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 polytechnique-maths-a__mp

44 maths questions

Q1a Invariant lines and eigenvalues and vectors Diagonalize a matrix explicitly View

Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_{n}(\mathbb{R})$ $$M_{x} = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right)$$ Show that the matrix $-M_{0}$ is diagonalizable and determine its eigenvalues and eigenspaces.

Q1b Matrices Determinant and Rank Computation View

Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_{n}(\mathbb{R})$ $$M_{x} = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right)$$ Deduce that for all $x \in \mathbb{R}$, we have $$\sum_{\sigma \in \mathfrak{S}_{n}} \varepsilon(\sigma) x^{\nu(\sigma)} = (x-1)^{n-1}(x+n-1)$$

Q2 Permutations & Arrangements Combinatorial Structures on Permutation Matrices/Groups View

Calculate $$\sum_{\sigma \in \mathfrak{S}_{n}} \varepsilon(\sigma), \quad \sum_{\sigma \in \mathfrak{S}_{n}} \varepsilon(\sigma) \nu(\sigma) \quad \text{and} \quad \sum_{\sigma \in \mathfrak{S}_{n}} \frac{\varepsilon(\sigma)}{\nu(\sigma)+1}.$$

Q3 Permutations & Arrangements Combinatorial Structures on Permutation Matrices/Groups View

Establish that $$\operatorname{Card}\left\{\sigma \in \mathfrak{S}_{n} : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{S}_{n} : \varepsilon(\sigma) = -1\right\}$$ and deduce the probability that a permutation of $\mathfrak{S}_{n}$ drawn uniformly at random has a prescribed signature.

Q4 Permutations & Arrangements Combinatorial Structures on Permutation Matrices/Groups View

For $\sigma \in \mathfrak{S}_{n}$, specify the condition on $\nu(\sigma)$ for which $\sigma \in \mathfrak{D}_{n}$. Deduce that $$\operatorname{Card}\left\{\sigma \in \mathfrak{D}_{n} : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{D}_{n} : \varepsilon(\sigma) = -1\right\} + (-1)^{n-1}(n-1).$$

Q5a Matrices Linear Transformation and Endomorphism Properties View

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Justify that the families $\left(1, X, \ldots, X^{m}\right)$ and $\left(1, (X-1), \ldots, (X-1)^{m}\right)$ are bases of $\mathbb{R}_{m}[X]$.

Q5b Matrices Linear Transformation and Endomorphism Properties View

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Show that the transpose of $M$ is the matrix of the linear map identity $$\begin{array}{ccc} \mathbb{R}_{m}[X] & \longrightarrow & \mathbb{R}_{m}[X] \\ P & \longmapsto & P \end{array}$$ in the bases $\left(1, X, \ldots, X^{m}\right)$ at the start and $\left(1, (X-1), \ldots, (X-1)^{m}\right)$ at the end.

Q5c Matrices Linear System and Inverse Existence View

Q5d Matrices Matrix Entry and Coefficient Identities View

Let $m \in \mathbb{N}$. We consider the matrix $$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$ Deduce that for all $\left(u_{0}, \ldots, u_{m}\right), \left(v_{0}, \ldots, v_{m}\right) \in \mathbb{R}^{m+1}$, $$\text{if} \quad \forall k \leqslant m, \quad u_{k} = \sum_{\ell=0}^{k} \binom{k}{\ell} v_{\ell}, \quad \text{then} \quad \forall k \leqslant m, \quad v_{k} = \sum_{\ell=0}^{k} (-1)^{k-\ell} \binom{k}{\ell} u_{\ell}$$

Q6 Sequences and Series Evaluation of a Finite or Infinite Sum View

Show that for any non-zero natural integer $n$, $$D_{n} = n! \sum_{k=0}^{n} \frac{(-1)^{k}}{k!}$$

Q7a Discrete Probability Distributions Proof of Distributional Properties or Symmetry View

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_{n}, \mathscr{P}(\mathfrak{D}_{n}))$ equipped with the uniform probability. We define a random variable $Y_{n}$ by $Y_{n}(\sigma) = \varepsilon(\sigma)$.
Explicitly state the distribution of $Y_{n}$.

