The sequence $(x_n)$ is defined as follows: $$x_0=2,\, x_1=1,\, x_{n+2}=x_{n+1}+x_n$$ for every non-negative integer $n$. a. For each $n\geq 1$, prove that $x_n$ is a prime number only if $n$ is a prime number or $n$ has no odd prime divisors b. Find all non-negative pairs of integers $(m,n)$ such that $x_m|x_n$.

The function $f(x)=x^2+ \sin x$ and the sequence of positive numbers $\{ a_n \}$ satisfy $a_1=1$, $f(a_n)=a_{n-1}$, where $n \geq 2$. Prove that there exists a positive integer $n$ such that $a_1+a_2+ \dots + a_n > 2020$.

[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality $$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$ holds. [b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?

A positive integer \( r \) is given, find the largest real number \( C \) such that there exists a geometric sequence $\{ a_n \}_{n\ge 1}$ with common ratio \( r \) satisfying $$ \| a_n \| \ge C $$ for all positive integers \( n \). Here, $\| x \|$ denotes the distance from the real number \( x \) to the nearest integer.

A numerical sequence is called lusophone if it satisfies the following three conditions: i) The first term of the sequence is number $1$. ii) To obtain the next term of the sequence we can multiply the previous term by a positive prime number ($2,3,5,7,11, ...$) or add $1$. (iii) The last term of the sequence is the number $2016$. For example: $1\overset{{\times 11}}{\to}11 \overset{{\times 61}}{\to} 671 \overset{{+1}}{\to}672 \overset{{\times 3}}{\to}2016$ How many Lusophone sequences exist in which (as in the example above) the add $1$ operation was used exactly once and not multiplied twice by the same prime number?

Let $p>3$ and $p+2$ are prime numbers,and define sequence $$a_{1}=2,a_{n}=a_{n-1}+\lfloor \dfrac{pa_{n-1}}{n}\rfloor$$ show that:for any $n=3,4,\cdots,p-1$ have $$n|pa_{n-1}+1$$

Determine all sequences $a_1, a_2, a_3, \dots$ of nonnegative integers such that $a_1 < a_2 < a_3 < \dots$ and $a_n$ divides $a_{n - 1} + n$ for all $n \geq 2$.

Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for which the following property holds: If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$), then $x_1 = . . . = x_n = 0$. Proposed by [i]Vincent Jugé and Théo Lenoir, France[/i]

For a sequence of integers $a_1,a_2,a_3,...$ with $0<a_1<a_2<a_3<...$ applies: $$a_n=4a_{n-1}-a_{n-2} \,\,\, for \,\,\, n > 2$$ It is further given that $a_4 = 194$. Calculate $a_5$.

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Let $\{a_n \}_n$ be a sequence of real numbers such there there are countably infinite distinct subsequences converging to the same point. We call two subsequences distinct if they do not have a common term. Which of the following statements always holds: (A) $\{a_n \}_n$ is bounded (B) $\{a_n \}_n$ is unbounded (C) The set of convergent subsequence $\{a_n \}_n$ is countable (D) None of these

Rational numbers are written in the following sequence: $\frac{1}{1},\frac{2}{1},\frac{1}{2},\frac{3}{1},\frac{2}{2},\frac{1}{3},\frac{4}{1},\frac{3}{2},\frac{2}{3},\frac{1}{4}, . . .$ In which position of this sequence is $\frac{2005}{2004}$ ?

For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is fibonatic when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not fibonatic integers.

A finite sequence of natural numbers $a_1, a_2, \dots, a_n$ is given. A sub-sequence $a_{k+1}, a_{k+2}, \dots, a_l$ will be called a [i]repetition[/i] if there exists a natural number $p\leq \frac{l-k}2$ such that $a_i=a_{i+p}$ for $k+1\leq i\leq l-p$, but $a_i\neq a_{i+p}$ for $i=k$ (if $k>0$) and $i=l-p+1$ (if $l<n$). Show that the sequence contains less than $n$ repetitions.