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 mines-ponts-maths1__pc

21 maths questions

Q1 Matrices Symplectic and Orthogonal Group Properties View
Show that a matrix $S \in S_n(\mathrm{R})$ belongs to $S_n^+(\mathrm{R})$ if, and only if, $\mathrm{Sp}(S) \subset \mathbf{R}_+$.
Similarly, we will admit in the rest of the problem that: $S \in S_n^{++}(\mathrm{R})$ if, and only if, $\operatorname{Sp}(S) \subset \mathbf{R}_+^\star$.
Q2 Matrices Structured Matrix Characterization View
Show that $S_n^+(\mathrm{R})$ and $S_n^{++}(\mathrm{R})$ are convex subsets of $M_n(\mathrm{R})$. Are they vector subspaces of $M_n(\mathrm{R})$?
Q3 Matrices Matrix Decomposition and Factorization View
Show that, if $A \in S_n^{++}(\mathrm{R})$, there exists $S \in S_n^{++}(\mathrm{R})$ such that $A = S^2$.
Q4 Proof Prove a general algebraic or analytic statement by induction View
Let $I$ be an interval of $\mathbf{R}$. Let $f : I \rightarrow \mathbf{R}$ be a convex function. Show that, for all $p \in \mathbf{N}^\star$, for all $(\lambda_1, \ldots, \lambda_p) \in (\mathbf{R}_+)^p$ such that $\sum_{i=1}^p \lambda_i = 1$ and for all $(x_1, \ldots, x_p) \in I^p$, we have: $$f\left(\sum_{i=1}^p \lambda_i x_i\right) \leq \sum_{i=1}^p \lambda_i f(x_i)$$ Hint: You may proceed by induction on $p$.
Q5 Proof Prove a general algebraic or analytic statement by induction View
Let $M \in S_n^+(\mathbf{R})$ be a non-zero matrix. Show the inequality $\frac{\operatorname{Tr}(M)}{n} \geq \operatorname{det}^{1/n}(M)$.
Hint: You may show that $x \mapsto -\ln(x)$ is convex on $\mathbf{R}_+^\star$.
Q7 Proof Matrix Norm, Convergence, and Inequality View
Let $M \in S_n^+(\mathbf{R})$ be a non-zero matrix. You may use without proving it the inequality: $$\forall (x_1, \ldots, x_n) \in (\mathbf{R}_+)^n, \quad 2\max\{x_1, \ldots, x_n\}\left(\frac{1}{n}\sum_{k=1}^n x_k - \prod_{k=1}^n x_k^{1/n}\right) \geq \frac{1}{n}\sum_{k=1}^n \left(x_k - \prod_{j=1}^n x_j^{1/n}\right)^2.$$ Deduce that $$\frac{\operatorname{Tr}(M)}{n} - \operatorname{det}^{1/n}(M) \geq \frac{\left\|M - \operatorname{det}^{1/n}(M) I_n\right\|_2^2}{2n\|M\|_2}.$$
Q9 Curve Sketching Convexity and inflection point analysis View
Study the convexity of the function $t \mapsto \ln(1 + \mathrm{e}^t)$.
Q10 Proof Direct Proof of an Inequality View
Show the inequality $$\forall (A, B) \in S_n^{++}(\mathrm{R})^2, \quad \operatorname{det}^{1/n}(A + B) \geq \operatorname{det}^{1/n}(A) + \operatorname{det}^{1/n}(B)$$
Q11 Proof Determinant of Parametric or Structured Matrix View
Show that, if $A$ and $B$ belong to $S_n^{++}(\mathbf{R})$, then: $$\forall t \in [0,1], \quad \operatorname{det}((1-t)A + tB) \geq \operatorname{det}(A)^{1-t} \operatorname{det}(B)^t$$ Justify that this inequality remains valid for $A$ and $B$ only in $S_n^+(\mathbf{R})$.
Q12 Proof Determinant of Parametric or Structured Matrix View
What can be deduced about the function $\ln \circ \det$ on $S_n^{++}(\mathrm{R})$?
Q13 Proof Determinant of Parametric or Structured Matrix View
Let $A \in S_n^{++}(\mathbf{R})$ and let $g : t \in \mathbf{R} \mapsto \operatorname{det}(I_n + tA)$. Express, for all $t \in \mathbf{R}$, $g(t)$ using the eigenvalues of $A$. Deduce that $g$ is of class $C^\infty$ on $\mathbf{R}$.
Q14 Proof Prove inequality or sign of transcendental expression View
Let $A \in S_n^{++}(\mathbf{R})$ and let $f : t \mapsto \ln(\operatorname{det}(I_n + tA))$. Show that $$\forall t \in \mathbf{R}_+, \quad \ln(\operatorname{det}(I_n + tA)) \leq \operatorname{Tr}(A) t.$$
Q15 Proof Regularity and smoothness of transcendental functions View
Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Show that $f_A$ is of class $C^\infty$ on $\mathbf{R}$.
Q17 Proof Computation of a Limit, Value, or Explicit Formula View
Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Show that $f_A(t) \underset{t \rightarrow 0}{=} \operatorname{det}(A) + \operatorname{det}(A) \operatorname{Tr}(A^{-1}M) t + o(t)$.
Hint: You may begin by treating the case where $A = I_n$.
Q18 Proof Determinant and Rank Computation View
Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Determine $f_A'(t)$ for all $t \in ]-\varepsilon_0, \varepsilon_0[$.
Q19 Proof Linear System and Inverse Existence View
Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. We admit that the function $\Phi : t \mapsto (A + tM)^{-1}$ is of class $C^1$ on $]-\varepsilon_0, \varepsilon_0[$. By noting that $\Phi(t) \times (A + tM) = I_n$, show that $$\Phi(t) \underset{t \rightarrow 0}{=} A^{-1} - A^{-1}MA^{-1} t + o(t).$$
Q20 Proof Determinant and Rank Computation View
Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$. We define the application $\varphi_\alpha$ by $$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM).$$ Show that $\varphi_\alpha$ is differentiable on $]-\varepsilon_0, \varepsilon_0[$ and that $$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha'(t) = -\operatorname{Tr}\left((A + tM)^{-1}M\right) \operatorname{det}^{-\alpha}(A + tM).$$
Q21 Invariant lines and eigenvalues and vectors Determinant of Parametric or Structured Matrix View
Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Show that $\varphi_\alpha$ is twice differentiable at 0 and that $$\varphi_\alpha''(0) = \operatorname{det}^{-\alpha}(A)\left(\alpha \operatorname{Tr}^2(A^{-1}M) + \operatorname{Tr}\left((A^{-1}M)^2\right)\right).$$
Q22 Invariant lines and eigenvalues and vectors Diagonalizability and Similarity View
Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Show that $A^{-1}M$ is similar to a real symmetric matrix.
Hint: You may use question 3.
Q23 Invariant lines and eigenvalues and vectors Eigenvalue and Characteristic Polynomial Analysis View
Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Using the fact that $A^{-1}M$ is similar to a real symmetric matrix, deduce that $\varphi_\alpha''(0) \geq 0$.
Q24 Stationary points and optimisation Determinant of Parametric or Structured Matrix View
Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$ and $\varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM)$. Show that, if $\varphi_\alpha''(0) > 0$, then there exists $\eta > 0$ such that for all $t \in ]-\eta, \eta[$, $$\frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM) \geq \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A) - \operatorname{Tr}(A^{-1}M) \operatorname{det}^{-\alpha}(A) t.$$