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
2020 x-ens-maths-b__mp_cpge

19 maths questions

Q1 Discrete Probability Distributions Probability Bounds and Inequalities for Discrete Variables View
Let $Z$ be a discrete real random variable such that $\exp(\lambda Z)$ has finite expectation for all $\lambda > 0$. Show that for all $\lambda > 0$ and $t \in \mathbb{R}$, $$P[Z \geqslant t] \leqslant \exp(-\lambda t) E[\exp(\lambda Z)].$$
Q2 Discrete Probability Distributions Probability Bounds and Inequalities for Discrete Variables View
Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ We define $$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$ Show that $P[S_n \geqslant 0] \geqslant \frac{1}{2}$.
Q3 Central limit theorem Probability Bounds and Inequalities for Discrete Variables View
Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ We define $$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$ as well as, for all $\lambda \in \mathbb{R}$, $$\psi(\lambda) = \log\left(\frac{1}{2}e^{\lambda} + \frac{1}{2}e^{-\lambda}\right)$$ Show that for all $t \in \mathbb{R}$, we have $$\frac{1}{n} \log P[S_n \geqslant t] \leqslant \inf_{\lambda \geqslant 0} (\psi(\lambda) - \lambda t).$$
Q4 Moment generating functions Characteristic/Moment Generating Function Derivation View
Let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ For each $\lambda \geqslant 0$, we set $$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$ Show that the function $m$ is strictly increasing on $\mathbb{R}_{\geqslant 0}$, and that for all $t \in [0,1]$, there exists a unique $\lambda \geqslant 0$ such that $m(\lambda) = t$.
Q5 Moment generating functions Expectation and Moment Inequality Proof View
Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ We define $$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$ as well as, for all $\lambda \in \mathbb{R}$, $$\psi(\lambda) = \log\left(\frac{1}{2}e^{\lambda} + \frac{1}{2}e^{-\lambda}\right)$$ For each $\lambda \geqslant 0$, we set $$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$ as well as $$D_n(\lambda) = \exp(\lambda n S_n - n \psi(\lambda))$$
(a) For $n \geqslant 2$ and $\lambda \geqslant 0$, show that $$E[(X_1 - m(\lambda))(X_2 - m(\lambda)) D_n(\lambda)] = 0$$
(b) Deduce that, for $n \geqslant 1$ and $\lambda \geqslant 0$, $$E[(S_n - m(\lambda))^2 D_n(\lambda)] \leqslant \frac{4}{n}.$$
Q6 Moment generating functions Probability Bounds and Inequalities for Discrete Variables View
Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ We define $$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$ For each $\lambda \geqslant 0$, we set $$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$ For all $n \geqslant 1$, $\lambda \geqslant 0$ and $\varepsilon > 0$, we denote by $I_n(\lambda, \varepsilon)$ the random variable defined by $$I_n(\lambda, \varepsilon) = \begin{cases} 1 & \text{if } |S_n - m(\lambda)| \leqslant \varepsilon \\ 0 & \text{otherwise.} \end{cases}$$ Show that $$P[|S_n - m(\lambda)| \leqslant \varepsilon] \geqslant E[I_n(\lambda, \varepsilon) \exp(\lambda n(S_n - m(\lambda) - \varepsilon))],$$
Q7 Moment generating functions Probability Bounds and Inequalities for Discrete Variables View
Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ We define $$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$ as well as, for all $\lambda \in \mathbb{R}$, $$\psi(\lambda) = \log\left(\frac{1}{2}e^{\lambda} + \frac{1}{2}e^{-\lambda}\right)$$ For each $\lambda \geqslant 0$, we set $$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$ as well as $$D_n(\lambda) = \exp(\lambda n S_n - n \psi(\lambda))$$ For all $n \geqslant 1$, $\lambda \geqslant 0$ and $\varepsilon > 0$, we denote by $I_n(\lambda, \varepsilon)$ the random variable defined by $$I_n(\lambda, \varepsilon) = \begin{cases} 1 & \text{if } |S_n - m(\lambda)| \leqslant \varepsilon \\ 0 & \text{otherwise.} \end{cases}$$ Show that $$E[I_n(\lambda, \varepsilon) D_n(\lambda)] \geqslant 1 - \frac{4}{n\varepsilon^2}$$
Q8 Moment generating functions Concentration inequality via MGF and Markov's inequality (Chernoff method) View
Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ We define $$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$ as well as, for all $\lambda \in \mathbb{R}$, $$\psi(\lambda) = \log\left(\frac{1}{2}e^{\lambda} + \frac{1}{2}e^{-\lambda}\right)$$ For each $\lambda \geqslant 0$, we set $$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$ as well as $$D_n(\lambda) = \exp(\lambda n S_n - n \psi(\lambda))$$ For all $n \geqslant 1$, $\lambda \geqslant 0$ and $\varepsilon > 0$, we denote by $I_n(\lambda, \varepsilon)$ the random variable defined by $$I_n(\lambda, \varepsilon) = \begin{cases} 1 & \text{if } |S_n - m(\lambda)| \leqslant \varepsilon \\ 0 & \text{otherwise.} \end{cases}$$
(a) Deduce, for each $\lambda \geqslant 0$ and $\varepsilon > 0$, the existence of a sequence $(u_n(\varepsilon))_{n \geqslant 1}$ that tends to 0 as $n$ tends to infinity and such that $$\frac{1}{n} \log P[S_n \geqslant m(\lambda) - \varepsilon] \geqslant \psi(\lambda) - \lambda m(\lambda) - \lambda \varepsilon + u_n(\varepsilon)$$
(b) Conclude that for all $t \in [0,1[$, $$\lim_{n \rightarrow \infty} \frac{1}{n} \log P[S_n \geqslant t] = \inf_{\lambda \geqslant 0} (\psi(\lambda) - \lambda t).$$
(c) Is the preceding formula still valid for $t = 1$?
Q9 Proof Existence or properties of extrema via abstract/theoretical argument View
Let $a < b$ be two real numbers and $f : [a,b] \rightarrow \mathbb{R}$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x_0 \in [a,b]$ where $f$ attains its maximum, we have $a < x_0 < b$, and $f''(x_0) \neq 0$.
Show that under hypothesis (H), we have $f''(x_0) < 0$.
Q10 Proof Substitution to Evaluate Limit of an Integral Expression View
Let $a < b$ be two real numbers and $f : [a,b] \rightarrow \mathbb{R}$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x_0 \in [a,b]$ where $f$ attains its maximum, we have $a < x_0 < b$, and $f''(x_0) \neq 0$.
Under hypothesis (H), show that for all $\delta > 0$ such that $\delta < \min(x_0 - a, b - x_0)$, we have the asymptotic equivalence, as $t \rightarrow +\infty$, $$\int_a^b e^{tf(x)} \mathrm{d}x \sim \int_{x_0 - \delta}^{x_0 + \delta} e^{tf(x)} \mathrm{d}x.$$
Q11 Proof Substitution within a Multi-Part Proof or Derivation View
Let $a < b$ be two real numbers and $f : [a,b] \rightarrow \mathbb{R}$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x_0 \in [a,b]$ where $f$ attains its maximum, we have $a < x_0 < b$, and $f''(x_0) \neq 0$.
We admit the identity $\int_{-\infty}^{+\infty} \exp(-x^2) \mathrm{d}x = \sqrt{\pi}$.
Under hypothesis (H), show the asymptotic equivalence, as $t \rightarrow +\infty$, $$\int_a^b e^{tf(x)} \mathrm{d}x \sim e^{tf(x_0)} \sqrt{\frac{2\pi}{t|f''(x_0)|}}$$
Q12 Proof Compute a Base Case or Specific Value of a Parametric Integral View
We admit the identity $\int_{-\infty}^{+\infty} \exp(-x^2) \mathrm{d}x = \sqrt{\pi}$.
