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
2020 x-ens-maths__pc

18 maths questions

Q1 Matrices Projection and Orthogonality View
Show that for all matrices $A$ and $B$ in $\operatorname{Sym}^+(p)$ and all non-negative real numbers $a$ and $b$, we have $aA + bB \in \operatorname{Sym}^+(p)$.
Q2 Matrices Projection and Orthogonality View
Show that if $v \in \mathbf{R}^p$ then the matrix $A = \left(A_{ij}\right)_{(i,j) \in \llbracket 1,p \rrbracket^2}$ defined by $A = vv^T$ is in $\mathrm{Sym}^+(p)$.
Q3 Matrices Matrix Decomposition and Factorization View
(a) Show that for all $u, v \in \mathbf{R}^p$, we have $\left(uu^T\right) \odot \left(vv^T\right) = (u \odot v)(u \odot v)^T$.
(b) Let $A \in \operatorname{Sym}^+(p)$. We denote $\lambda_1, \ldots, \lambda_p$ the eigenvalues (with multiplicity) of $A$ and $\left(u_1, \ldots, u_p\right)$ an orthonormal family of associated eigenvectors. Show that $\lambda_k \geq 0$ for all $k \in \llbracket 1, p \rrbracket$ and that $A = \sum_{k=1}^{p} \lambda_k u_k u_k^T$.
(c) Deduce that if $A, B \in \operatorname{Sym}^+(p)$ then $A \odot B \in \operatorname{Sym}^+(p)$.
Q4 Matrices Matrix Algebra and Product Properties View
Let $n \in \mathbf{N}$ and $P : \mathbf{R} \rightarrow \mathbf{R}$ defined by $P(x) = \sum_{k=0}^{n} a_k x^k$ where $a_k \geq 0$ for all $k \in \llbracket 0, n \rrbracket$ a polynomial with non-negative coefficients.
(a) Verify that $P[A] = \sum_{k=0}^{n} a_k A^{(k)}$ for all matrices $A \in \mathcal{M}_p(\mathbf{R})$.
(b) Show that if $A \in \operatorname{Sym}^+(p)$ then $P[A] \in \operatorname{Sym}^+(p)$.
Q5 Matrices Matrix Norm, Convergence, and Inequality View
We set, for all $n \geq 0$ and all $x \in \mathbf{R}$, $P_n(x) = \sum_{k=0}^{n} \frac{x^k}{k!}$ where $k!$ denotes the factorial of $k$.
Let $A \in \operatorname{Sym}^+(p)$.
(a) Show that for all $(i,j) \in \llbracket 1,p \rrbracket^2$, we have $$\lim_{n \rightarrow +\infty} P_n[A]_{ij} = \exp\left(A_{ij}\right)$$
(b) Show that $\exp[A] \in \operatorname{Sym}^+(p)$.
(c) Let $u \in \mathbf{R}^p$. Show that $\exp[A] \odot \left(uu^T\right) \in \operatorname{Sym}^+(p)$.
Q6 Matrices Projection and Orthogonality View
Let $d \in \mathbf{N}_*$. We consider a $p$-tuple $\left(x_i\right)_{1 \leq i \leq p}$ of elements of $\mathbf{R}^d$ and the matrix $$A = \left(\left\langle x_i, x_j \right\rangle\right)_{(i,j) \in \llbracket 1,p \rrbracket^2}$$ where $\langle a, b \rangle$ denotes the usual inner product between two vectors $a$ and $b$ of $\mathbf{R}^d$. We denote $|a| = \sqrt{\langle a, a \rangle}$ the norm of $a$.
(a) Show that $A \in \operatorname{Sym}^+(p)$.
(b) We denote $u \in \mathbf{R}^p$ the vector with coordinates $\left(\exp\left(-\frac{|x_1|^2}{2}\right), \ldots, \exp\left(-\frac{|x_p|^2}{2}\right)\right)$. Show that $\left(\exp[A] \odot \left(uu^T\right)\right)_{ij} = \exp\left(-\frac{|x_i - x_j|^2}{2}\right)$ for all $(i,j) \in \llbracket 1,p \rrbracket^2$.
