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

2021 centrale-maths2__psi

37 maths questions

Q1 Geometric Sequences and Series Proof of a Structural Property of Geometric Sequences View

Show that a geometric sequence is hypergeometric.

Q2 Sequences and Series Recurrence Relations and Sequence Properties View

Let $p \in \mathbb{N}$. Show that the sequence with general term $u_n = \binom{n}{p}$ is hypergeometric.

Q3 Sequences and Series Recurrence Relations and Sequence Properties View

Prove that the set of sequences satisfying relation $$P(n) u_n = Q(n) u_{n+1}$$ with $$P(X) = X(X-1)(X-2) \quad \text{and} \quad Q(X) = X(X-2)$$ is a vector space for which we will specify a basis and the dimension.

Q4 Sequences and series, recurrence and convergence Convergence proof and limit determination View

Let $\left(u_n\right)_{n \in \mathbb{N}}$ be a hypergeometric sequence with associated polynomials $P$ and $Q$. Suppose that there exists a natural integer $n_0$ such that $P\left(n_0\right) = 0$ and, $\forall n \geqslant n_0, Q(n) \neq 0$. Justify that the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ is zero from a certain rank onwards.

Q5 Proof Existence Proof View

We define, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x) = \int_0^{+\infty} t^{x-1} \mathrm{e}^{-t} \, \mathrm{d}t$$ Justify that we thus define a function on $\mathbb{R}^{+*}$.

Q6 Proof Proof That a Map Has a Specific Property View

We define, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x) = \int_0^{+\infty} t^{x-1} \mathrm{e}^{-t} \, \mathrm{d}t$$ Show that the function $\Gamma$ is continuous and strictly positive on $\mathbb{R}^{+*}$.

Q7 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View

We define, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x) = \int_0^{+\infty} t^{x-1} \mathrm{e}^{-t} \, \mathrm{d}t$$ Show that, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x+1) = x\Gamma(x)$$

Q8 Reduction Formulae Evaluate a Closed-Form Expression Using the Reduction Formula View

We define, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x) = \int_0^{+\infty} t^{x-1} \mathrm{e}^{-t} \, \mathrm{d}t$$ Determine the value of $\Gamma(n)$, for $n \in \mathbb{N}^*$.

Q9 Sequences and Series Recurrence Relations and Sequence Properties View

The Pochhammer symbol is defined, for any real number $a$ and any natural integer $n$, by $$[a]_n = \begin{cases} 1 & \text{if } n = 0 \\ a(a+1)\cdots(a+n-1) = \prod_{k=0}^{n-1}(a+k) & \text{otherwise} \end{cases}$$ If $a$ is a negative or zero integer, justify that the sequence $\left([a]_n\right)_{n \in \mathbb{N}}$ is zero from a certain rank onwards.

Q10 Sequences and Series Recurrence Relations and Sequence Properties View

The Pochhammer symbol is defined, for any real number $a$ and any natural integer $n$, by $$[a]_n = \begin{cases} 1 & \text{if } n = 0 \\ a(a+1)\cdots(a+n-1) = \prod_{k=0}^{n-1}(a+k) & \text{otherwise} \end{cases}$$ Let $a \in \mathbb{R}$. Verify that, for any natural integer $n$, $[a]_{n+1} = a[a+1]_n$.

Q11 Sequences and Series Evaluation of a Finite or Infinite Sum View

The Pochhammer symbol is defined, for any real number $a$ and any natural integer $n$, by $$[a]_n = \begin{cases} 1 & \text{if } n = 0 \\ a(a+1)\cdots(a+n-1) = \prod_{k=0}^{n-1}(a+k) & \text{otherwise} \end{cases}$$ Let $n \in \mathbb{N}$. Give an expression of $[a]_n$

using factorials when $a \in \mathbb{N}^*$;
using two values of the function $\Gamma$, when $a \in D$.

Q12 Sequences and Series Recurrence Relations and Sequence Properties View

Given three real numbers $a, b$ and $c$, the Gauss hypergeometric function associated with the triplet $(a, b, c)$ is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Justify that, if $c \in D$, then $\frac{[a]_n [b]_n}{[c]_n}$ is well defined for any natural integer $n$.

Q13 Sequences and Series Power Series Expansion and Radius of Convergence View

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Show that the power series $\sum \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$ is hypergeometric and specify associated polynomials.

