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
2016 x-ens-maths__pc

15 maths questions

Q1 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
Verify that $\varphi$ is of class $\mathscr{C}^{0}$ on $[0, +\infty[$ and $\mathscr{C}^{\infty}$ on $]0, +\infty[$. Give the limit of the derivative $\varphi'(t)$ of $\varphi$ as $t$ tends to 0 in $]0, +\infty[$.
Where $\varphi$ is defined by $$\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$$
Q6 Central limit theorem View
Let $n$ be a strictly positive integer. We consider a family $(X_{k})_{1 \leqslant k \leqslant n}$ of $n$ random variables taking values in $\{1, \ldots, N\}$, pairwise independent and identically distributed, defined on a probability space $(\Omega, \mathscr{A}, \mathbf{P})$. We further assume that $\mathbf{P}(X_{1} = i) = p_{i}$ and that $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$. Show that for all $\epsilon > 0$, we have $\mathbf{P}\left(\left|\frac{1}{n} \ln\left(\prod_{k=1}^{n} p_{X_{k}}\right) + H_{N}(p)\right| \geqslant \epsilon\right)$ tends to 0 as $n$ tends to infinity.
Where $H_{N}(p) = \sum_{i=1}^{N} \varphi(p_{i})$ and $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$
Q7 Proof Existence Proof View
Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $$J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$$ the supremum of $J_{f}$ on $\Sigma_{N}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$ the set of $p$ in $\Sigma_{N}$ for which the supremum is attained.
Show that $\Sigma_{N}(f)$ is non-empty.
Q8 Proof Existence Proof View
Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Let $p \in \Sigma_{N}$.
(a) Suppose that $p_{1} = 0$ and $p_{2} > 0$. Show that there exists $p'$ in $\Sigma_{N}$ such that $J_{f}(p') > J_{f}(p)$ (you may look for $p'$ close to $p$).
(b) Deduce that if $p \in \Sigma_{N}(f)$, then $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$.
Q9 Proof Deduction or Consequence from Prior Results View
Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Let $p \in \Sigma_{N}$. We now assume that $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$. We denote $E_{0} = \{a \in \mathbb{R}^{N} \mid \sum_{i=1}^{N} a_{i} = 0\}$.
(a) Verify that $E_{0}$ is a vector subspace of $\mathbb{R}^{N}$ and give its dimension. Identify the orthogonal $E_{0}^{\perp}$ of $E_{0}$ for the canonical inner product on $\mathbb{R}^{N}$.
(b) Let $a \in E_{0}$ and $\tilde{p} : \mathbb{R} \rightarrow \mathbb{R}^{N}$ defined by $\tilde{p}(t) = p + ta$. Show that there exists $\epsilon > 0$ such that $\tilde{p}(t) \in \Sigma_{N}$ for all $t \in ]-\epsilon, \epsilon[$. Calculate the derivative of $\tilde{p}$ at 0.
(c) Suppose further that $p \in \Sigma_{N}(f)$. Show that for all $a \in E_{0}$, we have $\sum_{i=1}^{N} a_{i}(f_{i} - \ln(p_{i})) = 0$. Deduce that there exists $c \in \mathbb{R}$ such that $\ln(p_{i}) = f_{i} + c$ for all $i \in \{1, \ldots, N\}$.
Q10 Proof Computation of a Limit, Value, or Explicit Formula View
Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Identify $\Sigma_{N}(f)$. Show that $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$.
Q11 Proof Proof That a Map Has a Specific Property View
Let $f \in \mathbb{R}^{N}$ and $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$. We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$.
Show that $F$ is differentiable and calculate its derivative $F'$. Show further that for all $\beta \in ]0, +\infty[$, there exists $p(\beta) \in \Sigma_{N}(\beta f)$ such that $F'(\beta) = -\frac{1}{\beta^{2}} H_{N}(p(\beta))$.
Q12 Proof Computation of a Limit, Value, or Explicit Formula View
We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$ where $f \in \mathbb{R}^{N}$.
Study the limits of $F$ at 0 and at $+\infty$.
Q13 Matrices Matrix Algebra and Product Properties View
Let $(\Omega, \mathscr{A}, \mathbf{P})$ be a probability space and $X : \Omega \rightarrow \{1, \ldots, N\}$ a random variable with distribution $q \in \Sigma_{N}$. Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$ for $(i,j) \in \{1, \ldots, N\} \times \{1, \ldots, d\}$, $p \in \Sigma_{N}$ and $m \in \mathbb{R}^{d}$. We denote by $A \in \mathscr{M}_{d}(\mathbb{R})$ the square matrix of size $d \times d$ defined for all $(k,l) \in \{1, \ldots, d\}^{2}$ by $$A_{lk} = \sum_{i=1}^{N} p_{i}(M_{il} - m_{l})(M_{ik} - m_{k}).$$
Verify that if $Y : \Omega \rightarrow \{1, \ldots, N\}$ is a random variable with distribution $p$, then $A_{lk} = \mathbf{E}((g_{l}(Y) - m_{l})(g_{k}(Y) - m_{k}))$ and then that $A$ is a symmetric matrix such that $\theta^{T} A \theta \geqslant 0$ for all $\theta \in \mathbb{R}^{d}$.
