$(a_{n})_{n=1}^{\infty}$ is an integer sequence, $a_{1}=1$, $a_{2}=2$ and for $n\geq{1}$, $a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}$. $a)$ Prove that the set of primes that divides at least one term of the sequence can not be finite. $b)$ Find 3 different prime numbers that do not divide any terms of this sequence.

For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$

Given a circle $C:x^2+y^2=r^2$ ($r$ is an odd number). $P(u,v)\in C$, satisfying: $u=p^m, v=q^n$($p,q$ are prime numbers, $m,n$ are integers, $u>v$). Define $A,B,C,D,M,N:A(r,0),B(-r,0),C(0,-r),D(0,r),M(u,0),N(0,v)$. Prove that $|AM|=1,|BM|=9,|CN|=8,|DN|=2$.

All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.

Let $P = \{p_1,p_2,\ldots, p_{10}\}$ be a set of $10$ different prime numbers and let $A$ be the set of all the integers greater than $1$ so that their prime decomposition only contains primes of $P$. The elements of $A$ are colored in such a way that: [list] [*] each element of $P$ has a different color, [*] if $m,n \in A$, then $mn$ is the same color of $m$ or $n$, [*] for any pair of different colors $\mathcal{R}$ and $\mathcal{S}$, there are no $j,k,m,n\in A$ (not necessarily distinct from one another), with $j,k$ colored $\mathcal{R}$ and $m,n$ colored $\mathcal{S}$, so that $j$ is a divisor of $m$ and $n$ is a divisor of $k$, simultaneously. [/list] Prove that there exists a prime of $P$ so that all its multiples in $A$ are the same color.

There exists a block of 1000 consecutive positive integers containing no prime numbers, namely, $1001!+2,1001!+3,...,1001!+1001$. Does there exist a block of 1000 consecutive positive intgers containing exactly five prime numbers?

Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.

Find all functions $f : \mathbb N \to \mathbb N$ such that: i) $f^{2000}(m)=f(m)$ for all $m \in \mathbb N$, ii) $f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}$, for all $m,n\in \mathbb N$, and iii) $f(m)=1$ if and only if $m=1$.

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that $$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$ [i]Proposed by Ali Partofard[/i]

Find all prime numbers $p$ and $q$ such that $pq-p-q+3$ is a perfect square.

Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime. [i]Lucian Petrescu[/i]

Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$, there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k} $ is divisible by $p$. Prove that $n\leqslant 6$.

given $p_1,p_2,...$ be a sequence of integer and $p_1=2$, for positive integer $n$, $p_{n+1}$ is the least prime factor of $np_1^{1!}p_2^{2!}...p_n^{n!}+1 $ prove that all primes appear in the sequence (Proposed by Beatmania)

Nüx has three moira daughters, whose ages are three distinct prime numbers, and the sum of their squares is also a prime number. What is the age of the youngest moira?

Prove that for all odd prime numbers $p$ there exist a natural number $m<p$ and integers $x_1, x_2, x_3$ such that: \[mp=x_1^2+x_2^2+x_3^2.\]

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

Prove that for any positive integer $k,$ there exist finitely many sets $T$ satisfying the following two properties: $(1)T$ consists of finitely many prime numbers; $(2)\textup{ }\prod_{p\in T} (p+k)$ is divisible by $ \prod_{p\in T} p.$

Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.