Q14 Sequences and Series Power Series Expansion and Radius of Convergence View

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Conversely, prove that the set of hypergeometric series associated with the polynomials obtained in the previous question is a vector space for which we will give a basis and specify the dimension.

Q15 Sequences and Series Power Series Expansion and Radius of Convergence View

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Determine the radius of convergence of the power series $\sum \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$.

Q16 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Justify that $F_{a,b,c}$ is of class $\mathcal{C}^1$ on $]-1,1[$. Calculate its derivative and express it using a Gauss hypergeometric function.

Q17 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Justify that $F_{a,b,c}$ is of class $\mathcal{C}^\infty$ on $]-1,1[$ and express its $n$-th derivative using a Gauss hypergeometric function.

Q18 Sequences and Series Evaluation of a Finite or Infinite Sum View

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Express the function $x \mapsto F_{\frac{1}{2}, 1, \frac{3}{2}}\left(-x^2\right)$ using usual functions.

Q22 Combinations & Selection Combinatorial Identity or Bijection Proof View

We admit that Vandermonde's identity remains valid for all natural integers $u, v, N$: $$\binom{u+v}{N} = \sum_{k=0}^{N} \binom{u}{k} \binom{v}{N-k}.$$ Give a combinatorial interpretation of Vandermonde's identity.

Q23 Second order differential equations Solving homogeneous second-order linear ODE View

Let two real numbers $a$ and $c$ such that $c \in D$. Determine the solutions expandable as power series of the differential equation $$x y''(x) + (c - x) y'(x) - a y(x) = 0.$$ We will express these solutions using the Pochhammer symbol and specify the algebraic structure of their set.

Q24 Taylor series Construct Taylor/Maclaurin polynomial from derivative values View

Let $n \in \mathbb{N}$. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Determine $L_0, L_1, L_2$ and $L_3$.

Q25 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Using Leibniz's formula, prove that the function $L_n$ is polynomial of degree $n$. Determine the coefficients $c_{n,k}$ such that $L_n(x) = \sum_{k=0}^{n} c_{n,k} x^k$.

Q26 Differential equations Higher-Order and Special DEs (Proof/Theory) View

Q27 Differential equations Higher-Order and Special DEs (Proof/Theory) View

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Use the equality $\Phi_{n+1}^{(n+1)}(x) = \frac{\mathrm{d}^{n+1} x\Phi_n(x)}{\mathrm{d}x^{n+1}}$, which we will justify, to prove the equality $$L_{n+1}(x) = \left(1 - \frac{x}{n+1}\right) L_n(x) + \frac{x}{n+1} L_n'(x)$$ valid for any real number $x$.

Q28 Differential equations Higher-Order and Special DEs (Proof/Theory) View

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Use the equality $\Phi_{n+1}^{(n+2)}(x) = \frac{\mathrm{d}^{n+1} \Phi_{n+1}^{(1)}}{\mathrm{d}x^{n+1}}(x)$ to prove the equality $$L_{n+1}'(x) = L_n'(x) - L_n(x)$$ valid for any real number $x$.

Q29 Differential equations Verification that a Function Satisfies a DE View

Q30 Differential equations Higher-Order and Special DEs (Proof/Theory) View

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ The confluent hypergeometric function $M_{a,c}$ is the solution of $$x y''(x) + (c-x) y'(x) - a y(x) = 0$$ satisfying $M_{a,c}(0) = 1$. Show that $L_n$ is a confluent hypergeometric function.

Q31 Hypergeometric Distribution View

Let two natural integers $A$ and $n$ such that $n \leqslant A$ and $p$ a real number between 0 and 1. We assume $pA \in \mathbb{N}$ and we denote $q = 1-p$. Let $X$ be a discrete real random variable. We say that $X$ follows the hypergeometric distribution with parameters $n, p$ and $A$ when $$\left\{\begin{array}{l} X(\Omega) \subset \llbracket 0, n \rrbracket, \\ \mathbb{P}(X = k) = \frac{\binom{pA}{k}\binom{qA}{n-k}}{\binom{A}{n}} \quad \text{for all } k \in \llbracket 0, n \rrbracket. \end{array}\right.$$ Verify that we have indeed defined a probability distribution.

