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 mines-ponts-maths2__psi

22 maths questions

Q1 Groups Group Order and Structure Theorems View
Recall the cardinality of $\mathcal{S}_n$. Deduce that $R \geq 1$.
Q2 Combinations & Selection Combinatorial Identity or Bijection Proof View
For $k \in \llbracket 0, n \rrbracket$, show that the number of permutations of $\llbracket 1, n \rrbracket$ having exactly $k$ fixed points is $\binom{n}{k} d_{n-k}$.
Deduce that $P_n\left(X_n = k\right) = \frac{d_{n-k}}{k!(n-k)!}$.
Q3 Taylor series Identify a closed-form function from its Taylor series View
Show that $$\forall x \in ]-1,1[ \quad s(x)e^{x} = \frac{1}{1-x}$$ Deduce that $R = 1$.
Q4 Taylor series Extract derivative values from a given series View
Starting from the relation $(1-x)s(x) = e^{-x}$ for $x \in ]-1,1[$, express $\frac{d_n}{n!}$ for natural number $n$, in the form of a sum.
Q5 Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View
Show that the distribution of the random variable $X_n$ is given by $$\forall k \in \llbracket 0, n \rrbracket \quad P_n\left(X_n = k\right) = \frac{1}{k!} \sum_{i=0}^{n-k} \frac{(-1)^i}{i!}.$$
Q6 Discrete Probability Distributions Proof of Distributional Properties or Symmetry View
On the finite probability space $\left(\mathcal{S}_n, P_n\right)$, we define, for all $i \in \llbracket 1, n \rrbracket$, the random variable $U_i$ such that, for all $\sigma \in \mathcal{S}_n$, we have $U_i(\sigma) = 1$ if $\sigma(i) = i$, and $U_i(\sigma) = 0$ otherwise.
Show that $U_i$ follows a Bernoulli distribution with parameter $\frac{1}{n}$.
Show that, if $i \neq j$, the variable $U_i U_j$ follows a Bernoulli distribution whose parameter you will specify.
Q7 Discrete Random Variables Expectation and Variance of Sums of Independent Variables View
Express $X_n$ using the $U_i$, $1 \leq i \leq n$. Deduce the expectation $\mathrm{E}\left(X_n\right)$ and the variance $\mathrm{V}\left(X_n\right)$.
Q8 Poisson distribution View
In this question, we fix a natural number $k$. Determine $$y_k = \lim_{n \rightarrow +\infty} P_n\left(X_n = k\right).$$ Let $Y$ be a random variable on a probability space $(\Omega, \mathcal{A}, P)$, taking values in $\mathbf{N}$, and satisfying $$\forall k \in \mathbf{N} \quad P(Y = k) = y_k.$$ Identify the distribution of $Y$.
Q9 Probability Generating Functions Explicit computation of a PGF or characteristic function View
We denote by $G_{X_n}$ and $G_Y$ the generating functions of the variables $X_n$ and $Y$ from the previous question, respectively. Express $G_{X_n}(s)$ as a sum, for $s$ real, and verify that $$\forall s \in \mathbf{R} \quad \lim_{n \rightarrow +\infty} G_{X_n}(s) = G_Y(s)$$
Q10 Probability Definitions Proof of a Probability Identity or Inequality View
Let $x, y, z$ be three distributions on $\mathbf{N}$. Prove the properties: $$\begin{gathered} 0 \leq d_{VT}(x, y) \leq 1 \\ d_{VT}(x, y) = 0 \Longleftrightarrow x = y \\ d_{VT}(y, x) = d_{VT}(x, y) \\ d_{VT}(x, z) \leq d_{VT}(x, y) + d_{VT}(y, z) \end{gathered}$$
Q11 Discrete Random Variables Probability Distribution Construction and Parameter Determination View
Let $X$ and $Y$ be two Bernoulli random variables, having parameters $\lambda \in ]0,1[$ and $\mu \in ]0,1[$, respectively. Calculate $d_{VT}\left(p_X, p_Y\right)$.
