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
2023 x-ens-maths-b__mp

10 maths questions

Q16 Roots of polynomials Convergence proof and limit determination View
Prove Theorem 1: Let $n \in \mathbb { N }$ be a natural integer and let $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ be monic. Let $\lambda \in \mathbb { R }$ be a root of $P _ { \mid t = 0 }$ of multiplicity $d$. Then there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n - d } [ X ] \right)$ monic such that $P = F G$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.
Q18 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$ and we set $\chi = \operatorname { det } \left( X I _ { n } - M \right) \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$.
Show that $M _ { \mid t = 0 }$ admits a real eigenvalue.
Q19 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$ and we set $\chi = \operatorname { det } \left( X I _ { n } - M \right) \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$. By Theorem 1, there exists $\rho _ { 1 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 1 } \leqslant \rho$ such that $\chi$ factors in the form $\chi = F G$ with $F \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathbb { R } _ { n - d } [ X ] \right)$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.
Only in this question, we assume that $d = n$. Show that there exists a symmetric matrix $M _ { 0 } \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$ such that $M = \lambda I _ { n } + t M _ { 0 }$ for all $t \in U _ { \rho _ { 1 } }$.
Q20 Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$; we thus have $A , B \in \mathscr { D } _ { \rho _ { 1 } } \left( S _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 1 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$.
Show that there exist two matrices $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ such that:
  • $\operatorname { im } \left( B _ { 0 } U \right) = \operatorname { im } \left( B _ { 0 } \right)$,
  • $\operatorname { im } \left( A _ { 0 } V \right) = \operatorname { im } \left( A _ { 0 } \right)$ and
  • the block matrix $\left( B _ { 0 } U \mid A _ { 0 } V \right)$ is invertible.
Q21 Invariant lines and eigenvalues and vectors Linear System and Inverse Existence View
We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$ and $B = G ( M )$, and $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 1 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ where $U \in \mathscr { M } _ { n , d } ( \mathbb { R } )$ and $V \in \mathscr { M } _ { n , n - d } ( \mathbb { R } )$ are as in question 20.
Show that there exists $\rho _ { 2 } \in \mathbb { R } _ { + } ^ { * }$, $\rho _ { 2 } \leqslant \rho _ { 1 }$ such that $Q \in \operatorname { GL } _ { n } \left( \mathscr { D } _ { \rho _ { 2 } } ( \mathbb { R } ) \right)$. (One may use the result of question 6.)
Q22 Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$. For $a \in U _ { \rho _ { 2 } }$, we set $A _ { a } = A _ { \mid t = a }$ and $B _ { a } = B _ { \mid t = a }$. We consider a real number $a \in U _ { \rho _ { 2 } }$.
22a. Show that $\operatorname { im } \left( B _ { a } U \right) \oplus \operatorname { im } \left( A _ { a } V \right) = \mathbb { R } ^ { n }$.
22b. Show the equalities:
  • $\operatorname { im } \left( B _ { a } U \right) = \operatorname { im } \left( B _ { a } \right) = \operatorname { ker } \left( A _ { a } \right)$ and
  • $\operatorname { im } \left( A _ { a } V \right) = \operatorname { im } \left( A _ { a } \right) = \operatorname { ker } \left( B _ { a } \right)$.
(One may begin by showing the inclusions from left to right, then use a dimension argument.)
Q23 Invariant lines and eigenvalues and vectors Linear Transformation and Endomorphism Properties View
We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$.
Show that $Q ^ { - 1 } \cdot M \cdot Q = \operatorname { Diag } \left( M _ { 1 } , M _ { 2 } \right)$ with $M _ { 1 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { d } ( \mathbb { R } ) \right) , M _ { 2 } \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n - d } ( \mathbb { R } ) \right)$.
Q24 Invariant lines and eigenvalues and vectors Matrix Algebraic Properties and Abstract Reasoning View
We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $A = F ( M )$, $B = G ( M )$, $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$.
Show that, for all $a \in U _ { \rho _ { 2 } }$, the direct sum $\operatorname { im } \left( B _ { a } U \right) \oplus \operatorname { im } \left( A _ { a } V \right) = \mathbb { R } ^ { n }$ of question 22a is orthogonal for the standard inner product on $\mathbb { R } ^ { n }$.
Q25 Invariant lines and eigenvalues and vectors Matrix Algebraic Properties and Abstract Reasoning View
We consider $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. We fix a real eigenvalue $\lambda$ of $M _ { \mid t = 0 }$ and denote by $d$ its multiplicity as a root of $\chi _ { \mid t = 0 }$ where $\chi = \det(XI_n - M)$. We set $Q = ( B U \mid A V ) \in \mathscr { D } _ { \rho _ { 2 } } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ and $Q^{-1} \cdot M \cdot Q = \operatorname{Diag}(M_1, M_2)$.
Show that there exists $\rho _ { 3 } \in \mathbb { R } _ { + } ^ { * }$ such that $\rho _ { 3 } \leqslant \rho _ { 2 }$ and matrices $R _ { 1 } \in \mathrm { GL } _ { d } \left( \mathscr { D } _ { \rho _ { 3 } } ( \mathbb { R } ) \right) , R _ { 2 } \in \mathrm { GL } _ { n - d } \left( \mathscr { D } _ { \rho _ { 3 } } ( \mathbb { R } ) \right)$ such that the matrix $Q \cdot \operatorname { Diag } \left( R _ { 1 } , R _ { 2 } \right)$ is orthogonal. (One may use the result of question 17.)
Q26 Invariant lines and eigenvalues and vectors Matrix Algebraic Properties and Abstract Reasoning View
Prove Theorem 2: Let $M \in \mathscr { D } _ { \rho } \left( S _ { n } ( \mathbb { R } ) \right)$. Then there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and an orthogonal matrix $P \in \mathscr { D } _ { r } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ such that $P ^ { \mathrm { T } } \cdot M \cdot P$ is diagonal.