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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

2023 e3a-polytech-maths__mp

4 maths questions

Q1 Groups Decomposition and Basis Construction View

Let $n$ be a non-zero natural number. We denote $E _ { n } = \mathbb { R } _ { n } [ X ]$ and for all $k \in \llbracket 0 , n \rrbracket , P _ { k } = X ^ { k }$.
Let $\alpha$ be a real number.

Justify that the family $\mathcal { E } = \left( 1 , X - \alpha , \ldots , ( X - \alpha ) ^ { n } \right)$ is a basis of $E _ { n }$.
Let $P$ be a polynomial in $E _ { n }$. Give without proof the decomposition of $P$ in the basis $\mathcal { E }$ using the successive derivatives of the polynomial $P$.
Suppose that $\alpha$ is a root of order $r \in \llbracket 1 , n \rrbracket$ of $P$. Determine the quotient and remainder of the Euclidean division of $P$ by $( X - \alpha ) ^ { r }$.

To every polynomial $P$ of $E _ { n }$, we associate the polynomial $Q$ defined by: $$Q ( X ) = X P ( X ) - \frac { 1 } { n } \left( X ^ { 2 } - 1 \right) P ^ { \prime } ( X )$$ and we denote by $T$ the application that associates $Q$ to $P$.

Let $k \in \llbracket 0 , n \rrbracket$. Determine $T \left( P _ { k } \right)$.
Show that $T$ is an endomorphism of $E _ { n }$.
Write the matrix $M$ of $T$ in the basis $\mathscr { B } = \left( P _ { 0 } , P _ { 1 } , \ldots , P _ { n } \right)$ of $E _ { n }$.
Suppose that $\lambda$ is a real eigenvalue of the endomorphism $T$ and let $P$ be a monic polynomial, eigenvector associated with the eigenvalue $\lambda$.
1. [7.1.] Show that $P$ has degree $n$.
2. [7.2.] Let $z _ { 0 }$ be a complex root of $P$ with multiplicity order $r \in \mathbb { N } ^ { * }$. Prove that $z _ { 0 } ^ { 2 } - 1 = 0$.
3. [7.3.] Deduce an expression for $P$.
Determine the eigenvectors of the endomorphism $T$. Is the endomorphism $T$ diagonalisable?

Q2 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View

Let $a$ and $b$ be two real numbers with $a > 0$. Choose without justification the correct expression for $a ^ { b }$: $(A) : \mathrm { e } ^ { b \ln ( a ) }$ $(B) : \mathrm { e } ^ { a \ln ( b ) }$ $(C) : \mathrm { e } ^ { \ln ( a ) \ln ( b ) }$.
Let $x$ and $y$ be two real numbers such that $x < y$ and $t$ a real number in $]0,1[$. Compare $t ^ { x }$ and $t ^ { y }$.
Give, without proof, the power series expansion of the real exponential function and give its domain of validity.
We consider the function $\Gamma$ defined by $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$. We admit that this function is defined on $]0 , + \infty[$ and that, for all strictly positive real $x$: $$\Gamma ( x + 1 ) = x \Gamma ( x )$$ Calculate $\Gamma ( 1 )$ and deduce, by using a proof by induction, the value of $\Gamma ( n + 1 )$ for $n \in \mathbb { N }$.
For $x \in \mathbb { R }$, we denote, when it makes sense: $$F ( x ) = \int _ { 0 } ^ { 1 } t ^ { t ^ { x } } \mathrm { d } t$$ where, as is customary, $t ^ { t ^ { x } } = t^{(t^{x})}$.
1. [5.1.] Determine the domain of definition of $F$.
2. [5.2.] Determine the monotonicity of $F$.
3. [5.3.] Prove that for all non-negative real $x$, we have: $F ( x ) \geqslant \frac { 1 } { 2 }$.
4. [5.4.] Prove that $F$ is continuous on its domain of definition.
5. [5.5.] Determine $\lim _ { x \rightarrow + \infty } F ( x )$ and $\lim _ { x \rightarrow - \infty } F ( x )$. The theorems used will be cited with precision and we will ensure that their hypotheses are well verified.
6. [5.6.] Then carefully draw up the table of variations of $F$ and give a general sketch of its representative curve in an orthonormal coordinate system. We will admit that $F ^ { \prime } ( 0 ) = \frac { 1 } { 4 }$ and we will draw the tangent line at the point with abscissa $x = 0$.
Let $x$ be a strictly positive real number. For every natural number $n$, we denote by $g _ { n }$ the function defined on $]0,1]$ by $g _ { n } ( t ) = \frac { t ^ { n x } \ln ^ { n } ( t ) } { n ! }$.
1. [6.1.] Prove that the series of functions $\sum _ { n \in \mathbb { N } } g _ { n }$ converges pointwise on $]0,1]$ and give its sum.
2. [6.2.] Prove that, for every natural number $n , \int _ { 0 } ^ { 1 } \left| g _ { n } ( t ) \right| \mathrm { d } t = \frac { 1 } { n ! } \frac { \Gamma ( n + 1 ) } { ( n x + 1 ) ^ { n + 1 } }$.
3. [6.3.] Finally establish that we have: $$F ( x ) = \sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } } { ( 1 + n x ) ^ { n + 1 } }$$