Q14 Matrices Linear System and Inverse Existence View
Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $p \in \Sigma_{N}$, $m \in \mathbb{R}^{d}$, and $A \in \mathscr{M}_{d}(\mathbb{R})$ defined by $A_{lk} = \sum_{i=1}^{N} p_{i}(M_{il} - m_{l})(M_{ik} - m_{k})$. We denote by $\widetilde{M} = (M \mid \mathbf{1}) \in \mathscr{M}_{N,d+1}(\mathbb{R})$ the augmented matrix obtained by adding a column of 1s to the right of $M$.
Let $\theta \in \mathbb{R}^{d}$ such that $\theta^{T} A \theta = 0$. We assume that $p_{i} \neq 0$ for all $1 \leqslant i \leqslant N$.
(a) Show that there exists $c \in \mathbb{R}$, which you will specify, such that for all $i \in \{1, \ldots, N\}$, we have $\sum_{l=1}^{d} M_{il} \theta_{l} = c$.
(b) Show that if $\ker \widetilde{M} = \{0\}$ then $\theta = 0$.
Q15 Proof Proof That a Map Has a Specific Property View
Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$. For all $\theta \in \mathbb{R}^{d}$, let $f(\theta) = M\theta \in \mathbb{R}^{N}$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$ and $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N}$$ where $f(\theta) = (f_{1}(\theta), \ldots, f_{N}(\theta))$. The function $L : \mathbb{R}^{d} \rightarrow \mathbb{R}$ is defined by $$L(\theta) = \ln(Z(\theta)) - q^{T} M\theta.$$
Show that $L$ is of class $\mathscr{C}^{1}$ and calculate its gradient.
Q16 Proof Deduction or Consequence from Prior Results View
Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We denote $$\Sigma_{N}(\bar{g}, g) = \left\{p \in \Sigma_{N} \mid \sum_{i=1}^{N} p_{i} g_{k}(i) = \bar{g}_{k}, 1 \leqslant k \leqslant d\right\}.$$
Show that if $\theta$ is a critical point of $L$ (that is, a point where the gradient of $L$ vanishes) then $M^{T} p(\theta) = M^{T} q$ and $p(\theta) \in \Sigma_{N}(\bar{g}, g)$.
Q17 Proof Direct Proof of a Stated Identity or Equality View
Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$.
Show that $L$ is of class $\mathscr{C}^{2}$ and that for all integers $1 \leqslant l, k \leqslant d$ we have $$\frac{\partial^{2} L}{\partial \theta_{l} \partial \theta_{k}}(\theta) = \sum_{i=1}^{N} p_{i}(\theta)(M_{il} - m_{l}(\theta))(M_{ik} - m_{k}(\theta))$$ where $m(\theta) = M^{T} p(\theta)$.
Q18 Proof Deduction or Consequence from Prior Results View
Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We assume that $\ker \widetilde{M} = \{0\}$ where $\widetilde{M} = (M \mid \mathbf{1})$.
We are interested in this question in the number of points at which the function $L$ attains its minimum.
(a) Show that if $\theta$ and $\theta'$ are two distinct points of $\mathbb{R}^{N}$ such that $L$ has a critical point at $\theta$, then the derivative of $t \rightarrow L(t\theta + (1-t)\theta')$ is strictly increasing on $[0,1]$ and vanishes at $t = 1$.
(b) Deduce that there is at most one critical point for $L$ and conclude on the number of points at which $L$ attains its minimum.
Q19 Proof Deduction or Consequence from Prior Results View
Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We assume that $\ker \widetilde{M} = \{0\}$ and that the function $L$ has a global minimum attained at $\theta_{*}$. We denote $$\Sigma_{N}(\bar{g}, g) = \left\{p \in \Sigma_{N} \mid \sum_{i=1}^{N} p_{i} g_{k}(i) = \bar{g}_{k}, 1 \leqslant k \leqslant d\right\}.$$
(a) Show that $H_{N}(p(\theta_{*})) \geqslant H_{N}(q)$ and then that $H_{N}(p(\theta_{*}))$ is the maximum value of $H_{N}$ on $\Sigma_{N}(\bar{g}, g)$.
(b) Show that $p(\theta_{*})$ is the unique point of $\Sigma_{N}(\bar{g}, g)$ at which $H_{N}$ attains its maximum.