p ! ) ^ { 2 } }$$
Q19 Discrete Random Variables Expectation and Variance of Sums of Independent Variables View
Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Determine, for all $n \in \mathbf { N } ^ { * }$, $E \left( S _ { n } \right)$ and $V \left( S _ { n } \right)$.
Q20 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View
Let $S$ and $T$ be two independent random variables each taking a finite number of real values. Assume that $T$ and $- T$ follow the same distribution.
Show that:
$$E ( \cos ( S + T ) ) = E ( \cos ( S ) ) E ( \cos ( T ) )$$
Q21 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View
Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Deduce that for all $n \in \mathbf { N } ^ { * }$, and for all $t \in \mathbf { R }$:
$$E \left( \cos \left( t S _ { n } \right) \right) = ( \cos ( t ) ) ^ { n } .$$
Q22 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View
Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Let $a , b \in \mathbf { R }$ such that $a \neq 0$ and $| b | \leq | a |$. Show that
$$| a + b | = | a | + \operatorname { sign } ( a ) b$$
where $\operatorname { sign } ( x ) = x / | x |$ for nonzero real $x$. Deduce that:
$$\forall n \in \mathbf { N } ^ { * } , \quad E \left( \left| S _ { 2 n } \right| \right) = E \left( \left| S _ { 2 n - 1 } \right| \right)$$
Q23 Reduction Formulae Establish an Integral Identity or Representation View
Show that for all $s \in \mathbf { R }$
$$\int _ { 0 } ^ { + \infty } \frac { 1 - \cos ( s t ) } { t ^ { 2 } } \mathrm {~d} t = \frac { \pi } { 2 } | s |$$
Q24 Moment generating functions Extract moments from the MGF or characteristic function View
Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Deduce that for all $n \in \mathbf { N } ^ { * }$:
$$E \left( \left| S _ { n } \right| \right) = \frac { 2 } { \pi } \int _ { 0 } ^ { + \infty } \frac { 1 - ( \cos ( t ) ) ^ { n } } { t ^ { 2 } } \mathrm {~d} t$$
Q25 Moment generating functions Extract moments from the MGF or characteristic function View
Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Conclude that:
$$\forall n \in \mathbf { N } ^ { * } , \quad E \left( \left| S _ { 2 n } \right| \right) = E \left( \left| S _ { 2 n - 1 } \right| \right) = \frac { ( 2 n - 1 ) ! } { 2 ^ { 2 n - 2 } ( ( n - 1 ) ! ) ^ { 2 } }$$