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
2024 mines-ponts-maths1__psi

9 maths questions

Q1 Proof Proof of Inequalities Involving Series or Sequence Terms View
Show that every function bounded in absolute value by a polynomial function in $|x|$ has slow growth.
Q2 Proof Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
Show that $C^{0}(\mathbf{R}) \cap CL(\mathbf{R}) \subset L^{1}(\varphi)$.
We admit throughout the rest of the problem that $\int_{-\infty}^{+\infty} \varphi(t) \mathrm{d}t = 1$.
Q14 Differential equations Uniform or Pointwise Convergence of Function Series/Sequences View
Show that $S$ is continuous on $\mathbf{R}_{+}$, where $S(t) = \operatorname{Ent}_{\varphi}\left(P_{t}(f)\right)$.
Hint: You may first show that, if $x \in \mathbf{R}$, then $t \mapsto P_{t}(f)(x)$ is continuous on $\mathbf{R}_{+}$.
Q15 Differential equations Limit Evaluation Involving Sequences View
Verify that we have $S(0) = \operatorname{Ent}_{\varphi}(f)$ and $\lim_{t \rightarrow +\infty} S(t) = 0$, where $S(t) = \operatorname{Ent}_{\varphi}\left(P_{t}(f)\right)$.
Q16 Differential equations Higher-Order and Special DEs (Proof/Theory) View
We admit that $S$ is of class $C^{1}$ on $\mathbf{R}_{+}^{*}$ and that
$$\forall t \in \mathbf{R}_{+}^{*}, \quad S^{\prime}(t) = \int_{-\infty}^{+\infty} \frac{\partial P_{t}(f)(x)}{\partial t} \left(1 + \ln\left(P_{t}(f)(x)\right)\right) \varphi(x) \mathrm{d}x$$
Show that
$$\forall t \in \mathbf{R}_{+}^{*}, \quad S^{\prime}(t) = \int_{-\infty}^{+\infty} L\left(P_{t}(f)\right)(x) \left(1 + \ln\left(P_{t}(f)(x)\right)\right) \varphi(x) \mathrm{d}x$$
Q17 Differential equations Entropy, Information, or Log-Sobolev Functional Analysis View
By admitting that the result of question 7 is valid for the functions $P_{t}(f)$ and $1 + \ln\left(P_{t}(f)\right)$, show that
$$\forall t \in \mathbf{R}_{+}^{*}, \quad -S^{\prime}(t) = \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} \frac{P_{t}\left(f^{\prime}\right)(x)^{2}}{P_{t}(f)(x)} \varphi(x) \mathrm{d}x$$
Q18 Differential equations Expectation and Moment Inequality Proof View
By using the Cauchy-Schwarz inequality, show that
$$\forall t \in \mathbf{R}_{+}^{*}, \quad -S^{\prime}(t) \leq \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} P_{t}\left(\frac{f^{\prime 2}}{f}\right)(x) \varphi(x) \mathrm{d}x$$
Q19 Differential equations Change of Variable and Integral Evaluation View
Deduce that we have:
$$\forall t \in \mathbf{R}_{+}^{*}, \quad -S^{\prime}(t) \leq \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} \frac{f^{\prime 2}(x)}{f(x)} \varphi(x) \mathrm{d}x$$
Q20 Differential equations Entropy, Information, or Log-Sobolev Functional Analysis View
Establish the following inequality
$$\operatorname{Ent}_{\varphi}(f) \leq \frac{1}{2} \int_{-\infty}^{+\infty} \frac{f^{\prime 2}(x)}{f(x)} \varphi(x) \mathrm{d}x$$