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
2023 x-ens-maths-c__mp

20 maths questions

QI.1 Differential equations Solving Separable DEs with Initial Conditions View
We consider the Cauchy problem associated with $F_0$ defined by: $$\forall y \in ]0, +\infty[, \quad F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$$ with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$. We denote $\phi_0$ the solution of this problem on $[0, +\infty[$.
(a) Show that there exists $\varepsilon > 0$ such that for all $t \in ]0, \varepsilon]$ we have $y_{\text{init}} < \phi_0(t) < \theta$.
(b) By considering the function $z_0(t) = \ln\left(\phi_0(t)/\theta\right)$ find the expression of $\phi_0$.
(c) Deduce that $\phi_0$ satisfies $y_{\text{init}} < \phi_0(t) < \theta$ for all $t \in ]0, +\infty[$ and that moreover $\phi_0$ is strictly increasing.
QI.2 Differential equations Solving Separable DEs with Initial Conditions View
For $0 < \mu \leqslant 1$, we consider $F_\mu$ defined by: $$\forall y \in ]0, +\infty[, \quad F_\mu(y) = \frac{a}{\mu} y\left(1 - \left(\frac{y}{\theta}\right)^\mu\right)$$ with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$. By considering the function $z_\mu(t) = \phi_\mu(t)^{-\mu}$ find the expression of the solution $\phi_\mu$ on $[0, +\infty[$ associated with $F_\mu$.
QI.3 Differential equations Qualitative Analysis of DE Solutions View
For $0 < \mu \leqslant 1$, we consider $F_\mu$ defined by: $$\forall y \in ]0, +\infty[, \quad F_\mu(y) = \frac{a}{\mu} y\left(1 - \left(\frac{y}{\theta}\right)^\mu\right)$$ and $F_0$ defined by $F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$, with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$.
(a) Show that $F_\mu$ converges pointwise to $F_0$ as $\mu$ tends to 0.
(b) Show that $\phi_\mu$ converges pointwise to $\phi_0$ as $\mu$ tends to 0.
QII.1 Proof Proof That a Map Has a Specific Property View
Let $K$ be a compact set of $\mathbb{R}$. Let $k > 0$ and $B$ the set of functions from $K$ to $\mathbb{R}^d$ that are $k$-Lipschitz. Show that $B$ is equicontinuous.
QII.2 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. Show that a subset $A \subset C(K, \mathbb{R}^d)$ is relatively compact if and only if every sequence $(f_n)_{n \in \mathbb{N}} \in A^{\mathbb{N}}$ admits a subsequence that converges uniformly to a limit $f \in C(K, \mathbb{R}^d)$.
QII.3 Proof Deduction or Consequence from Prior Results View
Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. By reasoning by contradiction, show that if $A$ is relatively compact then $A$ is equicontinuous.
QII.4 Proof Deduction or Consequence from Prior Results View
Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. We seek to show the following theorem:
Theorem 1: The following two properties are equivalent: - (P1) $A$ is relatively compact. - (P2) $A$ is equicontinuous and for all $x \in K$, the set $A(x) = \{f(x) \mid f \in A\}$ is bounded.
Show that $(P1) \Rightarrow (P2)$.
QII.5 Proof Existence Proof View
We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$. Let $(x_p)_{p \geqslant 0}$ be a sequence of elements of $K$.
(a) Show that there exists a sequence $(\varphi_p)_{p \in \mathbb{N}}$ of strictly increasing functions from $\mathbb{N}$ to $\mathbb{N}$ such that for all $p \geqslant 0$, $f_{\psi_p(n)}(x_p)$ converges as $n$ tends to infinity with $\psi_0 = \varphi_0$ and $\psi_p = \psi_{p-1} \circ \varphi_p$ for $p \geqslant 1$.
(b) Show that for all $p \geqslant 0$, $f_{\psi_n(n)}(x_p)$ converges as $n$ tends to infinity.
QII.6 Proof Existence Proof View
We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$.
(a) Show that we can extract from the sequence $(f_n)_{n \in \mathbb{N}}$ a subsequence that converges pointwise on $\mathbb{Q} \cap K$. We denote $(g_n)_{n \in \mathbb{N}}$ this extraction.
(b) For $x \in K$, show that $(g_n(x))_{n \in \mathbb{N}}$ admits a unique cluster value denoted $g(x)$ and conclude on the pointwise convergence of the sequence $(g_n)_{n \in \mathbb{N}}$ on $K$ to $g$.
QII.7 Proof Deduction or Consequence from Prior Results View
We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$, and $(g_n)_{n \in \mathbb{N}}$ a subsequence converging pointwise on $K$ to $g$.
(a) Show that $g$ is continuous on $K$.
(b) Show that the sequence $(g_n)_{n \in \mathbb{N}}$ converges uniformly to $g$ on $K$. (Hint: you may reason by contradiction.)
(c) Deduce that $(P2) \Rightarrow (P1)$.
QIII.1 Differential equations Euler's Method Approximation View
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. For all $r \geqslant 0$, we denote $B_r$ the closed ball with center $y_{\text{init}}$ and radius $r$.
Show that we can choose $r > 0$ and $T > 0$ such that $B_r \subset \Omega$ and such that for all $N \in \mathbb{N}^*$, we can define by recursion, by setting $\Delta t = \frac{T}{N}$, a sequence $(y_n)_{0 \leqslant n \leqslant N}$ taking values in $B_r$ such that: $$y_0 = y_{\text{init}}, \quad y_{n+1} = y_n + \Delta t F(y_n), \forall n \in \{0, \cdots, N-1\}$$
QIII.2 Differential equations Convergence and Approximation of DE Solutions View
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With $r > 0$, $T > 0$, $\Delta t = \frac{T}{N}$ and the sequence $(y_n)_{0 \leqslant n \leqslant N}$ as defined in question III.1, show that we can construct a unique function $\phi_N$ continuous on $[0, T]$, affine on each interval $[n\Delta t, (n+1)\Delta t]$ for all $n \in \{0, \cdots, N-1\}$ and such that $\phi_N(n\Delta t) = y_n$ for all $n \in \{0, \cdots, N\}$.
