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
2025 polytechnique-maths-a__mp

27 maths questions

Q7 Proof Existence Proof View
Prove that $\varphi$ admits an extension $\psi$ to $V$ compatible with $u$.
Q8a Matrices Linear Transformation and Endomorphism Properties View
We choose a vector $v_0$ such that $u^{n-1}(v_0)$ is nonzero. Verify that the family $(v_0, u(v_0), \ldots, u^{n-1}(v_0))$ is free and that the subspace $W$ it spans contains $v_0$ and is stable by $u$. Write the matrix of the induced endomorphism $u_W$ in this basis.
Q8b Proof Existence Proof View
Prove that there exists an injective linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. According to Part III, this linear application $\varphi$ admits an extension $\psi : V \rightarrow \mathcal{D}$ compatible with $u$.
Q8c Proof Proof of Set Membership, Containment, or Structural Property View
Verify that the image of $\psi$ is contained in the kernel of $\xi^n$.
Q8d Proof Proof of Stability or Invariance View
Prove that the kernel of $\psi$ is a complement of $W$ stable by $u$.
Q9 Matrices Linear Transformation and Endomorphism Properties View
Let $u$ be a nilpotent endomorphism of a finite-dimensional vector space $V$. Prove that there exists a basis of $V$, a natural integer $s$ and nonzero natural integers $r_1 \geqslant \cdots \geqslant r_s$ in which the matrix of $u$ is block diagonal and whose diagonal blocks are Jordan blocks $J_{r_1}, \ldots, J_{r_s}$ of respective sizes $r_1, \ldots, r_s$.
Q10 Matrices Linear Transformation and Endomorphism Properties View
Prove that the number $s$ and the sizes of the blocks $r_1, \ldots, r_s$ that appear in question $9^\circ$ depend only on $u$ and not on the choice of basis. One may use question $2^\circ$.
Q11a Matrices Diagonalizability and Similarity View
Prove that $h$ is diagonalizable.
Q11b Matrices Linear Transformation and Endomorphism Properties View
Let $j$ be a natural integer strictly less than $N$. By denoting $V_j = \ker(h - \zeta^j \mathrm{id}_V)$ and $V_N = V_0$, verify that $u(V_j) \subset V_{j+1}$.
Q11c Matrices Matrix Algebra and Product Properties View
Calculate, for $k$ relative integer, $h^k \circ u \circ h^{-k}$ and, for $l$ natural integer, $h \circ u^l \circ h^{-1}$.
Q12a Matrices Projection and Orthogonality View
Let $W$ be a vector subspace of $V$ stable by $u$ and $h$. We assume that $W$ admits a complement $W'$ stable by $u$ and we seek a complement of $W$ stable by $u$ and $h$. Let $p$ be the projector onto $W$ parallel to $W'$. Verify that $u$ and $p$ commute.
Q12b Matrices Projection and Orthogonality View
We denote $$\bar{p} = \frac{1}{N} \sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$ Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$.
Q12c Matrices Projection and Orthogonality View
Deduce that $\bar{p}$ is a projector and that its image is $W$.
Q12d Matrices Projection and Orthogonality View
Prove carefully that $\bar{p}$ commutes with $u$ and $h$.
Q12e Matrices Projection and Orthogonality View
Deduce that the kernel of $\bar{p}$ is a complement of $W$ and that it is stable under $u$ and $h$.
Q13a Matrices Eigenvalue and Characteristic Polynomial Analysis View
Let $n$ be the index of $u$, that is, the integer such that $u^{n-1} \neq 0$ and $u^n = 0$. Prove that there exists a vector $v$ such that $v$ is an eigenvector of $h$ and $u^{n-1}(v) \neq 0$.
Q13b Matrices Diagonalizability and Similarity View
Prove that there exists a basis of $V$ in which the matrices of $u$ and $h$ are block diagonal and the diagonal blocks are respectively of the form $$J_r \quad \text{and} \quad D_{r,a} = \operatorname{diag}(\zeta^a, \zeta^{a+1}, \ldots, \zeta^{a+r-1})$$ for $r \in \mathbb{N}^*$ and $a \in \{0, \ldots, N-1\}$ suitable.
Q14a Matrices Linear Transformation and Endomorphism Properties View
In this question, we further assume that $N = 4$ and $\ker(h - \mathrm{id}_V) = \{0\}$. Verify that $u^3 = 0$.
