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
2013 centrale-maths1__psi

29 maths questions

QI.A.1 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Justify the equality
$$\forall t \in \mathbb { R } \quad G _ { x } ( t ) = e ^ { i x \sin t } = \sum _ { n = - \infty } ^ { + \infty } \varphi _ { n } ( x ) e ^ { i n t }$$
What can be said about the convergence of the Fourier series of $G _ { x }$ ?
QI.A.2 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
Show that for all $k$ in $\mathbb { N } ^ { * } , \left| \varphi _ { n } ( x ) \right| = o \left( \frac { 1 } { n ^ { k } } \right)$ as $n$ tends to $+ \infty$.
Use Fourier series of successive derivatives of $G _ { x }$.
QI.B Sequences and Series Functional Equations and Identities via Series View
By expressing $G _ { x } ( - t )$ in terms of $G _ { x } ( t )$, show that for $n$ in $\mathbb { Z } , \varphi _ { n } ( x ) \in \mathbb { R }$.
QI.C Sequences and Series Functional Equations and Identities via Series View
Express $G _ { x } ( t + \pi )$ and deduce the following equalities for $n$ in $\mathbb { Z }$ :
$$\varphi _ { n } ( - x ) = ( - 1 ) ^ { n } \varphi _ { n } ( x ) = \varphi _ { - n } ( x )$$
What can be said about the parity of $\varphi _ { n }$ for $n \in \mathbb { Z }$ ?
QI.D Sequences and Series Evaluation of a Finite or Infinite Sum View
Calculate $\sum _ { n = - \infty } ^ { + \infty } \left| \varphi _ { n } ( x ) \right| ^ { 2 }$.
QII.A Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Justify that for real $x$, $\left| \varphi _ { n } ( x ) \right| \leqslant 1$.
QII.B Sequences and Series Power Series Expansion and Radius of Convergence View
Show that for real $x$,
$$\varphi _ { n } ( x ) = \sum _ { k = 0 } ^ { + \infty } \frac { x ^ { k } } { k ! } I _ { n , k } \quad \text { with } \quad I _ { n , k } = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } i ^ { k } e ^ { - i n t } ( \sin t ) ^ { k } \mathrm {~d} t$$
QII.C.1 Sequences and Series Functional Equations and Identities via Series View
Using Euler's formula, justify that for $( n , k )$ in $\mathbb { N } \times \mathbb { N }$,
$$I _ { n , k } = \sum _ { m = 0 } ^ { k } \frac { A _ { m , k } } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { i t ( 2 m - k - n ) } \mathrm { d } t$$
with $A _ { m , k }$ constants to be determined.
QII.C.2 Sequences and Series Evaluation of a Finite or Infinite Sum View
Verify that
$$\begin{cases} I _ { n , k } = 0 & \text { if } n > k \text { or if } k - n \text { is odd } \\ I _ { n , k } = \frac { ( - 1 ) ^ { p } } { 2 ^ { n + 2 p } } \binom { n + 2 p } { n + p } & \text { if } k = n + 2 p \text { with } p \geqslant 0 \end{cases}$$
QII.C.3 Sequences and Series Power Series Expansion and Radius of Convergence View
Deduce the power series development, for $n \geqslant 0$ and $x \in \mathbb { R }$ :
$$\varphi _ { n } ( x ) = \sum _ { p = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { p } } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$
Specify the radius of convergence.
QII.C.4 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Show that $\varphi _ { n }$ is of class $\mathcal { C } ^ { \infty }$ on $\mathbb { R }$.
QII.D Second order differential equations Verifying a particular solution satisfies a second-order ODE View
Let $n$ in $\mathbb { N } ^ { * }$, verify that for real $x$
$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( x ^ { n } \varphi _ { n } ( x ) \right) = x ^ { n } \varphi _ { n - 1 } ( x )$$
QII.E.1 Sequences and Series Limit Evaluation Involving Sequences View
We approximate $\varphi _ { n } ( x )$ using partial sums
$$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$
From which value $p _ { 0 }$ of $p$ is the sequence $\left( a _ { p } \right) _ { p \in \mathbb { N } }$ decreasing?
