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-maths2__psi

21 maths questions

Q1 Proof Proof of Set Membership, Containment, or Structural Property View
Let $f : [a, b] \longrightarrow \mathbf{R}$ be a continuous function. Prove that the restriction $g$ of the function $f$ to the interval $]a, b[$ belongs to the set $\mathscr{D}_{a,b}$.
Q2 Proof True/False Justification View
By setting for every integer $k \geqslant 1$, $a_k = \frac{1}{k} - \frac{1}{2^{k+1}}$ and $b_k = \frac{1}{k} + \frac{1}{2^{k+1}}$, show that we can choose an integer $k_0 \geqslant 1$ such that: $$\forall k \geqslant k_0, \quad b_{k+1} < a_k.$$ Deduce that the function $f : ]0,1[ \longrightarrow \mathbf{R}$ defined by: $$f : t \longmapsto \begin{cases} k^2 \cdot 2^{k+1} \cdot (t - a_k), & \text{if there exists an integer } k \geqslant k_0 \text{ such that } t \in \left[a_k, a_k + \frac{1}{2^{k+1}}\right] \\ k^2 \cdot 2^{k+1} \cdot (b_k - t), & \text{if there exists an integer } k \geqslant k_0 \text{ such that } t \in \left[a_k + \frac{1}{2^{k+1}}, b_k\right] \\ 0, & \text{otherwise} \end{cases}$$ is a well-defined and continuous function on $]0,1[$, integrable on $]0,1[$ and that this function $f$ does not belong to the set $\mathscr{D}_{0,1}$.
Q3 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Show that the function $\varphi : t \longmapsto \frac{1}{\sqrt{t}}$ is integrable on $]0,1[$, then show that the function $\varphi$ belongs to $\mathscr{D}_{0,1}$.
Q4 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
We denote by $\tilde{h}$ the restriction of the function $h : ]0,1[ \longrightarrow \mathbf{R},\; t \longmapsto \frac{1}{\sqrt{t(1-t)}}$ to the interval $\left]0, \frac{1}{2}\right]$. Verify that the function $\tilde{h}$ is decreasing on $]0, \frac{1}{2}[$, then show that the function $\tilde{h}$ belongs to $\mathscr{D}_{0, \frac{1}{2}}$.
Q5 Reduction Formulae Prove Convergence or Determine Domain of Convergence of an Integral View
Consider the function $h : ]0,1[ \longrightarrow \mathbf{R},\; t \longmapsto \frac{1}{\sqrt{t(1-t)}}$, and let $\tilde{h}$ denote its restriction to $\left]0, \frac{1}{2}\right]$. Show that the function $h$ is integrable on $]0,1[$ and that: $$\int_0^1 h(t)\, dt = 2\int_0^{\frac{1}{2}} \tilde{h}(t)\, dt.$$
Q6 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
Consider the function $h : ]0,1[ \longrightarrow \mathbf{R},\; t \longmapsto \frac{1}{\sqrt{t(1-t)}}$. Prove that: $$\lim_{n \longrightarrow +\infty} \sum_{k=1}^{2n-1} \frac{1}{2n} h\!\left(\frac{k}{2n}\right) = \int_0^1 h(t)\, dt.$$
Q7 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
Consider the function $h : ]0,1[ \longrightarrow \mathbf{R},\; t \longmapsto \frac{1}{\sqrt{t(1-t)}}$, and let $\tilde{h}$ denote its restriction to $\left]0, \frac{1}{2}\right]$. Show that: $$\lim_{n \rightarrow +\infty} \sum_{k=1}^{n} \frac{1}{2n+1} h\!\left(\frac{k}{2n+1}\right) = \int_0^{\frac{1}{2}} h(t)\, dt.$$ Deduce that: $$\lim_{n \rightarrow +\infty} \sum_{k=1}^{2n} \frac{1}{2n+1} h\!\left(\frac{k}{2n+1}\right) = \int_0^1 h(t)\, dt.$$
Q8 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Consider the function $h : ]0,1[ \longrightarrow \mathbf{R},\; t \longmapsto \frac{1}{\sqrt{t(1-t)}}$. Deduce from the previous questions that the function $h$ belongs to $\mathscr{D}_{0,1}$.
Q9 Integration using inverse trig and hyperbolic functions View
Consider the function $h : ]0,1[ \longrightarrow \mathbf{R},\; t \longmapsto \frac{1}{\sqrt{t(1-t)}}$. Show that: $$\int_0^1 h(t)\, dt = \pi.$$
Q10 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
Show that when $n$ tends to $+\infty$, we have an equivalent of the form: $$\sum_{k=1}^{n} \frac{1}{\sqrt{k}} \underset{n \to +\infty}{\sim} \lambda \sqrt{n},$$ where the constant $\lambda$ is to be determined.
Q11 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
Deduce the limit: $$\lim_{n \rightarrow +\infty} \sum_{i=1}^{n-1} \frac{1}{\sqrt{i(n-i)}}.$$
Q12 Sequences and Series Limit Evaluation Involving Sequences View
Consider a sequence $(\varepsilon_n)_{n \in \mathbb{N}}$ of real numbers strictly greater than $-1$, convergent with limit zero. Show that: $$\lim_{n \rightarrow +\infty} \sum_{i=1}^{n-1} \frac{|\varepsilon_i|}{\sqrt{i(n-i)}} = 0.$$
Q13 Sequences and Series Limit Evaluation Involving Sequences View
Consider a sequence $(\varepsilon_n)_{n \in \mathbb{N}}$ of real numbers strictly greater than $-1$, convergent with limit zero. Deduce that: $$\lim_{n \rightarrow +\infty} \sum_{i=1}^{n-1} \frac{1}{\sqrt{i(n-i)}} \cdot \left(\frac{(1+\varepsilon_i)(1+\varepsilon_{n-i})}{1+\varepsilon_n} - 1\right) = 0.$$
Q14 Discrete Probability Distributions Proof of Distributional Properties or Symmetry View
We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. Show that for every $n \in \mathbf{N}^*$, the random variable $\frac{1+X_n}{2}$ follows a Bernoulli distribution with parameter $\frac{1}{2}$.