(a) Show that for all integer $n \in \mathbb{N}$, we have $$n! = \int_0^{+\infty} e^{-t} t^n \mathrm{d}t$$
(b) Using the preceding results, recover Stirling's formula giving an asymptotic equivalent of $n!$.
Q13 Proof Convergence and Evaluation of Improper Integrals View
Show that $$\lim_{a \rightarrow +\infty} \int_0^a |\sin(x^2)| \mathrm{d}x = +\infty$$
Q14 Taylor series Derive series via differentiation or integration of a known series View
Show that for all $a \in \mathbb{R}$, $$\int_0^a \sin(x^2) \mathrm{d}x = \sum_{n=0}^{+\infty} (-1)^n \frac{a^{4n+3}}{(2n+1)!(4n+3)}$$
Q15 Taylor series Alternating series estimation or partial sum approximation View
Show that the limits $$\lim_{a \rightarrow +\infty} \int_0^a \sin(x^2) \mathrm{d}x \text{ and } \lim_{a \rightarrow +\infty} \int_0^a \cos(x^2) \mathrm{d}x$$ exist and are finite.
Q16 Taylor series Derive a Reduction/Recurrence Formula via Integration by Parts View
We admit the identities: $$\lim_{a \rightarrow +\infty} \int_0^a \sin(x^2) \mathrm{d}x = \lim_{a \rightarrow +\infty} \int_0^a \cos(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4}$$
Show that there exist real numbers $c, c' \in \mathbb{R}$ such that, as $a \rightarrow +\infty$, $$\int_0^a \sin(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4} + \frac{c}{a} \cos(a^2) + \frac{c'}{a^3} \sin(a^2) + O\left(\frac{1}{a^5}\right)$$
Q17 Taylor series Bound or Estimate a Parametric Integral View
From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$. We are also given an infinitely differentiable function $g : [0,1] \rightarrow \mathbb{R}$.
We admit that there exist real numbers $d, d' \in \mathbb{R}$ such that, as $a \rightarrow +\infty$, $$\int_0^a \cos(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4} + \frac{d}{a} \sin(a^2) + \frac{d'}{a^3} \cos(a^2) + O\left(\frac{1}{a^5}\right)$$
Show that, as $t \rightarrow +\infty$, $$\int_{x_0}^1 g(x) \sin(tf(x)) \mathrm{d}x = g(x_0) \int_{x_0}^1 \sin(tf(x)) \mathrm{d}x + O\left(\frac{1}{t}\right)$$
Q18 Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View
From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$.
For all $x \in [x_0, 1]$, we define $$h(x) = \sqrt{|f(x) - f(x_0)|}$$
(a) Show that the function $h$ defines a bijection from $[x_0, 1]$ to $[0, h(1)]$.
(b) Show that the application $h$ is differentiable at $x_0$ from the right, and that $h'_+(x_0) = \sqrt{\frac{f''(x_0)}{2}}$.
Q20 Taylor series Taylor's formula with integral remainder or asymptotic expansion View
From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$. We are also given an infinitely differentiable function $g : [0,1] \rightarrow \mathbb{R}$.
For all $x \in [x_0, 1]$, we define $$h(x) = \sqrt{|f(x) - f(x_0)|}$$ We admit that the bijection $h : [x_0, 1] \rightarrow [0, h(1)]$ admits an inverse application $h^{-1} : [0, h(1)] \rightarrow [x_0, 1]$ that is infinitely differentiable.
We assume that $x_0 \in ]0,1[$. Show that, as $t \rightarrow +\infty$, $$\int_0^1 g(x) \sin(tf(x)) \mathrm{d}x = g(x_0) \sin\left(tf(x_0) + \frac{\pi}{4}\right) \sqrt{\frac{2\pi}{tf''(x_0)}} + O\left(\frac{1}{t}\right)$$