Q3 Differential equations Integral Equations Reducible to DEs View

We denote by $E$ the vector space of functions with real values continuous on $\mathbb { R } _ { + }$. For every element $f$ of $E$ and all $x \in \mathbb { R } _ { + }$ we set $F ( x ) = \int _ { 0 } ^ { x } f ( u ) \mathrm { d } u$.

Justify that $F$ is of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$ and give for all $x \in \mathbb { R } _ { + }$ the expression of $F ^ { \prime } ( x )$.
Let $\Psi : f \in E \mapsto \Psi ( f )$ defined by: $\forall x \in \mathbb { R } _ { + } , \Psi ( f ) ( x ) = \int _ { 0 } ^ { 1 } f ( x t ) \mathrm { d } t$.
Express, for all strictly positive real $x$, $\Psi ( f ) ( x )$ using $F ( x )$.
Justify that the function $\Psi ( f )$ is continuous on $\mathbb { R } _ { + }$ and give the value of $\Psi ( f ) ( 0 )$.
Show that $\Psi$ is an endomorphism of $E$.
Surjectivity of $\Psi$
Let $h : x \in \mathbb { R } _ { + } \longmapsto h ( x ) = \left\{ \begin{array} { l l } x \sin \left( \frac { 1 } { x } \right) & \text { for } x > 0 \\ 0 & \text { for } x = 0 \end{array} \right.$.
1. [5.1.] Show that the function $h$ is continuous on $\mathbb { R } _ { + }$.
2. [5.2.] Is the function $h$ of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$?
3. [5.3.] Let $g \in \operatorname { Im } ( \Psi )$. Show that the function $x \mapsto x g ( x )$ is of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$.
4. [5.4.] Do we have $h \in \operatorname { Im } ( \Psi )$?
5. [5.5.] Conclude.
Show that $\Psi$ is injective.
Search for the eigenvectors of $\Psi$
1. [7.1.] Justify that 0 is not an eigenvalue of $\Psi$.
  Let $\mu \in \mathbb { R }$. We consider the differential equation $(L)$ on $\mathbb { R } _ { + } ^ { * }$: $$y ^ { \prime } + \frac { \mu } { x } y = 0$$
2. [7.2.] Solve $(L)$ on $\mathbb { R } _ { + } ^ { * }$.
3. [7.3.] Determine the solutions of $(L)$ that can be extended by continuity on $\mathbb { R } _ { + }$.
4. [7.4.] Then determine the eigenvalues of $\Psi$ and the associated eigenspaces.
Let $n \in \mathbb { N } , n > 1$. For $i \in \llbracket 1 , n \rrbracket$, we set: $$f _ { i } : x \in \mathbb { R } _ { + } \longmapsto f _ { i } ( x ) = x ^ { i } \text { and } g _ { i } : x \in \mathbb { R } _ { + } \longmapsto g _ { i } ( x ) = \begin{cases} x ^ { i } \ln ( x ) & \text { for } x > 0 \\ 0 & \text { for } x = 0 \end{cases}$$ We denote $\mathscr { B } = \left( f _ { 1 } , \ldots , f _ { n } , g _ { 1 } , \ldots , g _ { n } \right)$ and $F _ { n }$ the vector subspace of $E$ generated by $\mathscr { B }$.
1. [8.1.] We want to show that the family $\mathcal { B } = \left( f _ { 1 } , \ldots , f _ { n } , g _ { 1 } , \ldots , g _ { n } \right)$ is a basis of $F _ { n }$.
  Let $\left( \alpha _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket }$ and $\left( \beta _ { j } \right) _ { j \in \llbracket 1 , n \rrbracket }$ be scalars such that $\sum _ { i = 1 } ^ { n } \alpha _ { i } f _ { i } + \sum _ { j = 1 } ^ { n } \beta _ { j } g _ { j } = 0$.
  1. [8.1.1.] Show that $\alpha _ { 1 } = \beta _ { 1 } = 0$. One may simplify the expression (*) by $x$ when $x$ is non-zero.
  2. [8.1.2.] Let $p \in \llbracket 1 , n - 1 \rrbracket$. Suppose that $\alpha _ { 1 } = \cdots = \alpha _ { p } = \beta _ { 1 } = \cdots = \beta _ { p } = 0$. Prove that $\alpha _ { p + 1 } = \beta _ { p + 1 } = 0$.
  3. [8.1.3.] Conclude and determine the dimension of the vector space $F _ { n }$.
2. [8.2.] Where we prove that $\Psi$ induces an endomorphism on $F _ { n }$
  1. [8.2.1.] Let $x > 0$ and $p \in \mathbb { N } ^ { * }$. Show that the integral $\int _ { 0 } ^ { x } t ^ { p } \ln ( t ) \mathrm { d } t$ is convergent and calculate it.
  2. [8.2.2.] Deduce that $\Psi$ induces an endomorphism $\Psi _ { n }$ on $F _ { n }$.
3. [8.3.] Give the matrix of the application $\Psi _ { n }$ in the basis $\mathcal { B }$.
4. [8.4.] Prove that $\Psi _ { n }$ is an automorphism of $F _ { n }$.
5. [8.5.] Let $z : x \in \mathbb { R } _ { + } \longmapsto z ( x ) = \left\{ \begin{array} { l l } \left( x + x ^ { 2 } \right) \ln ( x ) & \text { for } x > 0 \\ 0 & \text { for } x = 0 \end{array} \right.$. After verifying that $z \in F _ { n }$, determine $\Psi _ { n } ^ { - 1 } ( z )$.

