Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number. Find how many finite non-empty sets $S\in [p] \times [p]$ are such that $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$

Let $S$ be the set of all the odd positive integers that are not multiples of $5$ and that are less than $30m$, $m$ being an arbitrary positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two different integers, one of which divides the other?

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

Let $x = a_1a_2 \dots a_n$ be an n-digit number, where $a_1, a_2, \dots , an (a_1 \neq 0)$ are the digits. The $n$ numbers $ x_1 = x = a_1 a_2 ... a_n, $ $ x_2 = a_n a_1 ... a_{n-1}, $ $ x_3 = a_{n-1} a_n a _1 ... a_{n-2} $ , $ x_4 = a_{n-2} a_{n-1} a_n a_1 , ... a_{n-3} , $ $ ... , x_n = a_2 a_3 ... a_n a_1$ are said to be obtained from $x$ by the cyclic permutation of digits. [For example, if $n = 5$ and $x = 37001$, then the numbers are $x_1 = 37001, x_2 = 13700, $ $x_3 = 01370(= 1370), x_4 = 00137(= 137), $ $ x_5 = 70013.]$ Find, with proof, (i) the smallest natural number n for which there exists an n-digit number x such that the n numbers obtained from x by the cyclic permutation of digits are all divisible by 1989; and (ii) the smallest natural number x with this property.

Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?

Prove that there are infinitely many coprime, positive integers $a,b$ such that $a$ divides $b^2 -5$ and $b$ divides $a^2 -5.$

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

Let $m$ and $n$ be positive integers. Prove that the number $2n-1$ is divisible by $(2^m-1)^2$ if and only if $n$ is divisible by $m(2^m-1)$.

Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Find all functions $f : N \to N$ such that $f(m)+f(n)$ divides $m+n$ for all positive integers $m$ and $n$.

Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.

Let $p > 3$ be a prime. Prove that if \[ \sum_{i=1 }^{p-1}{1\over i^p} = {n\over m}, \] with $\gdc(n,m) = 1$, then $p^3$ divides $n$.

Given an infinite positive integer sequence $\{x_i\}$ such that $$x_{n+2}=x_nx_{n+1}+1$$ Prove that for any positive integer $i$ there exists a positive integer $j$ such that $x_j^j$ is divisible by $x_i^i$. [i]Remark: Unfortunately, there was a mistake in the problem statement during the contest itself. In the last sentence, it should say "for any positive integer $i>1$ ..."[/i]

Let $P(x)$ be a polynomial with integer coefficients for which there exists a positive integer n such that the real parts of all roots of $P(x)$ are less than $n- \frac{1}{2}$ , polynomial $x-n+1$ does not divide $P(x)$, and $P(n)$ is a prime number. Prove that the polynomial $P(x)$ is irreducible (over $Z[x]$).

Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then $$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$ is not an integer.

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.

Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.

A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

Determine if for any positive integers $a,b,c$ there exist pairwise distinct non-negative integers $A,B,C$ which are greater than $2019$ such that $a+A,b+B$ and $c+C$ divide $ABC$.

Let $m$ be a positive integer and $x_0, y_0$ integers such that $x_0, y_0$ are relatively prime, $y_0$ divides $x_0^2+m$, and $x_0$ divides $y_0^2+m$. Prove that there exist positive integers $x$ and $y$ such that $x$ and $y$ are relatively prime, $y$ divides $x^2 + m$, $x$ divides $y^2 + m$, and $x + y \leq m+ 1.$

Let $m,n \geq 2$ be integers with gcd$(m,n-1) = $gcd$(m,n) = 1$. Prove that among $a_1, a_2, \ldots, a_{m-1}$, where $a_1 = mn+1, a_{k+1} = na_k + 1$, there is at least one composite number.

Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]