Q32 Hypergeometric Distribution View

Let two natural integers $A$ and $n$ such that $n \leqslant A$ and $p$ a real number between 0 and 1. We assume $pA \in \mathbb{N}$ and we denote $q = 1-p$. Let $X$ be a random variable such that $X \hookrightarrow \mathcal{H}(n, p, A)$, i.e. $$\mathbb{P}(X = k) = \frac{\binom{pA}{k}\binom{qA}{n-k}}{\binom{A}{n}} \quad \text{for all } k \in \llbracket 0, n \rrbracket.$$ We recall that, for all non-zero natural integers $k$ and $N$, $k\binom{N}{k} = N\binom{N-1}{k-1}$. Calculate the expectation of $X$.

Q33 Hypergeometric Distribution View

Let two natural integers $A$ and $n$ such that $n \leqslant A$ and $p$ a real number between 0 and 1. We assume $pA \in \mathbb{N}$ and we denote $q = 1-p$. Let $X$ be a random variable such that $X \hookrightarrow \mathcal{H}(n, p, A)$, i.e. $$\mathbb{P}(X = k) = \frac{\binom{pA}{k}\binom{qA}{n-k}}{\binom{A}{n}} \quad \text{for all } k \in \llbracket 0, n \rrbracket.$$ Show that the sequence $(\mathbb{P}(X = k))_{k \in \mathbb{N}}$ is hypergeometric. Deduce an expression of the generating function of $X$ using a hypergeometric function.

Q34 Binomial Distribution Compute Expectation, Variance, or Standard Deviation View

We consider two urns each containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously, in an equiprobable manner, $n$ balls from the first urn. We denote $Y$ the number of white balls obtained. We also draw, in an equiprobable manner, $n$ balls from the second urn, but successively and with replacement. We denote $Z$ the number of white balls obtained. What is the distribution of the variable $Z$? Give the expectation and variance of $Z$.

Q35 Hypergeometric Distribution View

Q36 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View

We consider an urn containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously $n$ balls from the urn. We number from 1 to $pA$ each of the white balls and, for any natural integer $i \in \llbracket 1, pA \rrbracket$, we define $$Y_i = \begin{cases} 1 & \text{if the ball numbered } i \text{ was drawn,} \\ 0 & \text{otherwise.} \end{cases}$$ Let $Y$ denote the number of white balls drawn. Express $Y$ using the $Y_i$ and recover the value of the expectation of $Y$. Compare it to that of $Z$ (where $Z \sim \mathcal{B}(n,p)$ is the number of white balls in $n$ draws with replacement).

Q37 Discrete Probability Distributions Proof of Distributional Properties or Symmetry View

We consider an urn containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously $n$ balls from the urn. We number from 1 to $pA$ each of the white balls and, for any natural integer $i \in \llbracket 1, pA \rrbracket$, we define $$Y_i = \begin{cases} 1 & \text{if the ball numbered } i \text{ was drawn,} \\ 0 & \text{otherwise.} \end{cases}$$ For $1 \leqslant i < j \leqslant pA$, prove that the random variable $Y_i Y_j$ follows a Bernoulli distribution whose parameter we will specify.

Q38 Discrete Random Variables Expectation and Variance via Combinatorial Counting View

We consider an urn containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously $n$ balls from the urn. We number from 1 to $pA$ each of the white balls and, for any natural integer $i \in \llbracket 1, pA \rrbracket$, we define $$Y_i = \begin{cases} 1 & \text{if the ball numbered } i \text{ was drawn,} \\ 0 & \text{otherwise.} \end{cases}$$ Let $Y = \sum_{i=1}^{pA} Y_i$ be the number of white balls drawn. Deduce the value of the variance of $Y$. Compare it to that of $Z$ (where $Z \sim \mathcal{B}(n,p)$).

Q39 Hypergeometric Distribution View

Let $X$ be a random variable following the distribution $\mathcal{H}(n, p, A)$, i.e. $$\mathbb{P}(X = k) = \frac{\binom{pA}{k}\binom{qA}{n-k}}{\binom{A}{n}} \quad \text{for all } k \in \llbracket 0, n \rrbracket.$$ We fix $n$ and $p$. Let $k \in \llbracket 0, n \rrbracket$. Show that $$\lim_{A \to +\infty} \mathbb{P}(X = k) = \binom{n}{k} p^k (1-p)^{n-k}.$$

Q40 Hypergeometric Distribution View

Let $X$ be a random variable following the distribution $\mathcal{H}(n, p, A)$. We fix $n$ and $p$. We have shown that $$\lim_{A \to +\infty} \mathbb{P}(X = k) = \binom{n}{k} p^k (1-p)^{n-k}.$$ Interpret this result in connection with those obtained for the expectation and variance of $Y$.