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 mines-ponts-maths2__mp

28 maths questions

Q1 Roots of polynomials Factored form and root structure from polynomial identities View
Show that $p_0$, the reciprocal polynomial of $p$, satisfies $$\forall x \in \mathbf{R}^* \quad p_0(x) = x^n p(1/x)$$ and deduce that $$p_0 = a_n \prod_{j=1}^{n} \left(1 - \alpha_j X\right)$$
Q2 Roots of polynomials Divisibility and minimal polynomial arguments View
Show that $p \wedge p_0 = 1$ if and only if $p$ has no stable root.
Q3 Roots of polynomials Reciprocal and antireciprocal polynomial properties View
Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Justify that there exists $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
Q4 Roots of polynomials Coefficient and structural properties of special polynomial families View
Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Let $h$ be the polynomial of degree $n$ defined by $h(X) = X p'$, where $p'$ is the derivative polynomial of $p$. We denote by $h_0$ and $(p')_0$ the reciprocal polynomials of $h$ and $p'$ respectively.
Show that $h = np - \lambda (p')_0$, then that $h_0 = \lambda(np - Xp')$.
Q5 Roots of polynomials Multiplicity and derivative analysis of roots View
Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Verify that $p'$ is split over $\mathbf{R}$ then show that $h \wedge h_0 = 1$ and deduce that $p'$ has no stable root.
Q6 Roots of polynomials Proof of polynomial identity or inequality involving roots View
For every integer $j \in \llbracket 1, n \rrbracket$, we denote by $f_j$ the polynomial $$f_j = a_n \prod_{k=j+1}^{n}\left(1 - \alpha_k X\right) \prod_{k=1}^{j-1}\left(X - \alpha_k\right)$$ with, according to standard conventions, $\prod_{k=n+1}^{n}(1-\alpha_k X) = \prod_{k=1}^{0}(X - \alpha_k) = 1$.
Show that if there exist two integers $i, k$ such that $1 \leq i < k \leq n$ and $\alpha_i \alpha_k = 1$, then $\alpha_i$ is a root of each polynomial $f_j$, where $j \in \llbracket 1, n \rrbracket$, and that the family $(f_1, \ldots, f_n)$ is linearly dependent.
Q7 Matrices Linear Transformation and Endomorphism Properties View
Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$, we define the rational function $g_j \in E$ by $$g_j = \frac{f_j}{\prod_{i=1}^{n}(1 - \alpha_i X)}$$ and the map $P_j$, which associates to a rational function $f \in E$ the rational function $$P_j(f) = \frac{(1 - \alpha_j X)f - (1 - \alpha_j^2)f(\alpha_j)}{X - \alpha_j}$$
Show that for every $j \in \llbracket 1, n \rrbracket$, the map $P_j$ is an endomorphism of $E$ and determine its kernel.
Q8 Matrices Linear Transformation and Endomorphism Properties View
Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$ and every $g \in E$, compute $P_j\left(\frac{(X - \alpha_j)g}{1 - \alpha_j X}\right)$.
Q9 Matrices Linear Transformation and Endomorphism Properties View
Until the end of part B, we assume that no root of $p$ is stable.
Deduce that the family $(f_1, \ldots, f_n)$ is linearly independent.
Q10 Matrices Linear Transformation and Endomorphism Properties View
Show that the family $\left((S^\top)^i U\right)_{0 \leq i \leq n-1}$ is a basis of $\mathcal{M}_{n,1}(\mathbf{R})$. The matrices $S$ and $U$ were defined in the preliminary part of the problem.
Q11 Matrices Matrix Algebra and Product Properties View
For every integer $j \in \llbracket 1, n \rrbracket$, we define the matrices $$B_j = S - \alpha_j I_n \quad \text{and} \quad C_j = I_n - \alpha_j S$$
Prove that $$J(p) = \sum_{j=1}^{n} f_j(S)^\top \left(C_j^\top C_j - B_j^\top B_j\right) f_j(S)$$
Q12 Matrices Matrix Algebra and Product Properties View
For every integer $j \in \llbracket 1, n \rrbracket$, we define the matrices $$B_j = S - \alpha_j I_n \quad \text{and} \quad C_j = I_n - \alpha_j S$$
Let $j \in \llbracket 1, n \rrbracket$. Show that $C_j^\top C_j - B_j^\top B_j = (1 - \alpha_j^2) U U^\top$.
Q13 Matrices Matrix Decomposition and Factorization View
We denote by $D$ the diagonal matrix of size $n$: $$D = \operatorname{Diag}\left((1 - \alpha_j^2)_{1 \leq j \leq n}\right)$$ and $V \in \mathcal{M}_n(\mathbf{R})$ the matrix such that for every $j \in \llbracket 1, n \rrbracket$, the $j$-th column of $V$ is $V_j = f_j(S^\top) U$. Show that $$J(p) = V D V^\top.$$
Q14 Matrices Determinant and Rank Computation View
Deduce, using question 6, that if $p$ has a stable root then $J(p)$ is not invertible.