Q7b Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities View

Q8a Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_{n}, \mathscr{P}(\mathfrak{S}_{n}))$ equipped with the uniform probability. We define a random variable $Z_{n}$ by $Z_{n}(\sigma) = \nu(\sigma)$.
Explicitly state the distribution of $Z_{n}$.

Q8b Sequences and series, recurrence and convergence Convergence proof and limit determination View

Q8c Discrete Random Variables Expectation and Variance via Combinatorial Counting View

Q9 Discrete Random Variables Expectation and Variance via Combinatorial Counting View

Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. We thus obtain a map $\omega : \mathfrak{S}_{n} \rightarrow \mathbb{N}$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_{n}$ such that $\omega(\sigma) = k$. We then consider, on the probability space $(\mathfrak{S}_{n}, \mathscr{P}(\mathfrak{S}_{n}))$ equipped with the uniform probability, the random variable $X_{n}$ defined by $X_{n}(\sigma) = \omega(\sigma)$.
Calculate, for $n \in \{2, 3, 4\}$, the quantity $\frac{1}{n!} \sum_{\sigma \in \mathfrak{S}_{n}} \omega(\sigma)$.

Q10 Permutations & Arrangements Combinatorial Structures on Permutation Matrices/Groups View

Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_{n}$ such that $\omega(\sigma) = k$.
Specify $s(n,n)$ and $s(n,1)$ then show that, for $2 \leqslant k \leqslant n-1$, we have $$s(n,k) = s(n-1, k-1) + (n-1) s(n-1, k)$$ For $\sigma \in \mathfrak{S}_{n}$, one may distinguish the cases $\sigma(1) = 1$ and $\sigma(1) \neq 1$.

Q11 Sequences and Series Functional Equations and Identities via Series View

Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_{n}$ such that $\omega(\sigma) = k$.
Establish that, for any real $x$, $\prod_{i=0}^{n-1}(x+i) = \sum_{k=1}^{n} s(n,k) x^{k}$.

Q12 Discrete Random Variables Convergence of Expectations or Moments View

Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. We consider, on the probability space $(\mathfrak{S}_{n}, \mathscr{P}(\mathfrak{S}_{n}))$ equipped with the uniform probability, the random variable $X_{n}$ defined by $X_{n}(\sigma) = \omega(\sigma)$.
Prove that $\mathbb{E}\left[X_{n}\right] \underset{n \rightarrow +\infty}{=} \ln(n) + \gamma + O\left(\frac{1}{n}\right)$.

Q13a Sequences and Series Evaluation of a Finite or Infinite Sum View

Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_{n}$ such that $\omega(\sigma) = k$.
Show that $$\frac{1}{n!} \sum_{k=1}^{n} k(k-1) s(n,k) = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{1}{ij} - \sum_{i=1}^{n} \frac{1}{i^{2}}.$$

Q13b Discrete Random Variables Expectation and Variance via Combinatorial Counting View

Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_{n}$ such that $\omega(\sigma) = k$. We consider, on the probability space $(\mathfrak{S}_{n}, \mathscr{P}(\mathfrak{S}_{n}))$ equipped with the uniform probability, the random variable $X_{n}$ defined by $X_{n}(\sigma) = \omega(\sigma)$.
Deduce that $$\frac{1}{n!} \sum_{k=1}^{n} k^{2} s(n,k) = \mathbb{E}\left[X_{n}\right] + \left(\sum_{i=1}^{n} \sum_{j=1}^{n} \frac{1}{ij} - \sum_{i=1}^{n} \frac{1}{i^{2}}\right).$$

Q14a Discrete Random Variables Convergence of Expectations or Moments View

Q14b Discrete Random Variables Convergence of Expectations or Moments View

Q15 Central limit theorem View

Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. We consider, on the probability space $(\mathfrak{S}_{n}, \mathscr{P}(\mathfrak{S}_{n}))$ equipped with the uniform probability, the random variable $X_{n}$ defined by $X_{n}(\sigma) = \omega(\sigma)$.
Justify that there exists a positive real number $C > 0$ such that, for any real $\varepsilon > 0$ and any integer $n \geqslant 1$, we have $$\mathbb{P}\left(\left|X_{n} - \ln(n)\right| > \varepsilon \ln(n)\right) \leqslant \frac{C}{\varepsilon^{2} \ln(n)}$$