Q12 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Let $X$ be a Bernoulli random variable with parameter $\lambda \in ]0,1[$. Show that $$d_{VT}\left(p_X, \pi_\lambda\right) = \lambda\left(1 - e^{-\lambda}\right).$$ Deduce that $$d_{VT}\left(p_X, \pi_\lambda\right) \leq \lambda^2.$$
Q13 Poisson distribution View
Verify the relation, for all non-zero natural number $n$, $$2 d_{VT}\left(p_{X_n}, \pi_1\right) = \sum_{k=0}^{n} \frac{1}{k!} \left|\sum_{i=n-k+1}^{+\infty} \frac{(-1)^i}{i!}\right| + e^{-1} \sum_{k=n+1}^{+\infty} \frac{1}{k!}.$$
Q14 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
For natural number $n$, we set $r_n = \sum_{k=n+1}^{+\infty} \frac{1}{k!}$. Prove the bound $$r_n \leq \frac{1}{(n+1)!} \sum_{k=0}^{+\infty} \frac{1}{(n+2)^k}$$ Deduce a simple equivalent of $r_n$ as $n$ tends to $+\infty$.
Q15 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
By continuing to bound the right-hand side of the equality in question 13, establish the estimate $$d_{VT}\left(p_{X_n}, \pi_1\right) \underset{n \rightarrow +\infty}{=} O\left(\frac{2^n}{(n+1)!}\right)$$ One may use binomial coefficients.
Q16 Discrete Random Variables Probability Distribution Construction and Parameter Determination View
If $x$ and $y$ are two probability distributions on $\mathbf{N}$, we define the application $x * y : \mathbf{N} \rightarrow \mathbf{R}_+$ by $$\forall k \in \mathbf{N} \quad (x * y)(k) = \sum_{i=0}^{k} x(i) y(k-i) = \sum_{i+j=k} x(i) y(j)$$ Show that $x * y$ is a distribution on $\mathbf{N}$.
Q17 Discrete Random Variables Expectation and Variance of Sums of Independent Variables View
Let $X$ and $Y$ be two independent random variables, taking values in $\mathbf{N}$, defined on the same probability space $(\Omega, \mathcal{A}, P)$. Prove the relation $$p_{X+Y} = p_X * p_Y$$
Q18 Probability Definitions Proof of a Probability Identity or Inequality View
Let $(x, y, u, v) \in \left(\mathcal{D}_{\mathbf{N}}\right)^4$. Show that, for all natural number $k$, $$|(x * y)(k) - (u * v)(k)| \leq \sum_{i+j=k} y(j)|x(i) - u(i)| + \sum_{i+j=k} u(i)|y(j) - v(j)|.$$
Q19 Probability Definitions Proof of a Probability Identity or Inequality View
With the notation of the previous question, establish the inequality $$d_{VT}(x * y, u * v) \leq d_{VT}(x, u) + d_{VT}(y, v)$$
Q20 Approximating the Binomial to the Poisson distribution View
Let $U$ be a binomial random variable with parameters $n \in \mathbf{N}^*$ and $\lambda \in ]0,1[$. Prove the inequality $$d_{VT}\left(p_U, \pi_{n\lambda}\right) \leq n\lambda^2.$$
Q21 Approximating the Binomial to the Poisson distribution View
Let $\alpha$ be a strictly positive real number. For all natural number $n$ such that $n > \lfloor \alpha \rfloor$, we denote by $B_n$ a binomial random variable with parameters $n$ and $\frac{\alpha}{n}$. For all natural number $k$, determine $$\lim_{n \rightarrow +\infty} P\left(B_n = k\right)$$ One may use the previous question.
Q22 Approximating the Binomial to the Poisson distribution View
Let $\alpha$ and $\beta$ be two strictly positive real numbers. Using the results and methods above, show that $$d_{VT}\left(\pi_\alpha, \pi_\beta\right) \leq |\beta - \alpha|.$$