Q4 Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View

Let $n$ be a non-zero natural number.

Let $p$ be a vector projection of rank $r \in \mathbb { N }$.
1. [1.1.] Give, as a function of $r$, a matrix $W$ of $p$ in an adapted basis.
2. [1.2.] Give the possible spectra of $W$.
3. [1.3.] Compare $\boldsymbol { \operatorname { rg } } ( W )$ and $\boldsymbol { \operatorname { tr } } ( W )$.
4. [1.4.] Calculate $\boldsymbol { \operatorname { det } } ( W )$.

We consider the family $X _ { 1 } , \ldots , X _ { n }$ of independent random variables defined on the same probability space $( \Omega , \mathscr { A } , \mathbb { P } )$ all following the Bernoulli distribution with parameter $p \in ]0,1[$.
Let $M$ be a discrete random variable from $\Omega$ to $\mathscr { M } _ { n } ( \mathbb { R } )$ such that for all $\omega$ in $\Omega , M ( \omega )$ is diagonalisable and similar to $\Delta ( \omega ) = \operatorname { diag } \left( X _ { 1 } ( \omega ) , \ldots , X _ { n } ( \omega ) \right)$.

We denote by $T$ the random variable $\mathbf { tr } ( M )$.
1. [2.1.] Determine $T ( \Omega )$, that is the set of values taken by the random variable $T$.
2. [2.2.] Give the probability distribution of $T$ and the expectation of the random variable $T$.
Deduce the probability distribution of the random variable $R = \mathbf { rg } ( M )$.
We denote by $D$ the random variable $\boldsymbol { \operatorname { det } } ( M )$.
1. [4.1.] Determine $D ( \Omega )$.
2. [4.2.] Give the probability distribution of $D$ and calculate the expectation of the random variable $D$.
We propose to determine the probability of the event $Z$: ``the eigenspaces of the matrix $M$ all have the same dimension''.
1. [5.1.] We denote by $V$ the event: ``$M$ has only one eigenvalue''. Calculate $\mathbb { P } ( V )$.
2. [5.2.] Suppose $n$ is odd. Determine $\mathbb { P } ( Z )$.
3. [5.3.] Suppose $n$ is even and set $n = 2 r$. Calculate $\mathbb { P } ( T = r )$. Deduce $\mathbb { P } ( Z )$.
For all $\omega \in \Omega$, we denote $U ( \omega ) = \left( \begin{array} { c } X _ { 1 } ( \omega ) \\ \vdots \\ X _ { n } ( \omega ) \end{array} \right) \in \mathscr { M } _ { n , 1 } ( \mathbb { R } )$ and $A ( \omega ) = U ( \omega ) \times ( U ( \omega ) ) ^ { \top } = \left( a _ { i j } ( \omega ) \right) _ { ( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } }$.
1. [6.1.] Let $\omega \in \Omega$. Determine, for all pairs $( i , j ) \in \llbracket 1 , n \rrbracket ^ { 2 } , a _ { i j } ( \omega )$.
2. [6.2.] Give the probability distribution of each random variable $a _ { i j }$.
3. [6.3.] Show that $\operatorname { tr } ( A ) = \sum _ { i = 1 } ^ { n } X _ { i }$.
4. [6.4.] Determine the values taken by the random variable $\boldsymbol { \operatorname { rg } } ( A )$.
5. [6.5.] For all $\omega$ in $\Omega$, give the eigenvalues of the matrix $A ( \omega )$.
6. [6.6.] Determine the probability distribution of the random variable $\mathbf { rg } ( A )$.