Q16 Sequences and Series Evaluation of a Finite or Infinite Sum View

Let $\left(a_{n}\right)_{n \geqslant 2}$ be a sequence of real numbers. For $t \in \mathbb{R}$, we set $A(t) = \sum_{2 \leqslant k \leqslant t} a_{k}$. Let $b : [2, +\infty[ \rightarrow \mathbb{R}$ be a function of class $\mathscr{C}^{1}$. Show that for any integer $n \geqslant 2$, $$\sum_{k=2}^{n} a_{k} b(k) = A(n) b(n) - \int_{2}^{n} b^{\prime}(t) A(t) \mathrm{d}t.$$

Q17a Number Theory Prime Counting and Distribution View

The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^{n}$.
Handle the cases $n \in \{1, 2, 3\}$.

Q17b Number Theory Prime Counting and Distribution View

Q17c Number Theory Prime Counting and Distribution View

The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^{n}$.
We now assume $n \geqslant 4$ and the result is known at rank $k$ for any integer $k$ between 1 and $n-1$. Let $n = 2m+1$ with $m \in \mathbb{N}$. Justify that $\prod_{\substack{m+1 < p \leqslant 2m+1 \\ p \text{ prime}}} p$ divides $\binom{2m+1}{m}$ and show that $\binom{2m+1}{m} \leqslant 4^{m}$.

Q17d Number Theory Prime Counting and Distribution View

The objective of this question is to prove that if $n$ is a non-zero natural integer, then $\prod_{\substack{p \leqslant n \\ p \text{ prime}}} p \leqslant 4^{n}$.
Conclude.

Q18 Number Theory Divisibility and Divisor Analysis View

Let $n$ be a non-zero natural integer and let $p$ be a prime number. Justify the formula $\nu_{p}(n!) = \sum_{k=1}^{+\infty} E\left(\frac{n}{p^{k}}\right)$ and show that $$\frac{n}{p} - 1 < \nu_{p}(n!) \leqslant \frac{n}{p} + \frac{n}{p(p-1)}$$

Q19a Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View

By comparison with an integral, establish that $$\sum_{k=1}^{n} \ln(k) \underset{n \rightarrow +\infty}{=} n\ln(n) - n + O(\ln(n))$$

Q19b Number Theory Prime Counting and Distribution View

Justify that $n! = \prod_{\substack{p \leqslant n \\ p \text{ prime}}} p^{\nu_{p}(n!)}$ and deduce that $$n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} - n\ln(4) < \ln(n!) \leqslant n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} + n \sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p(p-1)}.$$

Q19c Sequences and Series Convergence/Divergence Determination of Numerical Series View

Justify that the series $\sum_{k \geqslant 2} \frac{\ln(k)}{k(k-1)}$ converges.

Q19d Number Theory Prime Counting and Distribution View

Conclude that $\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{\ln(p)}{p} \underset{n \rightarrow +\infty}{=} \ln(n) + O(1)$.

Q20a Number Theory Prime Counting and Distribution View

For all real $t \geqslant 2$, we define $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Show, using the result from question 16, that $$\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{1}{p} = 1 + \ln_{2}(n) - \ln_{2}(2) + \frac{R(n)}{\ln(n)} + \int_{2}^{n} \frac{R(t)}{t(\ln(t))^{2}} \mathrm{~d}t$$

Q20b Sequences and Series Convergence/Divergence Determination of Numerical Series View

For all real $t \geqslant 2$, we define $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Justify that the function $t \mapsto \frac{R(t)}{t(\ln(t))^{2}}$ is integrable on $[2, +\infty[$.

