Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$m+f(n) \mid f(m)^2 - nf(n)$$ for all positive integers $m$ and $n$. (Here, $f(m)^2$ denotes $\left(f(m)\right)^2$.)

Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

Let $n > 1$ be an integer and $X = \{1, 2, \cdots , n^2 \}$. If there exist $x, y$ such that $x^2\mid y$ in all subsets of $X$ with $k$ elements, find the least possible value of $k$.

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 $P(x)$ 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 $Z[x]$).

Are there infinitely many positive integers $n$ such that $19|1+2^n+3^n+4^n$? Justify your claim.

Find all integers $m$ and $n$ such that $\frac{m^2+n}{n^2-m}$ and $\frac{n^2+m}{m^2-n}$ are both integers.

Consider the sequence defined by \(a_1 = 2\), \(a_2 = 3\), and \[ a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1}, \] for all integers \(k \geq 1\). Determine all positive integers \(n\) such that \[ \frac{a_n}{n} \] is an integer. Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.

Let $a,b,c\in\mathbb{Z^{+}}$ such that $$(a^2-1, b^2-1, c^2-1)=1$$ Prove that $$(ab+c, bc+a, ca+b)=(a,b,c)$$ (As usual, $(x,y,z)$ means the greatest common divisor of numbers $x,y,z$) [I]Proposed by A. Golovanov[/i]

Let $a_0, a_1, \ldots , a_k \ (k \geq 1)$ be positive integers. Find all positive integers $y$ such that \[a_0 | y, (a_0 + a_1) | (y + a1), \ldots , (a_0 + a_n) | (y + a_n).\]

Determine the smallest positive integer $n\geq 2$ for which there exists a positive integer $m$ such that $mn$ divides $m^{2023} + n^{2023} + n$.

Let $k$ be a positive integer. Define $n_k$ to be the number with decimal representation $70...01$ where there are exactly $k$ zeroes. Prove the following assertions: a) None of the numbers $n_k$ is divisible by $13$. b) Infinitely many of the numbers $n_k$ are divisible by $17$.

Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.

Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

Let $k, n$ be odd positve integers greater than $1$. Prove that if there a exists natural number $a$ such that $k|2^a+1, \ n|2^a-1$, then there is no natural number $b$ satisfying $k|2^b-1, \ n|2^b+1$.

An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Find all non-negative integers $x, y$ such that\[x^3y+x+y=xy+2xy^2\]

Given positive odd integers $m$ and $n$ where the set of all prime factors of $m$ is the same as the set of all prime factors $n$, and $n \vert m$. Let $a$ be an arbitrary integer which is relatively prime to $m$ and $n$. Prove that: \[ o_m(a) = o_n(a) \times \frac{m}{\gcd(m, a^{o_n(a)}-1)} \] where $o_k(a)$ denotes the smallest positive integer such that $a^{o_k(a)} \equiv 1$ (mod $k$) holds for some natural number $k > 1$.

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.

Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2023$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2023$ loses. Who wins if every player wants to win? [i]Proposed by Mykhailo Shtandenko[/i]

Let $p$ be an odd prime and $n$ a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length $p^{n}$. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by $p^{n+1}$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

A pair of polynomials \(F(x, y)\) and \(G(x, y)\) with integer coefficients is called $\emph{important}$ if from the divisibility of both differences \(F(a, b) - F(c, d)\) and \(G(a, b) - G(c, d)\) by $100$, it follows that both \(a - c\) and \(b - d\) are divisible by 100. Does there exist such an important pair of polynomials \(P(x, y)\), \(Q(x, y)\), such that the pair \(P(x, y) - xy\) and \(Q(x, y) + xy\) is also important?

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