(c) Let $\lambda > 0$ and $K \in \mathcal{M}_p(\mathbf{R})$ the matrix defined by $K_{ij} = \exp\left(-\frac{|x_i - x_j|^2}{2\lambda}\right)$ for all $(i,j) \in \llbracket 1,p \rrbracket^2$. Show that $K \in \operatorname{Sym}^+(p)$.
Q7 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Let $\lambda > 0$ be fixed. We consider the space $\mathcal{C}(\mathbf{R}, \mathbf{R})$ of continuous functions from $\mathbf{R}$ to $\mathbf{R}$. We denote by $\mathcal{E}$ the vector subspace of $\mathcal{C}(\mathbf{R}, \mathbf{R})$ defined by $$\mathcal{E} = \left\{ f \in \mathcal{C}(\mathbf{R}, \mathbf{R}) \mid \exists (a, A) \in \left(\mathbf{R}_*^+\right)^2 \text{ such that } \forall y \in \mathbf{R},\ |f(y)| \leq A \exp\left(-y^2/a\right) \right\}$$
For all $(f, g) \in \mathcal{E}^2$, show that $fg$ is integrable on $\mathbf{R}$.
Q8 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View
For all $f, g \in \mathcal{E}$, we define $$(f \mid g) = \int_{-\infty}^{+\infty} f(y) g(y) \,\mathrm{d}y.$$
We define $\gamma_\lambda : \mathbf{R} \rightarrow \mathbf{R}$ by $\gamma_\lambda(y) = \exp\left(-y^2/\lambda\right)$ and for all $x \in \mathbf{R}$, $\tau_x(f)(y) = f(y-x)$.
(a) Show that for all $f \in \mathcal{E}$, we have $(f \mid f) \geq 0$ with equality if and only if $f = 0$.
(b) Show that for all $x \in \mathbf{R}$, $\tau_x\left(\gamma_\lambda\right)$ belongs to $\mathcal{E}$.
Q9 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View
We define $\gamma_\lambda(y) = \exp(-y^2/\lambda)$ and for all $x \in \mathbf{R}$, $\tau_x(f)(y) = f(y-x)$. For all $f, g \in \mathcal{E}$, $(f \mid g) = \int_{-\infty}^{+\infty} f(y)g(y)\,\mathrm{d}y$.
(a) Let $a > 0$. Show that there exists $c \geq 0$ such that for all $x \in \mathbf{R}$ we have $$\int_{-\infty}^{+\infty} \exp\left(-\frac{(y-x)^2}{\lambda}\right) \exp\left(-\frac{y^2}{a}\right) \mathrm{d}y = c \exp\left(-\frac{x^2}{a+\lambda}\right).$$ Hint: One may show the equality $$\frac{(y-x)^2}{\lambda} + \frac{y^2}{a} = \frac{a+\lambda}{a\lambda}\left(y - \frac{ax}{a+\lambda}\right)^2 + \frac{x^2}{a+\lambda}.$$
(b) Let $g \in \mathcal{E}$. We consider $C(g) : \mathbf{R} \rightarrow \mathbf{R}$ defined for all $x \in \mathbf{R}$ by $$C(g)(x) = \left(\tau_x(\gamma_\lambda) \mid g\right)$$ Show that $C(g) \in \mathcal{E}$.
(c) Show that $C : \mathcal{E} \rightarrow \mathcal{E}$ defines an endomorphism of $\mathcal{E}$.
Q10 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View
Let $\lambda > 0$ be fixed. We consider the set $\mathcal{G}$ of functions $g$ that can be written in the form $g = \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda)$ where $n$ is a strictly positive integer and $\left((x_i, \alpha_i)\right)_{1 \leq i \leq n}$ is a family of elements of $\mathbf{R}^2$: $$\mathcal{G} = \left\{ \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda) \mid n \in \mathbf{N}_*, \forall i \in \llbracket 1,n \rrbracket\ (x_i, \alpha_i) \in \mathbf{R} \times \mathbf{R} \right\}$$
Show that $\mathcal{G}$ is a vector subspace of $\mathcal{E}$ and that it is the smallest vector subspace of $\mathcal{E}$ that contains all functions $\tau_x(\gamma_\lambda)$ for arbitrary $x \in \mathbf{R}$.
