Olympiad problems

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Found problems: 54

1999 Tournament Of Towns, 2

Tags: composite , odd , number theory

Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite. (V Senderov)

2013 Costa Rica - Final Round, 2

Tags: number theory , composite number , odd , even , sum

Determine all even positive integers that can be written as the sum of odd composite positive integers.

2015 Thailand Mathematical Olympiad, 8

Tags: perfect square , odd , number theory

Let $m$ and $n$ be positive integers such that $m - n$ is odd. Show that $(m + 3n)(5m + 7n)$ is not a perfect square.

2002 Estonia National Olympiad, 3

Tags: number theory , odd , sum , product

John takes seven positive integers $a_1,a_2,...,a_7$ and writes the numbers $a_i a_j$, $a_i+a_j$ and $|a_i -a_j |$ for all $i \ne j$ on the blackboard. Find the greatest possible number of distinct odd integers on the blackboard.