QIII.3 Proof Existence Proof View
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With the functions $\phi_N$ constructed in question III.2, show, using Theorem 1, that there exists a subsequence of $\phi_N$ that converges uniformly on $[0, T]$ to a continuous function $\phi$.
QIII.4 Differential equations Convergence and Approximation of DE Solutions View
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. Show that we can define a sequence of step functions $\psi_N : [0, T] \rightarrow \mathbb{R}^d$ such that $\psi_N(t) = \phi_N(t)$ for $t \in \{n\Delta t \mid n \in \{0, \cdots, N\}\}$ and such that: $$\forall N \in \mathbb{N}^*, \phi_N(t) = y_{\text{init}} + \int_0^t F(\psi_N(s))\, ds \text{ for all } t \in [0, T]$$ We will specify the expression of the functions $\psi_N$.
QIII.5 Differential equations Convergence and Approximation of DE Solutions View
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With the step functions $\psi_N$ defined in question III.4, deduce that there exists a subsequence of $\psi_N$ that converges uniformly on $[0, T]$ and specify its limit.
QIII.6 Differential equations Convergence and Approximation of DE Solutions View
We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. Show that $(\phi, T)$ is a solution of the Cauchy problem $$\left\{\begin{array}{l} y'(t) = F(y(t)) \\ y(0) = y_{\text{init}} \end{array}\right.$$ and deduce the following theorem:
Theorem 2: If $F$ is a continuous function, then there exists at least one solution of the Cauchy problem (1).
QIII.7 Differential equations Qualitative Analysis of DE Solutions View
We consider the particular case for $d = 1$ given for all $y \in \mathbb{R}$ by $F(y) = 3|y|^{2/3}$ and $y_{\text{init}} = 0$. Show that this Cauchy problem admits infinitely many global solutions.
QIV.1 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We consider $\mathcal{F} : \mathbb{R}^d \rightarrow \mathcal{P}_c(\mathbb{R}^d)$ taking values in the set $\mathcal{P}_c(\mathbb{R}^d)$ of compact subsets of $\mathbb{R}^d$, and the differential inclusion problem: $$\left\{\begin{array}{l} y'(t) \in \mathcal{F}(y(t)) \\ y(0) = y_{\text{init}} \end{array}\right.$$
Show that if for every compact $K \subset \mathbb{R}^d$, there exists $C_K > 0$ such that $\mathcal{F}$ satisfies: $$\forall x, y \in K, \forall v_x \in \mathcal{F}(x), \forall v_y \in \mathcal{F}(y), \quad \langle v_x - v_y, x - y \rangle \leqslant C_K \|x - y\|^2$$ then problem (2) admits at most one maximal solution. (Hint: You may look at $\|X(t) - Y(t)\|^2$ for $X$ and $Y$ two solutions.)
QIV.2 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We consider the differential inclusion problem given by $d = 2$ and $\mathcal{F} : \mathbb{R}^2 \rightarrow \mathcal{P}_c(\mathbb{R}^2)$ defined for all $x = (x_1, x_2) \in \mathbb{R}^2$ by: $$\mathcal{F}(x) = \begin{cases} \{v^-\} & \text{if } x_1 < 0 \\ \{v^+\} & \text{if } x_1 > 0 \\ [v_1^+, v_1^-] \times [v_2^+, v_2^-] & \text{if } x_1 = 0 \end{cases}$$ where $v^- = (v_1^-, v_2^-) \in \mathbb{R}^2$ and $v^+ = (v_1^+, v_2^+) \in \mathbb{R}^2$ with $v_1^- \geqslant v_1^+$ and $v_2^- \geqslant v_2^+$.
We set $v^- = (1, 2)$ and $v^+ = (-1, 2)$.
(a) Show that $\mathcal{F}$ satisfies condition (3).
(b) We choose $y_{\text{init}} = (0, 0)$. Find all maximal solutions of problem (2).
(c) We choose $y_{\text{init}} = (1, 0)$. Find all maximal solutions of problem (2).
QIV.3 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We consider the differential inclusion problem given by $d = 2$ and $\mathcal{F} : \mathbb{R}^2 \rightarrow \mathcal{P}_c(\mathbb{R}^2)$ defined for all $x = (x_1, x_2) \in \mathbb{R}^2$ by: $$\mathcal{F}(x) = \begin{cases} \{v^-\} & \text{if } x_1 < 0 \\ \{v^+\} & \text{if } x_1 > 0 \\ [v_1^+, v_1^-] \times [v_2^+, v_2^-] & \text{if } x_1 = 0 \end{cases}$$ where $v^- = (v_1^-, v_2^-) \in \mathbb{R}^2$ and $v^+ = (v_1^+, v_2^+) \in \mathbb{R}^2$ with $v_1^- \geqslant v_1^+$ and $v_2^- \geqslant v_2^+$.
We set $v^- = (0, 1)$ and $v^+ = (1, 1)$.
(a) Show that $\mathcal{F}$ does not satisfy condition (3).
(b) We choose $y_{\text{init}} = (1, 0)$. Find all maximal solutions of problem (2).
(c) We choose $y_{\text{init}} = (0, 0)$. Find all maximal solutions of problem (2).