Q11 Continuous Probability Distributions and Random Variables Convolution Properties and Computation View
Let $\lambda > 0$ be fixed. We use the notation $\mathcal{G}$, $\mathcal{H} = C(\mathcal{G})$, $\gamma_\lambda$, $\tau_x$, and $(f \mid g) = \int_{-\infty}^{+\infty} f(y)g(y)\,\mathrm{d}y$ as defined previously.
(a) Show that there exists $c_\lambda > 0$ such that for all $(x, x') \in \mathbf{R} \times \mathbf{R}$ we have $$\left(\tau_x(\gamma_\lambda) \mid \tau_{x'}(\gamma_\lambda)\right) = c_\lambda \gamma_{2\lambda}(x - x')$$ Hint: One may note that $\frac{1}{\lambda}\left((y-x)^2 + (y-x')^2\right) = \frac{2}{\lambda}\left(y - (x+x')/2\right)^2 + \frac{1}{2\lambda}(x'-x)^2$.
(b) Deduce that for all $x \in \mathbf{R}$ $$C\left(\tau_x(\gamma_\lambda)\right) = c_\lambda \tau_x(\gamma_{2\lambda})$$ and that $$\mathcal{H} = \left\{ \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_{2\lambda}) \mid n \in \mathbf{N}_*, \forall i \in \llbracket 1,n \rrbracket\ (x_i, \alpha_i) \in \mathbf{R} \times \mathbf{R} \right\}$$
Q12 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View
We use the notation $\mathcal{G}$, $\mathcal{H}$, $\gamma_{2\lambda}$, $\tau_x$, $C$, $D$ as defined previously.
(a) Let $n \in \mathbf{N}_*$ and $(x_i)_{1 \leq i \leq n}$ a family of real numbers such that for all $i, j \in \llbracket 1,n \rrbracket$ we have $x_i \neq x_j$ when $i \neq j$. Show that the function $\sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_{2\lambda})$ is zero if and only if $\alpha_i = 0$ for all $1 \leq i \leq n$ (Hint: One may proceed by induction on $n$).
(b) Deduce that there exists a unique linear map $D$ from $\mathcal{H}$ to $\mathcal{G}$ such that $D \circ C(g) = g$ for all $g \in \mathcal{G}$ and $C \circ D(h) = h$ for all $h \in \mathcal{H}$.
(c) Show that for all $h \in \mathcal{H}$, we have for all $x \in \mathbf{R}$ that $h(x) = \left(\tau_x(\gamma_\lambda) \mid D(h)\right)$.
Q13 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View
For all $(h_1, h_2) \in \mathcal{H} \times \mathcal{H}$, we denote $(h_1 \mid h_2)_{\mathcal{H}} = c_\lambda \left(D(h_1) \mid D(h_2)\right)$ where $c_\lambda$ is introduced in question (11a).
(a) Verify that $(\mid)_{\mathcal{H}}$ defines an inner product on $\mathcal{H}$.
(b) Show that for all $x \in \mathbf{R}$ and $h \in \mathcal{H}$ we have $h(x) = \left(\tau_x(\gamma_{2\lambda}) \mid h\right)_{\mathcal{H}}$.
(c) Show that for all $h \in \mathcal{H}$ we have $$\|h\|_\infty \leq \|h\|_{\mathcal{H}}$$ where we have set $\|h\|_\infty = \sup_{x \in \mathbf{R}} |h(x)|$ and $\|h\|_{\mathcal{H}} = (h \mid h)_{\mathcal{H}}^{1/2}$.
