BG-MO 2012 Problem 2

Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled:

1. For every prime number $p$ and every natural number $n$ , the numbers $p^n,p^{n+1}$ and $p^{n+2}$ do not have the same colour.
2. There does not exist an infinite geometric sequence of natural numbers of the same colour.