Q20c Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View

For all real $t \geqslant 2$, we define $$R(t) = \sum_{\substack{p \leqslant t \\ p \text{ prime}}} \frac{\ln(p)}{p} - \ln(t)$$ Establish that $\sum_{\substack{p \leqslant n \\ p \text{ prime}}} \frac{1}{p} \underset{n \rightarrow +\infty}{=} \ln_{2}(n) + c_{1} + O\left(\frac{1}{\ln(n)}\right)$, for a real $c_{1} \in \mathbb{R}$ to be determined.

Q21a Number Theory Combinatorial Number Theory and Counting View

Let $x$ be a positive real number greater than or equal to 1 and $q \in \mathbb{N}^{*}$. Justify that the quantity $$\operatorname{Card}\{n \in \mathbb{N} \cap [1,x] : n \equiv 0 \pmod{q}\} - \frac{x}{q}$$ is bounded in absolute value by a real number independent of $x$ and $q$.

Q21b Number Theory Arithmetic Functions and Multiplicative Number Theory View

For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Prove, using a summation interchange, that $\frac{1}{x} \sum_{n \leqslant x} \omega(n) \underset{x \rightarrow +\infty}{=} \ln_{2}(x) + O(1)$.

Q22a Number Theory Arithmetic Functions and Multiplicative Number Theory View

For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Show that $$\frac{1}{x} \sum_{n \leqslant x} \left(\omega(n) - \ln_{2}(x)\right)^{2} \underset{x \rightarrow +\infty}{=} \frac{1}{x}\left(\sum_{n \leqslant x} \omega(n)^{2}\right) - \ln_{2}(x)^{2} + O\left(\ln_{2}(x)\right).$$

Q22b Number Theory Arithmetic Functions and Multiplicative Number Theory View

For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Show that $$\sum_{n \leqslant x} \omega(n)^{2} = \sum_{\substack{p_{1} \leqslant x \\ p_{1} \text{ prime}}} \sum_{\substack{p_{2} \leqslant x \\ p_{2} \text{ prime}}} \operatorname{Card}\left\{n \in \mathbb{N}^{*} : n \leqslant x, p_{1} \mid n \text{ and } p_{2} \mid n\right\}$$

Q22c Number Theory Prime Counting and Distribution View

For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Show that $$\left(\sum_{\substack{p_{1}, p_{2} \leqslant x \\ p_{1} \neq p_{2} \text{ prime}}} \operatorname{Card}\left\{n \in \mathbb{N}^{*} : n \leqslant x, p_{1} \mid n \text{ and } p_{2} \mid n\right\}\right) - x\ln_{2}(x)^{2} \underset{x \rightarrow +\infty}{=} O\left(x\ln_{2}(x)\right)$$ One may estimate the cardinality of the set of pairs of prime numbers $(p_{1}, p_{2})$ such that $p_{1} p_{2} \leqslant x$ as $x$ tends to $+\infty$.

Q22d Number Theory Arithmetic Functions and Multiplicative Number Theory View

For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ Conclude that $\frac{1}{x}\left(\sum_{n \leqslant x} \left(\omega(n) - \ln_{2}(x)\right)^{2}\right) \underset{x \rightarrow +\infty}{=} O\left(\ln_{2}(x)\right)$.

Q23 Number Theory Combinatorial Number Theory and Counting View

For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ We define $\mathscr{S} = \left\{n \geqslant 3 : \left|\frac{\omega(n) - \ln_{2}(n)}{\sqrt{\ln_{2}(n)}}\right| \geqslant \left(\ln_{2}(n)\right)^{1/4}\right\}$. Show that $$\lim_{x \rightarrow +\infty} \frac{1}{x} \operatorname{Card}\{n \leqslant x : n \in \mathscr{S}\} = 0$$ One may begin by writing $\operatorname{Card}(\mathscr{S} \cap [1,x]) \underset{x \rightarrow +\infty}{=} \operatorname{Card}(\mathscr{S} \cap [\sqrt{x}, x]) + O(\sqrt{x})$ and note that in the sum on the right-hand side, the difference $\left|\ln_{2}(n) - \ln_{2}(x)\right|$ remains bounded.