Q15 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. For every $n \in \mathbf{N}^*$, we denote $S_n = \sum_{k=1}^n X_k$. We fix the integer $n \geqslant 1$. A path is any $2n$-tuple $\gamma = (\varepsilon_1, \cdots, \varepsilon_{2n})$ whose components $\varepsilon_k$ equal $-1$ or $1$. An equality index of a path is any integer $k \in \llbracket 1, 2n \rrbracket$ such that $\sum_{i=1}^k \varepsilon_i = 0$. For every integer $i$ between $1$ and $n$, the event $A_i$ is defined by: $$A_i = \left\{\omega,\; 2i \text{ is an equality index of } (X_1(\omega), \cdots, X_{2n}(\omega))\right\}.$$ Calculate the probability $\mathbf{P}(A_i)$, for every integer $i$ between $1$ and $n$.
Q16 Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. For every $n \in \mathbf{N}^*$, we denote $S_n = \sum_{k=1}^n X_k$. Let $\ell \in \mathbf{Z}$ be an integer and $n \geqslant 1$ be another integer. By distinguishing the case where the integer $\ell - n$ is even or odd, calculate $\mathbf{P}(S_n = \ell)$.
Q17 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
Let $(c_n)_{n \in \mathbf{N}^*}$ and $(d_n)_{n \in \mathbf{N}^*}$ be two sequences of strictly positive real numbers such that: $c_n \underset{n \to +\infty}{\sim} d_n$ and the series $\sum_n c_n$ diverges.
We admit without proof the following result:
Theorem 1. Let $(a_n)_{n \in \mathbf{N}^*}$ and $(b_n)_{n \in \mathbf{N}^*}$ be two sequences of nonzero real numbers such that $a_n = o(b_n)$ as $n \to +\infty$ and the series $\sum_n |b_n|$ is divergent. Then: $$\sum_{k=1}^n a_k = o\!\left(\sum_{k=1}^n |b_k|\right) \text{ as } n \to +\infty.$$
By using this result, show that the series $\sum_n d_n$ is divergent and that: $$\sum_{k=1}^n c_k \underset{n \rightarrow +\infty}{\sim} \sum_{k=1}^n d_k.$$
Q18 Discrete Random Variables Expectation and Variance via Combinatorial Counting View
We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. For every $n \in \mathbf{N}^*$, we denote $S_n = \sum_{k=1}^n X_k$. We fix the integer $n \geqslant 1$. The random variable $N_n : \Omega \longrightarrow \mathbf{N}$ counts, for every $\omega \in \Omega$, the number of equality indices of the path $(X_1(\omega), \cdots, X_{2n}(\omega))$. For every integer $i$ between $1$ and $n$, the event $A_i$ is defined by: $$A_i = \left\{\omega,\; 2i \text{ is an equality index of } (X_1(\omega), \cdots, X_{2n}(\omega))\right\}.$$ Show that the random variable $N_n$ has finite expectation and that its expectation $\mathbb{E}(N_n)$ equals: $$\mathbb{E}(N_n) = \sum_{i=1}^n \frac{\binom{2i}{i}}{4^i}.$$ [Hint: one may express the variable $N_n$ using indicator functions associated with the events $A_i$.]
Q19 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. The random variable $N_n$ counts the number of equality indices of the path $(X_1(\omega), \cdots, X_{2n}(\omega))$, and it has been shown that: $$\mathbb{E}(N_n) = \sum_{i=1}^n \frac{\binom{2i}{i}}{4^i}.$$ Deduce the equivalent: $$\mathbb{E}(N_n) \underset{n \to +\infty}{\sim} \frac{2}{\sqrt{\pi}} \sqrt{n}.$$
Q20 Discrete Random Variables Expectation and Variance via Combinatorial Counting View
In an urn containing $n$ white balls and $n$ black balls, we proceed to draw balls without replacement, until the urn is completely empty. The draws are equally likely at each draw. For every integer $k$ between $1$ and $2n$, we say that the integer $k$ is an equality index if, after drawing the first $k$ balls without replacement, there remain as many black balls as white balls in the urn. We note that the integer $2n$ is always an equality index. We denote by $M_n$ the random variable counting the number of equality indices $k$ between $1$ and $2n$.
By using for example the events $B_i$: ``the integer $i$ is an equality index'', show that the variable $M_n$ has finite expectation equal to: $$\mathbb{E}(M_n) = \sum_{i=0}^{n-1} \frac{\binom{2i}{i} \cdot \binom{2n-2i}{n-i}}{\binom{2n}{n}}.$$
Q21 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
In an urn containing $n$ white balls and $n$ black balls, we proceed to draw balls without replacement, until the urn is completely empty. The random variable $M_n$ counts the number of equality indices $k$ between $1$ and $2n$, and it has been shown that: $$\mathbb{E}(M_n) = \sum_{i=0}^{n-1} \frac{\binom{2i}{i} \cdot \binom{2n-2i}{n-i}}{\binom{2n}{n}}.$$ Deduce the equivalent: $$\mathbb{E}(M_n) \underset{n \to +\infty}{\sim} \sqrt{\pi n}.$$