Q14 Matrices Linear System and Inverse Existence View
We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers. We assume that the $x_i$ are pairwise distinct. We denote $\mathcal{S} = \{h \in \mathcal{H} \mid h(x_i) = a_i\}$ the set of $h \in \mathcal{H}$ that equal $a_i$ at $x_i$ for all $i \in \llbracket 1,p \rrbracket$. We denote $J : \mathcal{H} \rightarrow \mathbf{R}$ defined by $J(h) = \frac{1}{2}\|h\|_{\mathcal{H}}^2$ and $J_* = \inf\{J(h) \mid h \in \mathcal{S}\}$. We denote $\mathcal{S}_* = \{h \in \mathcal{S} \mid J(h) = J_*\}$.
Show that $\mathcal{S}_*$ has at most one element.
Q15 Matrices Projection and Orthogonality View
We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers with the $x_i$ pairwise distinct. We use the notation $\mathcal{S}$, $J$, $J_*$, $\mathcal{S}_*$, $(\mid)_{\mathcal{H}}$ as defined previously.
Let $\mathcal{H}_0 = \{h \in \mathcal{H} \mid h(x_i) = 0\ \forall i \in \llbracket 1,p \rrbracket\}$ and $\tilde{h} \in \mathcal{S}_*$ (we assume here $\mathcal{S}_*$ non-empty).
Show that $\left(\tilde{h} \mid h_0\right)_{\mathcal{H}} = 0$ for all $h_0 \in \mathcal{H}_0$.
Q16 Matrices Projection and Orthogonality View
We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers with the $x_i$ pairwise distinct. We use the notation $\mathcal{S}$, $\mathcal{S}_*$, $\mathcal{H}_0$, $(\mid)_{\mathcal{H}}$, $\gamma_{2\lambda}$, $\tau_x$ as defined previously. We denote $\mathcal{H}_0^\perp = \{h \in \mathcal{H} \mid \forall h_0 \in \mathcal{H}_0\ (h \mid h_0)_{\mathcal{H}} = 0\}$ the orthogonal subspace to $\mathcal{H}_0$ in $\mathcal{H}$.
(a) Show that $\mathcal{S}_* = \mathcal{S} \cap \mathcal{H}_0^\perp$.
(b) Show that $\mathcal{H}_0^\perp$ contains the vector subspace of $\mathcal{H}$ spanned by the functions $\tau_{x_i}(\gamma_{2\lambda})$ for $i \in \llbracket 1,p \rrbracket$.
Q17 Matrices Linear System and Inverse Existence View
We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers with the $x_i$ pairwise distinct. Let $\alpha \in \mathbf{R}^p$ (resp. $a \in \mathbf{R}^p$) be the vector with coordinates $(\alpha_i)_{i \in \llbracket 1,p \rrbracket}$ (resp. $(a_i)_{i \in \llbracket 1,p \rrbracket}$) and $h_\alpha = \sum_{i=1}^p \alpha_i \tau_{x_i}(\gamma_{2\lambda})$. The matrix $K$ is defined by $K_{ij} = \exp\left(-\frac{|x_i - x_j|^2}{2\lambda}\right)$ (here in the case $d=1$).
(a) Show that $h_\alpha$ is an interpolant if and only if $K\alpha = a$ where $K$ is the matrix introduced in question (6) (here in the case $d = 1$).
(b) Show that $K$ is invertible.
Q18 Matrices Linear System and Inverse Existence View
We fix two $p$-tuples $(x_i)_{i \in \llbracket 1,p \rrbracket}$ and $(a_i)_{i \in \llbracket 1,p \rrbracket}$ of real numbers with the $x_i$ pairwise distinct. We use the notation $\alpha_*$, $h_\alpha$, $K$, $a$, $\mathcal{S}_*$, $J_*$, $(\mid)_{\mathcal{H}}$ as defined previously.
Deduce that there exists $\alpha_* \in \mathbf{R}^p$ such that $\mathcal{S}_* = \{h_{\alpha_*}\}$ and calculate the value of $J_*$ in terms of $K$ and $a$.