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
2020 mines-ponts-maths2__mp_cpge

21 maths questions

Q1 Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View
Let $n \in \mathbb{N}$. Using the factorization $$( X + 1 ) ^ { 2 n } = ( X + 1 ) ^ { n } ( X + 1 ) ^ { n }$$ show that $$\sum _ { k = 0 } ^ { n } \binom { n } { k } ^ { 2 } = \binom { 2 n } { n }$$
Q2 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
Recall Stirling's formula, then determine a real number $c > 0$ such that $$\binom { 2 n } { n } \underset { n \rightarrow + \infty } { \sim } c \frac { 4 ^ { n } } { \sqrt { n } }$$
Q3 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
If $\alpha$ is an element of $]0,1[$, show, for example by using a series-integral comparison, that $$\sum _ { k = 1 } ^ { n } \frac { 1 } { k ^ { \alpha } } \underset { n \rightarrow + \infty } { \sim } \frac { n ^ { 1 - \alpha } } { 1 - \alpha }$$ If $\alpha$ is an element of $]1 , + \infty[$, show similarly that $$\sum _ { k = n + 1 } ^ { + \infty } \frac { 1 } { k ^ { \alpha } } \underset { n \rightarrow + \infty } { \sim } \frac { 1 } { ( \alpha - 1 ) n ^ { \alpha - 1 } }$$
Q4 Integration by Parts Prove an Integral Inequality or Bound View
For $x \in [ 2 , + \infty[$, we set $$I ( x ) = \int _ { 2 } ^ { x } \frac { \mathrm{dt} } { \ln ( t ) }$$ Justify, for $x \in [ 2 , + \infty[$, the relation $$I ( x ) = \frac { x } { \ln ( x ) } - \frac { 2 } { \ln ( 2 ) } + \int _ { 2 } ^ { x } \frac { \mathrm{dt} } { ( \ln ( t ) ) ^ { 2 } }$$ Establish moreover the relation $$\int _ { 2 } ^ { x } \frac { \mathrm{dt} } { ( \ln ( t ) ) ^ { 2 } } \underset { x \rightarrow + \infty } { = } o ( I ( x ) )$$ Deduce finally an equivalent of $I ( x )$ as $x$ tends to $+ \infty$.
Q5 Generalised Binomial Theorem View
For $\alpha \in \mathbb{R}$, recall, without giving a proof, the power series expansion of $( 1 + x ) ^ { \alpha }$ on $]-1,1[$.
Justify the formula: $$\forall x \in ]-1,1[ , \quad \frac { 1 } { \sqrt { 1 - x } } = \sum _ { n = 0 } ^ { + \infty } \frac { \binom { 2 n } { n } } { 4 ^ { n } } x ^ { n }$$
Q6 Probability Generating Functions Radius of convergence and analytic properties of PGF View
We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ Show that the power series defining $F$ and $G$ have radius of convergence greater than or equal to 1. Justify then that the functions $F$ and $G$ are defined and of class $C^{\infty}$ on $]-1,1[$.
Show that $G$ is defined and continuous on $[-1,1]$ and that $$G ( 1 ) = P ( R \neq + \infty ) .$$
Q7 Probability Generating Functions Recursive or recurrence relation via PGF coefficients View
We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ If $k$ and $n$ are positive integers such that $k \leq n$, show that $$P \left( \left( S _ { n } = 0 _ { d } \right) \cap ( R = k ) \right) = P ( R = k ) P \left( S _ { n - k } = 0 _ { d } \right) .$$ Deduce that $$\forall n \in \mathbb{N}^{*} , \quad P \left( S _ { n } = 0 _ { d } \right) = \sum _ { k = 1 } ^ { n } P ( R = k ) P \left( S _ { n - k } = 0 _ { d } \right) .$$
Q8 Probability Generating Functions Compound or random-sum PGF View
We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ Show that $$\forall x \in ]-1,1[ , \quad F ( x ) = 1 + F ( x ) G ( x ) .$$ Determine the limit of $F ( x )$ as $x$ tends to $1^{-}$, discussing according to the value of $P ( R \neq + \infty )$.
Q9 Sequences and Series Limit Evaluation Involving Sequences View
Let $\left( c _ { k } \right) _ { k \in \mathbb{N} }$ be a sequence of elements of $\mathbb{R}^{+}$ such that the power series $\sum c _ { k } x ^ { k }$ has radius of convergence 1 and the series $\sum c _ { k }$ diverges. Show that $$\sum _ { k = 0 } ^ { + \infty } c _ { k } x ^ { k } \underset { x \rightarrow 1 ^ { - } } { \longrightarrow } + \infty$$ With the element $A$ of $\mathbb{R}^{+*}$ fixed, one will show that there exists $\alpha \in ]0,1[$ such that $$\forall x \in ]1 - \alpha , 1[ , \quad \sum _ { k = 0 } ^ { + \infty } c _ { k } x ^ { k } > A$$
Q10 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View
Show that the series $\sum P \left( S _ { n } = 0 _ { d } \right)$ is divergent if and only if $P ( R \neq + \infty ) = 1$.
