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
2024 mines-ponts-maths1__psi

19 maths questions

Q1 Sequences and Series 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 Continuous Probability Distributions and Random Variables 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 $\int_{-\infty}^{+\infty} \varphi(t) \mathrm{d}t = 1$.
Q3 Matrices Matrix Group and Subgroup Structure View
Show that $CL(\mathbf{R})$ is a vector space. Also show that $CL(\mathbf{R})$ is closed under multiplication.
Q4 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
Let $t \in \mathbf{R}_{+}$. Verify that the function $P_{t}(f)$ is well defined for $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ and verify that $P_{t}$ is linear on $C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$.
Recall that $\forall x \in \mathbf{R}, \quad P_{t}(f)(x) = \int_{-\infty}^{+\infty} f\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y.$
Q5 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View
Show that for all $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ and all $x \in \mathbf{R}$,
$$\lim_{t \rightarrow +\infty} P_{t}(f)(x) = \int_{-\infty}^{+\infty} f(y) \varphi(y) \mathrm{d}y$$
Q6 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
Let $t \in \mathbf{R}_{+}$. Show that if $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$, then $P_{t}(f) \in C^{0}(\mathbf{R})$. Also show that $P_{t}(f)$ is bounded in absolute value by a polynomial function in $|x|$ independent of $t$. Deduce that $P_{t}(f) \in L^{1}(\varphi)$.
Q7 Chain Rule Proof of Differentiability Class for Parameterized Integrals View
Show that for all functions $f, g \in C^{2}(\mathbf{R})$ such that the functions $f, f^{\prime}, f^{\prime\prime}$ and $g$ have slow growth, we have
$$\int_{-\infty}^{+\infty} L(f)(x) g(x) \varphi(x) \mathrm{d}x = -\int_{-\infty}^{+\infty} f^{\prime}(x) g^{\prime}(x) \varphi(x) \mathrm{d}x$$
where $\forall x \in \mathbf{R}, \quad L(f)(x) = f^{\prime\prime}(x) - x f^{\prime}(x).$
Q8 Chain Rule Proof of Differentiability Class for Parameterized Integrals View
Show that if $f \in C^{1}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime} \in CL(\mathbf{R})$ and $x \in \mathbf{R}$, then $t \in \mathbb{R}_{+} \mapsto P_{t}(f)(x)$ is of class $C^{1}$ on $\mathbb{R}_{+}$ and show that for all $t > 0$, we have
$$\frac{\partial P_{t}(f)(x)}{\partial t} = \int_{-\infty}^{+\infty} \left(-x \mathrm{e}^{-t} + \frac{\mathrm{e}^{-2t}}{\sqrt{1 - \mathrm{e}^{-2t}}} y\right) f^{\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y$$
Q9 Taylor series Prove smoothness or power series expandability of a function View
Let $f \in C^{2}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime}$ and $f^{\prime\prime}$ have slow growth and $t \in \mathbf{R}_{+}$.
Show that $x \in \mathbb{R} \mapsto P_{t}(f)(x)$ is of class $C^{2}$ on $\mathbf{R}$. Also show that
$$\forall x \in \mathbf{R}, \quad P_{t}(f)^{\prime}(x) = \mathrm{e}^{-t} \int_{-\infty}^{+\infty} f^{\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y$$
and
$$\forall x \in \mathbf{R}, \quad P_{t}(f)^{\prime\prime}(x) = \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} f^{\prime\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y.$$
Q10 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Deduce that for $f \in C^{2}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime}$ and $f^{\prime\prime}$ have slow growth, we have
$$\forall t \in \mathbf{R}_{+}^{*}, \forall x \in \mathbf{R}, \quad \frac{\partial P_{t}(f)(x)}{\partial t} = L\left(P_{t}(f)\right)(x)$$
where $\forall x \in \mathbf{R}, \quad L(f)(x) = f^{\prime\prime}(x) - x f^{\prime}(x).$
Q11 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Study the variations of the function $t \mapsto t \ln(t)$ on $\mathbf{R}_{+}^{*}$. Verify that the function can be extended by continuity at 0 and verify that the quantity $\operatorname{Ent}_{\varphi}(g)$ is well defined for all $g \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} g(x) \varphi(x) \mathrm{d}x = 1$.
Recall that for $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x = 1$, the entropy of $f$ with respect to $\varphi$ is defined by: $$\operatorname{Ent}_{\varphi}(f) = \int_{-\infty}^{+\infty} \ln(f(x)) f(x) \varphi(x) \mathrm{d}x$$
Q13 Continuous Probability Distributions and Random Variables Entropy, Information, or Log-Sobolev Functional Analysis View
For $t \in \mathbf{R}_{+}$, we set $S(t) = \operatorname{Ent}_{\varphi}\left(P_{t}(f)\right)$. Justify that $S(t)$ is well defined.
Here $f$ is an element of $C^{2}(\mathbf{R})$ with strictly positive values such that the functions $f, f^{\prime}, f^{\prime\prime}$ and $\frac{f^{\prime 2}}{f}$ have slow growth, and $\int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x = 1$.
Q14 Sequences and Series 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 Sequences and Series 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 Continuous Probability Distributions and Random Variables 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 Continuous Probability Distributions and Random Variables 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 Continuous Probability Distributions and Random Variables 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 Continuous Probability Distributions and Random Variables Entropy, Information, or Log-Sobolev Functional Analysis View
Establish the following inequality
$$\operatorname{Ent}_{\varphi}(f) \leq \frac{1}{2} \int_{-\infty}^{+\infty} \frac{f^{\prime 2}(x)}{f(x)} \varphi(x) \mathrm{d}x$$