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-d__mp

10 maths questions

QII.3 Groups Group Homomorphisms and Isomorphisms View
Let $M$ be an additive subgroup of $\mathbf { Z } ^ { n }$ with $n \in \mathbf { N }$ (we agree that $\mathbf { Z } ^ { 0 }$ is the trivial group). We propose to prove by induction on $n$ the following result: (*) There exists $r \in \mathbf { N }$ such that the abelian group $M$ is isomorphic to $\mathbf { Z } ^ { r }$. a) Verify the cases $n = 0$ and $n = 1$. We now assume the result is true for $n - 1$. Let $p : \mathbf { Z } ^ { n } \rightarrow \mathbf { Z }$ be the projection onto the first coordinate, we denote by $N$ the kernel of $p$ and $N _ { 1 } = M \cap N$, then we set $p ( M ) = a \mathbf { Z }$ with $a \in \mathbf { Z }$. We choose $e _ { 1 } \in M$ such that $p \left( e _ { 1 } \right) = a$. Show that if $a \neq 0$, then the application $$N _ { 1 } \times \mathbf { Z } \rightarrow M , ( x , m ) \mapsto x + m e _ { 1 }$$ is a group isomorphism. b) Deduce (*). c) Show that the integer $r$ such that $M$ is isomorphic to $\mathbf { Z } ^ { r }$ is unique (one may consider the rank of a family of vectors of $\mathbf { Z } ^ { r }$ in the $\mathbf { Q }$-vector space $\mathbf { Q } ^ { r }$).
QII.4 Groups Subgroup and Normal Subgroup Properties View
Show that if an abelian group $M$ has property (F), then every subgroup of $M$ also has it.
QII.5 Groups Ring and Field Structure View
We consider the ring $A = \mathbf { Z } [ X , Y ]$. Let $U$ be the set of elements of $A$ of the form $X Y ^ { k }$ with $k \in \mathbf { N }$, we set $B = \mathcal { A } ( U )$. Let $S$ be a finite subset of $B$. a) Show that there exists $m \in \mathbf { N } ^ { * }$ such that $\mathcal { A } ( S ) \subset \mathcal { A } \left( \left\{ X , X Y , \ldots , X Y ^ { m } \right\} \right)$. b) Show that there exists an integer $N > 0$ such that every element of $\mathcal { A } ( S )$ is a sum of monomials of the form $\alpha X ^ { i } Y ^ { j }$ with $\alpha \in \mathbf { Z }$ and $j \leq i N$. c) Deduce that the ring $B$ does not have property (TF).
QIII.1 Groups Ring and Field Structure View
Let $E$ be a finite subset of $M _ { n } ( A )$. Show that there exists a subring $B$ of $A$ such that: $B$ has property (TF) and for every matrix $M \in E$, all coefficients of $M$ belong to $B$.
QIII.2 Groups Ring and Field Structure View
Let $M$ be a matrix of $M _ { n } ( A )$. The purpose of this question is to generalize to an arbitrary commutative ring $A$ the two formulas recalled in the introduction when $A$ is a field. a) Show that if the ring $A$ is integral, then $M \widetilde { M } = \widetilde { M } M = ( \operatorname { det } M ) I _ { n }$. b) We no longer assume $A$ is integral. Show that the result of a) still holds if there exists a surjective ring morphism $B \rightarrow A$ with $B$ integral. c) Deduce that the result of a) still holds for every commutative ring $A$. d) Prove that if $M$ and $N$ are in $M _ { n } ( A )$, then we have $$\operatorname { det } ( M N ) = \operatorname { det } M \times \operatorname { det } N .$$
QIII.3 Groups Group Actions and Surjectivity/Injectivity of Maps View
Let $r$ and $s$ be strictly positive integers. Let $M \in M _ { s , r } ( A )$. We consider the application $u : A ^ { r } \rightarrow A ^ { s }$ defined by $u ( X ) = M X$, where we identify elements of $A ^ { r }$ and $A ^ { s }$ with column vectors. We assume that $u$ is surjective and that the ring $A$ is not reduced to $\{ 0 \}$. The purpose of this question is to prove that we then have $r \geq s$. For this, we reason by contradiction by assuming $r < s$. a) Show that there exists a matrix $N \in M _ { r , s } ( A )$ such that $M N = I _ { s }$. b) We define matrices of $M _ { s } ( A )$ by blocks: $$\begin{aligned} M _ { 1 } & = \left( \begin{array} { l l } M & 0 \end{array} \right) \\ N _ { 1 } & = \binom { N } { 0 } \end{aligned}$$ In other words, $M _ { 1 }$ is the matrix obtained by adding $s - r$ zero columns to $M$ and $N _ { 1 }$ is the matrix obtained by adding $s - r$ zero rows to $N$. Calculate $M _ { 1 } N _ { 1 }$. c) Reach a contradiction and conclude. d) We assume that $r = s$. Show the equivalence of the following properties: i) The application $u$ is surjective; ii) The determinant $\operatorname { det } M$ belongs to $A ^ { * }$; iii) There exists $N \in M _ { r } ( A )$ such that $M N = N M = I _ { r }$. iv) The application $u$ is bijective.
QIV.1 Groups Binary Operation Properties View
Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We denote by $G L _ { r } ( A )$ the set of matrices of $M _ { r } ( A )$ that satisfy the equivalent properties of question III.3.d). a) Show that matrix multiplication induces a group structure on $G L _ { r } ( A )$. b) We define a relation on $M _ { s , r } ( A )$ by $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$. Show that this is an equivalence relation.
QIV.2 Groups Group Homomorphisms and Isomorphisms View
Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We say that two matrices $M$ and $N$ of $M _ { s , r } ( A )$ are $A$-equivalent if $M \sim N$ (where $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$). If $M \in M _ { s , r } ( \mathbf { Z } )$ and $k$ is an integer at most equal to $\min ( r , s )$, we denote by $m _ { k } ( M )$ the gcd of the minors of size $k$ of $M$. Let $M$ and $N$ be two $\mathbf { Z }$-equivalent matrices of $M _ { s , r } ( \mathbf { Z } )$. Show that for all $k \leq \min ( r , s )$, we have $m _ { k } ( M ) = m _ { k } ( N )$ (one may begin by showing that $m _ { k } ( M )$ divides $m _ { k } ( N )$).
QIV.3 Groups True/False with Justification View
We consider two matrices of $M _ { 2 } ( \mathbf { Z } )$ that are $\mathbf { C }$-equivalent. Are they always $\mathbf { Z }$-equivalent?
QV.1 Groups Decomposition and Basis Construction View
Let $r$ and $m$ be strictly positive integers with $r \leq m$. We consider a subspace $V$ of the $\mathbf { C }$-vector space $M _ { m } ( \mathbf { C } )$. Throughout the following, we make the following hypothesis: every element of $V$ is a matrix of rank at most $r$. Show that we can assume that $V$ contains the block matrix: $$A = \left( \begin{array} { c c } I _ { r } & 0 \\ 0 & 0 \end{array} \right)$$