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
2020 x-ens-maths-a__mp_cpge

10 maths questions

Q1 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
Exhibit a matrix $M \in S _ { 2 } ( \mathbb { Q } )$ for which $\sqrt { 2 }$ is an eigenvalue.
Q2 Invariant lines and eigenvalues and vectors Congruence Reasoning and Parity Arguments View
The purpose of this question is to show that $\sqrt { 3 }$ is not an eigenvalue of a matrix in $S _ { 2 } ( \mathbb { Q } )$. We assume that there exists $M \in S _ { 2 } ( \mathbb { Q } )$ such that $\sqrt { 3 }$ is an eigenvalue of $M$.
2a. Using the irrationality of $\sqrt { 3 }$, show that the characteristic polynomial of $M$ is $X ^ { 2 } - 3$.
2b. Show that if $n \in \mathbb { Z }$, then $n ^ { 2 }$ is congruent to 0 or 1 modulo 3.
2c. Show that there does not exist a triple of integers $(x, y, z)$ that are coprime as a whole such that $x ^ { 2 } + y ^ { 2 } = 3 z ^ { 2 }$.
2d. Conclude.
Q3 Matrices Matrix Algebra and Product Properties View
3a. We are given $q \in \mathbb { Q }$, $n \in \mathbb { N } ^ { \star }$ and a matrix $A \in S _ { n } ( \mathbb { Q } )$ such that $A ^ { 2 } = q I _ { n }$. Construct a matrix $B \in S _ { 2 n } ( \mathbb { Q } )$ commuting with the matrix $\left( \begin{array} { c c } A & 0 \\ 0 & A \end{array} \right)$ and such that $B ^ { 2 } = ( q + 1 ) I _ { 2 n }$.
3b. Show that for all $d \geqslant 1$, there exist $n \in \mathbb { N } ^ { \star }$ and matrices $M _ { 1 } , \ldots , M _ { d } \in S _ { n } ( \mathbb { Q } )$ that commute pairwise and such that $M _ { k } ^ { 2 } = k I _ { n }$ for all integers $1 \leqslant k \leqslant d$.
3c. Let $d \geqslant 1$ be an integer. Deduce that if $q _ { 1 } , \ldots , q _ { d } \in \mathbb { Q }$, $q _ { i } > 0$, then there exist $n \in \mathbb { N } ^ { \star }$ and matrices $M _ { 1 } , \ldots , M _ { d } \in S _ { n } ( \mathbb { Q } )$ that commute pairwise and such that $M _ { i } ^ { 2 } = q _ { i } I _ { n }$ for all $1 \leqslant i \leqslant d$.
Q4 Invariant lines and eigenvalues and vectors Compute or factor the characteristic polynomial View
The purpose of this question is to show that $\sqrt [ 3 ] { 2 }$ is not an eigenvalue of a symmetric matrix with coefficients in $\mathbb { Q }$. We reason by contradiction, assuming the existence of a matrix $M \in S _ { n } ( \mathbb { Q } )$ (for some integer $n$) for which $\sqrt [ 3 ] { 2 }$ is an eigenvalue.
4a. Show that $X ^ { 3 } - 2$ divides the characteristic polynomial of $M$. (One may begin by proving that $\sqrt [ 3 ] { 2 } \notin \mathbb { Q }$.)
4b. Conclude.
Q5 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
For $n \in \mathbb { N } ^ { * }$, construct a matrix $M \in S _ { n } ( \mathbb { Q } )$ for which $\cos \left( \frac { 2 \pi } { n } \right)$ is an eigenvalue. (One may begin by constructing an orthogonal matrix with coefficients in $\mathbb { Q }$ that admits $e ^ { 2 i \pi / n }$ as an eigenvalue.)
Q6 Roots of polynomials Reciprocal and antireciprocal polynomial properties View
Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
Let $Q ( X )$ be the reciprocal polynomial of $P ( X )$ defined by $Q ( X ) = X ^ { d } P \left( \frac { 1 } { X } \right)$. Show that: $$\begin{aligned} Q ( X ) & = 1 + a _ { d - 1 } X + \cdots + a _ { 1 } X ^ { d - 1 } + a _ { 0 } X ^ { d } \\ & = \left( 1 - \lambda _ { 1 } X \right) \left( 1 - \lambda _ { 2 } X \right) \cdots \left( 1 - \lambda _ { d } X \right) \end{aligned}$$
Q7 Roots of polynomials Power Series Expansion and Radius of Convergence View
Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
We define the function $f : \mathbb { R } \backslash \left( \mathbb { R } \cap \left\{ \frac { 1 } { \lambda _ { 1 } } , \ldots , \frac { 1 } { \lambda _ { d } } \right\} \right) \rightarrow \mathbb { C }$ by $f ( x ) = \frac { Q ^ { \prime } ( x ) } { Q ( x ) }$.
Show that there exists $r > 0$ such that $f$ is expandable as a power series on $] - r , r [$, and that the power series expansion of $f$ at 0 is written: $$\forall x \in ] - r , r \left[ , \quad f ( x ) = - \sum _ { n = 0 } ^ { \infty } N _ { n + 1 } x ^ { n } \right.$$
Q8 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
8a. Show that if $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$, then $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$.
8b. Conversely, show that if $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$, then $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$.
8c. Deduce that if $\mu _ { 1 } , \ldots , \mu _ { d }$ are complex numbers and if $P ( X ) = \prod _ { i = 1 } ^ { d } \left( X - \mu _ { i } \right)$, then $P ( X ) \in \mathbb { Q } [ X ]$ if and only if $$\forall n \geqslant 1 , \quad \sum _ { i = 1 } ^ { d } \mu _ { i } ^ { n } \in \mathbb { Q }$$
Q9 Roots of polynomials Coefficient and structural properties of special polynomial families View
Let $n \geqslant 1$ and $m \geqslant 1$ be two integers and $\alpha _ { 1 } , \ldots , \alpha _ { n } , \beta _ { 1 } , \ldots , \beta _ { m }$ be complex numbers. We define: $$\begin{aligned} & A ( X ) = \left( X - \alpha _ { 1 } \right) \left( X - \alpha _ { 2 } \right) \cdots \left( X - \alpha _ { n } \right) \\ & B ( X ) = \left( X - \beta _ { 1 } \right) \left( X - \beta _ { 2 } \right) \cdots \left( X - \beta _ { m } \right) \end{aligned}$$
Show that if $A ( X )$ and $B ( X )$ have rational coefficients, then the polynomials $$\prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } \beta _ { j } \right) \text { and } \prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } - \beta _ { j } \right)$$ also have rational coefficients.
Q10 Invariant lines and eigenvalues and vectors Eigenvalue-root connection for matrices or linear operators View
We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that: (i) $z$ is a root of $P$, and (ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $M$ be a symmetric matrix with coefficients in $\mathbb { Q }$. Show that the eigenvalues of $M$ are totally real.