Q11 Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View
For $i \in \mathbb{N}^{*}$, let $Y _ { i }$ be the Bernoulli random variable indicating the event $$Y _ { i } = \mathbf{1} \left( S _ { i } \notin \left\{ S _ { k } , 0 \leq k \leq i - 1 \right\} \right) .$$ Show that, for $i \in \mathbb{N}^{*}$: $$P \left( Y _ { i } = 1 \right) = P ( R > i )$$ Deduce that, for $n \in \mathbb{N}^{*}$: $$E \left( N _ { n } \right) = 1 + \sum _ { i = 1 } ^ { n } P ( R > i )$$
Q12 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Conclude that $$\frac { E \left( N _ { n } \right) } { n } \underset { n \rightarrow + \infty } { \longrightarrow } P ( R = + \infty ) .$$ One may admit and use Cesàro's theorem: if $\left( u _ { n } \right) _ { n \in \mathbb{N}^{*} }$ is a real sequence converging to the real number $\ell$, then $$\frac { 1 } { n } \sum _ { k = 1 } ^ { n } u _ { k } \underset { n \rightarrow + \infty } { \longrightarrow } \ell .$$
Q13 Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View
In this part, $d$ equals 1 and we simply denote $0_d = 0$. Moreover, $p$ is an element of $]0,1[$, $q = 1 - p$ and the distribution of $X$ is given by $$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$ For $n \in \mathbb{N}$, determine $P \left( S _ { 2 n + 1 } = 0 \right)$ and justify the equality: $$P \left( S _ { 2 n } = 0 \right) = \binom { 2 n } { n } ( p q ) ^ { n }$$
Q14 Discrete Probability Distributions Properties of Named Discrete Distributions (Non-Binomial) View
In this part, $d$ equals 1 and we simply denote $0_d = 0$. Moreover, $p$ is an element of $]0,1[$, $q = 1 - p$ and the distribution of $X$ is given by $$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$ We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ For $x \in ]-1,1[$, give a simple expression for $G ( x )$.
Express $P ( R = + \infty )$ as a function of $| p - q |$.
Determine the distribution of $R$.
Q15 Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities View
In this part, $d$ equals 1 and we simply denote $0_d = 0$. Moreover, $p$ is an element of $]0,1[$, $q = 1 - p$ and the distribution of $X$ is given by $$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$ We assume that $$p = q = \frac { 1 } { 2 }$$ Give a simple equivalent of $P ( R = 2 n )$ as $n$ tends to $+ \infty$. Deduce a simple equivalent of $E \left( N _ { n } \right)$ as $n$ tends to $+ \infty$.
Q16 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Let $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ and $\left( b _ { n } \right) _ { n \in \mathbb{N} }$ be two sequences of elements of $\mathbb{R}^{+*}$. We assume that $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ is decreasing and that $$\forall n \in \mathbb{N} , \quad \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k } = 1 .$$ We set, for $n \in \mathbb{N}$: $$B _ { n } = \sum _ { k = 0 } ^ { n } b _ { k } .$$ Let $m$ and $n$ be two natural integers such that $m > n$. Show that $$a _ { n } \leq \frac { 1 } { B _ { n } } \quad \text{and} \quad 1 \leq a _ { n } B _ { m - n } + a _ { 0 } \left( B _ { m } - B _ { m - n } \right) .$$
Q17 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
Let $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ and $\left( b _ { n } \right) _ { n \in \mathbb{N} }$ be two sequences of elements of $\mathbb{R}^{+*}$. We assume that $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ is decreasing and that $$\forall n \in \mathbb{N} , \quad \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k } = 1 .$$ We set, for $n \in \mathbb{N}$: $$B _ { n } = \sum _ { k = 0 } ^ { n } b _ { k } .$$ We assume in this question that there exists a sequence $\left( m _ { n } \right) _ { n \in \mathbb{N} }$ satisfying $m _ { n } > n$ for $n$ large enough and $$B _ { m _ { n } - n } \underset { n \rightarrow + \infty } { \sim } B _ { n } \quad \text{and} \quad B _ { m _ { n } } - B _ { m _ { n } - n } \underset { n \rightarrow + \infty } { \longrightarrow } 0 .$$ Show that $$a _ { n } \underset { n \rightarrow + \infty } { \sim } \frac { 1 } { B _ { n } } .$$
Q18 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
Let $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ and $\left( b _ { n } \right) _ { n \in \mathbb{N} }$ be two sequences of elements of $\mathbb{R}^{+*}$. We assume that $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ is decreasing and that $$\forall n \in \mathbb{N} , \quad \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k } = 1 .$$ We set, for $n \in \mathbb{N}$: $$B _ { n } = \sum _ { k = 0 } ^ { n } b _ { k } .$$ We assume in this question that there exists $C > 0$ such that $$b _ { n } \underset { n \rightarrow + \infty } { \sim } \frac { C } { n }$$ Using question 17 for a well-chosen sequence $\left( m _ { n } \right) _ { n \in \mathbb{N} }$, show that $$a _ { n } \underset { n \rightarrow + \infty } { \sim } \frac { 1 } { C \ln ( n ) }$$
Q19 Discrete Probability Distributions Recurrence Relations and Sequences Involving Probabilities View
Let $n \in \mathbb{N}^{*}$. Show that $$1 = \sum _ { k = 0 } ^ { n } P \left( S _ { k } = 0 _ { d } \right) P ( R > n - k )$$
Q20 Permutations & Arrangements Lattice Path / Grid Route Counting View
We assume that $d = 2$ and that the distribution of $X$ is given by $$P ( X = ( 0,1 ) ) = P ( X = ( 0 , - 1 ) ) = P ( X = ( 1,0 ) ) = P ( X = ( - 1,0 ) ) = \frac { 1 } { 4 }$$ Let $n \in \mathbb{N}$. Establish the equality $$P \left( S _ { 2 n } = 0 _ { 2 } \right) = \left( \frac { \binom { 2 n } { n } } { 4 ^ { n } } \right) ^ { 2 }$$
Q21 Poisson distribution View
We assume that $d = 2$ and that the distribution of $X$ is given by $$P ( X = ( 0,1 ) ) = P ( X = ( 0 , - 1 ) ) = P ( X = ( 1,0 ) ) = P ( X = ( - 1,0 ) ) = \frac { 1 } { 4 }$$ Give a simple equivalent of $E \left( N _ { n } \right)$ as $n$ tends to $+ \infty$.