In each unit cell of a finite set of cells of an infinite checkered board, an integer is written so that the sum of the numbers in each row, as well as in each column, is divided by $2002$. Prove that every number $\alpha$ can be replaced by a certain number $\alpha'$ , divisible by $2002$ so that $|\alpha-\alpha'| <2002$ and the sum of the numbers in all rows, and in all columns will not change.

[b]a)[/b] Show that if two non-negative integers $ p,q $ satisfy the property that both $ \sqrt{2p-q} $ and $ \sqrt{2p+q} $ are non-negative integers, then $ q $ is even. [b]b)[/b] Determine how many natural numbers $ m $ are there such that $ \sqrt{2m-4030} $ and $ \sqrt{2m+4030} $ are both natural.

Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.

Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.

Determine all triplets of prime numbers $p<q<r$, such that $p+q=r$ and $(r-p)(q-p)-27p$ is a square.

Prove that for any given positive integer $m$ and $n$, there is always a positive integer $k$ so that $2^k-m$ has at least $n$ different prime divisors.

A box contains answer $4032$ scripts out of which exactly half have odd number of marks. We choose 2 scripts randomly and, if the scores on both of them are odd number, we add one mark to one of them, put the script back in the box and keep the other script outside. If both scripts have even scores, we put back one of the scripts and keep the other outside. If there is one script with even score and the other with odd score, we put back the script with the odd score and keep the other script outside. After following this procedure a number of times, there are 3 scripts left among which there is at least one script each with odd and even scores. Find, with proof, the number of scripts with odd scores among the three left.

Find the number of positive integers less than $2019$ that are neither multiples of $3$ nor have any digits that are multiples of $3$.

Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n \minus{} 1.$

For any natural number $n > 1$ write the finite decimal expansion of $\frac{1}{n}$ (for example we write $\frac{1}{2}=0.4\overline{9}$ as its infinite decimal expansion not $0.5)$. Determine the length of non-periodic part of the (infinite) decimal expansion of $\frac{1}{n}$.

Find all integer polynomials $f$ such that there are infinitely many pairs of relatively prime natural numbers $(a,b)$ so that $a+b \mid f(a)+f(b)$.

Determine all four-digit numbers $\overline{abcd}$ which are perfect squares and for which the equality holds: $\overline{ab}=3 \cdot \overline{cd} + 1$.

Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.

Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$, where $a,\,b,\,c,\,d$ are positive integers.

Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

Two players play a game as follows: there $n > 1$ rounds and $d \geq 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \not = m_1$. In round $k$ (for $k = 2, \ldots , n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} < n_k \leq n_{k-1} + d$. A gets $\gcd(m_k,n_{k-1})$ points and B gets $\gcd(m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses. For each $(n, d)$ which player has a winning strategy?

Let $n$ be a positive integer such that the number \[\frac{1^k + 2^k + \dots + n^k}{n}\] is an integer for any $k \in \{1, 2, \dots, 99\}$. Prove that $n$ has no divisors between 2 and 100, inclusive.

$p$ is a prime. Find the all $(m,n,p)$ positive integer triples satisfy $m^3+7p^2=2^n$.

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 $n\in \Bbb{N}, n \geq 4.$ Determine all sets $ A = \{a_1, a_2, . . . , a_n\} \subset \Bbb{N}$ containing $2015$ and having the property that $ |a_i - a_j|$ is prime, for all distinct $i, j\in \{1, 2, . . . , n\}.$

Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$. A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$, compute $100m+n$. [i]Proposed by Eugene Chen[/i]

Let $a_1, a_2, . . . , a_n$ be positive integers, with $n \ge 2$, such that $$ \lfloor \sqrt{a_1 \cdot a_2\cdot\cdot\cdot a_n} \rfloor = \lfloor \sqrt{a_1} \rfloor \cdot \lfloor \sqrt{a_2} \rfloor \cdot\cdot\cdot \lfloor \sqrt{a_n} \rfloor.$$ Prove that at least $n - 1$ of these numbers are perfect squares. Clarification: Given a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$. For example $\lfloor \sqrt2\rfloor$ and $\lfloor 3\rfloor =3$.

Find all pairs $(x, y)$ of integers satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ . (S Zlobin)

Prove that if $n$ is a natural number such that both $3n+1$ and $4n+1$ are squares, then $n$ is divisible by $56$.

Each box in the equation $$\square \times \square \times \square - \square \times \square \times \square = 9$$ is filled in with a different number in the list 2, $3, 4, 5, 6, 7, 8$ so that the equation is true. Which number in the list is not used to fill in a box?