Olympiad problems

2011 May Olympiad, 5

We consider all $14$-digit positive integers, divisible by $18$, whose digits are exclusively $ 1$ and $2$, but there are no consecutive digits $2$. How many of these numbers are there?

1964 Poland - Second Round, 3

Tags: arithmetic sequence , number theory , arithmetic progression , divisible

Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.

2020 Swedish Mathematical Competition, 5

Tags: number theory , combinatorics , divide , divisible

Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides ${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$+ ...+$ {p-1 \choose p-3} a^{p-5} .$

1995 Bulgaria National Olympiad, 1

Tags: number theory , divide , divisible

Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.

2011 Belarus Team Selection Test, 1

Tags: number theory , odd , divisible , prime

Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with $a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$). Prove that a) $(a-1)\vdots p_i$ for some $i=1,..,n$ b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$? I. Bliznets

2006 Switzerland - Final Round, 10

Tags: number theory , divisible , divide

Decide whether there is an integer $n > 1$ with the following properties: (a) $n$ is not a prime number. (b) For all integers $a$, $a^n - a$ is divisible by $n$

1956 Polish MO Finals, 4

Tags: number theory , divide , divisible

Prove that if the natural numbers $ a $, $ b $, $ c $ satisfy the equation $$ a^2 + b^2 = c^2,$$ then: 1) at least one of the numbers $ a $ and $ b $ is divisible by $ 3 $, 2) at least one of the numbers $ a $ and $ b $ is divisible by $ 4 $, 3) at least one of the numbers $ a $, $ b $, $ c $ is divisible by $ 5 $.

2002 Estonia National Olympiad, 1

Tags: gcd , lcm , divisible , divide , number theory

The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.