Q14b Matrices Diagonalizability and Similarity View
Construct pairs $(u, h)$ that give rise to six different types of pairs of diagonal blocks $(J_r, D_{r,a})$ in the graded version of the decomposition theorem.
Q14c Matrices Determinant and Rank Computation View
Prove that the number of blocks of each type is determined by the data of the three dimensions $d_j = \dim V_j$ ($1 \leqslant j \leqslant 3$) and the three ranks $r_1 = \operatorname{rg} u_1$, $r_2 = \operatorname{rg} u_2$ and $r_{21} = \operatorname{rg}(u_2 \circ u_1)$.
Q15 Matrices Diagonalizability and Similarity View
Let $(A, B)$ and $(A', B')$ be two pairs of matrices in $\mathcal{M}_{n,m}(\mathbb{C}) \times \mathcal{M}_{m,n}(\mathbb{C})$. Prove that the following conditions are equivalent: (i) $(A, B)$ and $(A', B')$ are simultaneously equivalent; (ii) there exist $P \in \mathrm{GL}_m(\mathbb{C})$ and $Q \in \mathrm{GL}_n(\mathbb{C})$ such that $A' = QAP^{-1}$ and $B' = PBQ^{-1}$; (iii) there exists $R \in \mathrm{GL}_{m+n}(\mathbb{C})$ such that $M_{A',B'} = RM_{A,B}R^{-1}$ and $H = RHR^{-1}$.
Q16a Matrices Matrix Algebra and Product Properties View
Calculate $H^2$, $HMH^{-1}$ and, for a polynomial $P$ of $\mathbb{C}[X]$, calculate $HP(M)H^{-1}$.
Q16b Matrices Eigenvalue and Characteristic Polynomial Analysis View
Prove that if a complex number $\lambda$ is an eigenvalue of $M$, then $-\lambda$ is also an eigenvalue of $M$ with the same multiplicity.
Q16c Matrices Linear Transformation and Endomorphism Properties View
Let $\chi_M$ be the characteristic polynomial of $M$. We write it as $\chi_M = X^r Q$ where $r$ is an integer and $Q$ is a polynomial whose constant coefficient is nonzero. Briefly justify that $$\mathbb{C}^{m+n} = \ker M^r \oplus \ker Q(M)$$ and verify that these subspaces are stable under $H$.
Q17 Matrices Diagonalizability and Similarity View
In this question, we assume that $M$ is nilpotent. Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are respectively of the form $$\left(\begin{array}{cc} 0_r & B_0 \\ A_0 & 0_s \end{array}\right) \quad \text{and} \quad \left(\begin{array}{cc} \mathrm{I}_r & 0 \\ 0 & -\mathrm{I}_s \end{array}\right),$$ where $r$ and $s$ are natural integers with $|r - s| \leqslant 1$ and $A_0$ and $B_0$ form one of the following pairs: $$A_0 = \left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)_{s \times (s+1)} \quad \text{and} \quad B_0 = \left(\begin{array}{ccccc} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right)_{(s+1) \times s};$$ $$A_0 = \mathrm{I}_r \quad \text{and} \quad B_0 = J_r;$$ $$A_0 = J_r \quad \text{and} \quad B_0 = \mathrm{I}_r;$$ $$A_0 = \left(\begin{array}{cccc} 0 & \cdots & \cdots & 0 \\ 1 & \ddots & & \vdots \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right)_{(r+1) \times r} \quad \text{and} \quad B_0 = \left(\begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0 \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots \\ 0 & \cdots & 0 & 1 & 0 \end{array}\right)_{r \times (r+1)}.$$
Q18a Matrices Determinant and Rank Computation View
In this question, we assume that $M$ is invertible. Prove that $m = n$ and that $A$ and $B$ are invertible.
Q18b Matrices Diagonalizability and Similarity View
Prove that $(M, H)$ is simultaneously similar to a pair of block diagonal matrices whose diagonal blocks are of even size and are respectively of the form $$\left(\begin{array}{cc} 0_r & B_1 \\ A_1 & 0_r \end{array}\right) \quad \text{and} \quad \left(\begin{array}{cc} \mathrm{I}_r & 0_r \\ 0_r & -\mathrm{I}_r \end{array}\right),$$ where $$A_1 = \mathrm{I}_r \quad \text{and} \quad B_1 = \lambda \mathrm{I}_r + J_r$$ for $r$ nonzero integer and $\lambda$ nonzero complex suitable.