Q15 Matrices Determinant and Rank Computation View
Let two matrices $A, B \in \mathcal{M}_n(\mathbf{R})$ such that there exists a matrix $P \in GL_n(\mathbf{R})$ satisfying $A = P^\top B P$. Show that $d(B) \geq d(A)$ then that $d(B) = d(A)$.
Q16 Matrices Projection and Orthogonality View
For every matrix $M \in S_n(\mathbf{R})$ construct a vector subspace $F_M$ of $\mathcal{M}_{n,1}(\mathbf{R})$ of dimension $\pi(M)$ satisfying condition $(\mathcal{C}_M)$: $$\forall X \in F \setminus \{0_{n,1}\} \quad X^\top M X > 0.$$ We thus have $d(M) \geq \pi(M)$.
Q17 Matrices Projection and Orthogonality View
We want to show that for every matrix $M \in S_n(\mathbf{R})$ we have $\pi(M) = d(M)$. By contradiction, assuming the existence of a vector subspace $G$ of $\mathcal{M}_{n,1}(\mathbf{R})$ of dimension $\dim G > \pi(M)$ satisfying condition $(\mathcal{C}_M)$, show $\dim(F_M^\perp \cap G) \geq 1$, deduce a contradiction and conclude.
Q18 Matrices Eigenvalue and Characteristic Polynomial Analysis View
Prove the Schur-Cohn criterion: If $J(p)$ is invertible then $p$ has no stable root and $\sigma(p) = \pi(J(p))$.
Q19 Matrices Linear Transformation and Endomorphism Properties View
Show, using questions 9 and 13, that if $p$ has no stable root and if $J(p)$ is not invertible then there exists a non-zero polynomial $q$ with real coefficients of degree at most $n-1$ such that $q(S^\top) U = 0_{n,1}$.
Q20 Matrices Determinant and Rank Computation View
Deduce that the matrix $J(p)$ is invertible if and only if $p$ has no stable root.
Q21 Matrices Determinant and Rank Computation View
We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
Show that $J(h)$ is invertible.
Q22 Roots of polynomials Location and bounds on roots View
We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
Show that there exists a real number $\eta > 0$ such that for every $r \in ]1-\eta; 1[$, the polynomial $p(rX)$ is split, has exactly $\sigma(p)$ roots inside the interval $]-1; 1[$ and has no stable root.
Q23 Roots of polynomials Limiting behavior involving polynomial roots or values View
We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
For every real number $r > 0$, we denote by $F(r) = J(p(rX))$.
Show that $$\lim_{r \rightarrow 1^-} \pi\left(\frac{n}{2(r-1)} F(r)\right) = n - \sigma(p)$$
Q24 Applied differentiation Partial derivatives and multivariable differentiation View
We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
For every real number $r > 0$, we denote by $F(r) = J(p(rX))$.
Justify that the map $F : \mathbf{R}_+^* \rightarrow S_n(\mathbf{R})$ is differentiable and that $$F'(1) = 2n(p(S))^\top p(S) - 2S^\top (p'(S))^\top p(S) - 2(p(S))^\top p'(S) S.$$
Q25 Applied differentiation Limit evaluation involving derivatives or asymptotic analysis View
We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
For every real number $r > 0$, we denote by $F(r) = J(p(rX))$.
Deduce, using the results of question 4, that $$\frac{n}{2(r-1)} F(r) \underset{r \rightarrow 1}{=} J(h) + o(1)$$
Q26 Roots of polynomials Eigenvalue-root connection for matrices or linear operators View
We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
We admit that the map defined on $S_n(\mathbf{R})$ with values in $\mathbf{R}^n$ which associates to a symmetric matrix the $n$-tuple of its real eigenvalues counted with their multiplicities, arranged in decreasing order, is continuous.
Deduce that $\sigma(p) = n - 1 - \pi(J(p'))$.
Q27 Roots of polynomials Existence or counting of roots with specified properties View
We consider the general case, without having information on the stability and multiplicity of the roots of $p$, and we seek to calculate $\sigma(p)$.
We construct the two polynomials $f$ and $g$ satisfying $f = p \wedge p_0$ and $p = fg$.
Show that $\sigma(g) = \pi(J(g))$.
Q28 Roots of polynomials Existence or counting of roots with specified properties View
We consider the general case, without having information on the stability and multiplicity of the roots of $p$, and we seek to calculate $\sigma(p)$.
We construct the two polynomials $f$ and $g$ satisfying $f = p \wedge p_0$ and $p = fg$.
Propose a method allowing us to construct a finite number (possibly zero) of polynomials $g_1, \ldots, g_\ell$, whose roots are stable and of multiplicity 1, such that $f = g_1 g_2 \cdots g_\ell$. Express $\sigma(p)$ using $n$, $\deg g$, $\pi(J(g))$, $\ell$, $\pi(J(g))$ as well as $\pi(J(g_1')), \ldots, \pi(J(g_\ell'))$.