QII.E.2 Sequences and Series Estimation or Bounding of a Sum View
We approximate $\varphi _ { n } ( x )$ using partial sums
$$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$
Assume $N > p _ { 0 }$. Bound $\left| R _ { N } \right|$ as a function of $(N, n, x)$ with
$$R _ { N } = \sum _ { p = N + 1 } ^ { + \infty } ( - 1 ) ^ { p } a _ { p }$$
Deduce, for fixed $\varepsilon > 0$, a sufficient condition on $N$ for $\left| \varphi _ { n } ( x ) - S _ { N } \right| < \varepsilon$.
QII.E.3 Sequences and Series Algorithmic/Computational Implementation for Sequences and Series View
We approximate $\varphi _ { n } ( x )$ using partial sums
$$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$
Write a Maple or Mathematica function, CalculPhi, with arguments $(n, x, \varepsilon)$ returning an approximate value of $\varphi _ { n } ( x )$ to within $\varepsilon$. The coefficients $a _ { p }$ will be calculated by recursion.
QIII.A.1 Second order differential equations Verifying a particular solution satisfies a second-order ODE View
We study the differential equation with unknown $y$
$$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$
We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$.
Using the power series development of $\varphi _ { n }$ (II.C.3), show that $\varphi _ { n }$ is a solution on $[ 0 , + \infty [$ of (III.1).
QIII.A.2 Second order differential equations Solving homogeneous second-order linear ODE View
We study the differential equation with unknown $y$
$$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$
We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$.
Let $y$ be a solution in $E$ of (III.1). We set $z ( x ) = \sqrt { x } y ( x )$ for all $x \in ] 0 , + \infty [$.
Show that $z$ is a solution in $E$ of a differential equation of the type
$$z ^ { \prime \prime } + q z = 0 \tag{III.2}$$
with $q \in E$. Specify the expression of the function $q$ and verify that $\lim _ { x \rightarrow + \infty } q ( x ) = 1$.
QIII.A.3 Second order differential equations Properties of special function solutions View
We study the differential equation with unknown $y$
$$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$
We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$. Let $z$ be a solution of $z^{\prime\prime} + qz = 0$ (III.2).
Justify that if $z$ is a non-zero solution of (III.2), then for $x > 0 , \left( z ( x ) , z ^ { \prime } ( x ) \right) \neq ( 0,0 )$.
Deduce that if $\alpha$ is a zero of $z$, then there exists a strictly positive real $\eta$ such that $\alpha$ is the only point where $z$ vanishes on $I = ] \alpha - \eta , \alpha + \eta [$. In this case, we say that $\alpha$ is an isolated zero of $z$.
QIII.A.4 Second order differential equations Properties of special function solutions View
Verify that the zeros of $\varphi _ { n }$ on $] 0 , + \infty [$ are isolated.
QIII.B.1 Second order differential equations Solving non-homogeneous second-order linear ODE View
We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Let $x _ { 0 }$ in $] 0 , + \infty [$.
By considering the differential equation (III.3) in the form $z ^ { \prime \prime } + z = g$ with $g ( x ) = \frac { - \lambda } { x ^ { 2 } } z ( x )$, solve it on $] 0 , + \infty [$ using the method of variation of constants. Deduce that there exist two real numbers $A$ and $B$ such that
$$\forall x \in ] 0 , + \infty \left[ \quad z ( x ) = A \cos ( x ) + B \sin ( x ) + \lambda \int _ { x _ { 0 } } ^ { x } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } \right.$$
QIII.B.2 Second order differential equations Qualitative and asymptotic analysis of solutions View
We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Let $x _ { 0 }$ in $] 0 , + \infty [$. We set for $x > 0$
$$h ( x ) = \int _ { x _ { 0 } } ^ { x } | z ( u ) | \frac { \mathrm { d } u } { u ^ { 2 } }$$
a) Show that there exist real constants $\mu$ and $M$ such that $h$ satisfies the differential inequality for $x \geqslant x _ { 0 }$
$$h ^ { \prime } ( x ) - \frac { \mu } { x ^ { 2 } } h ( x ) \leqslant \frac { M } { x ^ { 2 } }$$
Specify the constants $\mu$ and $M$ in terms of $A , B$ and $\lambda$.
b) Deduce that $h$ is bounded on $\left[ x _ { 0 } , + \infty [ \right.$ and then that $z$ is bounded on the same interval.
Multiply by $e ^ { \mu / x }$ and integrate the inequality from the previous question.
QIII.B.3 Second order differential equations Qualitative and asymptotic analysis of solutions View
We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Justify that
$$\int _ { x } ^ { + \infty } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } = O \left( \frac { 1 } { x } \right)$$
near $+ \infty$. Deduce the existence of constants $\alpha$ and $\beta$ such that near $+ \infty$,
$$z ( x ) = \alpha \cos ( x - \beta ) + O \left( \frac { 1 } { x } \right)$$
QIII.B.4 Second order differential equations Qualitative and asymptotic analysis of solutions View
Let $n \in \mathbb { N }$. Show that there exists a pair of real numbers $( \alpha _ { n } , \beta _ { n } )$ such that for $x \rightarrow + \infty$,
$$\varphi _ { n } ( x ) = \frac { \alpha _ { n } } { \sqrt { x } } \cos \left( x - \beta _ { n } \right) + O \left( \frac { 1 } { x \sqrt { x } } \right)$$
QIV.A Second order differential equations Properties of special function solutions View
We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
Using the bound from question II.E.2, show that $\varphi _ { 0 } ( 3 ) < 0$. Deduce that $\varphi _ { 0 }$ has a zero $\left. \alpha _ { 0 } \in \right] 0,3 [$.
We admit that this is the first zero of $\varphi _ { 0 }$, that is, $\varphi _ { 0 }$ does not vanish on $] 0 , \alpha _ { 0 } [$.
QIV.B Second order differential equations Properties of special function solutions View
We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
Using question II.D, show by induction that for all integer $n \geqslant 1$ the function $\varphi _ { n }$ is strictly positive on $] 0 , \alpha _ { 0 } [$.
QIV.C.1 Second order differential equations Properties of special function solutions View
We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$.
Justify that there exists a real $A > 0$ such that for $x > A , q ( x ) > c ^ { 2 }$ ($q$ defined in III.A.2).
QIV.C.2 Second order differential equations Properties of special function solutions View
We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$.
Let $a > A$. We set for $x > 0 , z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, solution of (IV.1). We define the function $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$.
Verify that for $x > 0 , W ^ { \prime } ( x ) = \left( q ( x ) - c ^ { 2 } \right) z ( x ) z _ { 1 } ( x )$.
QIV.C.3 Second order differential equations Properties of special function solutions View
We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$. Let $a > A$, $z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, and $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$.
We denote $\left. I _ { a } = \right] a , a + \pi / c \left[ \right.$ and assume that $\varphi _ { n }$ has no zeros on $I _ { a }$.
Determine the signs of $W ( a ) , W ( a + \pi / c )$ and of $W ^ { \prime }$ on $I _ { a }$ and reach a contradiction. Deduce that $\varphi _ { n }$ has a zero in every interval $I _ { a }$ with $a > A$.
One may distinguish cases according to the sign of $\varphi _ { n }$ on $I _ { a }$.
QIV.D.1 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Let $n \in \mathbb { N }$.
Show that we can order the zeros of $\varphi _ { n }$, that is, there exists a strictly increasing sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ of zeros of $\varphi _ { n }$ such that $\varphi _ { n }$ does not vanish on $] 0 , \alpha _ { 0 } ^ { ( n ) } [$ and on every interval $] \alpha _ { k } ^ { ( n ) } , \alpha _ { k + 1 } ^ { ( n ) } [$ with $k$ in $\mathbb { N }$ and that $\lim _ { k \rightarrow \infty } \alpha _ { k } ^ { ( n ) } = + \infty$.
Construct the sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ by induction on $k$ by showing that the set $\mathcal { Z } _ { k }$ of zeros of $\varphi _ { n }$ in the interval $] \alpha _ { k } ^ { ( n ) } , + \infty [$ has a smallest element.