NL-NWO 2011 Problem 5

The number devil has coloured the integer numbers: every integer is coloured either black or white. The number $1$ is coloured white. For every two white numbers $a$ and $b$ ( $a$ and $b$ are allowed to be equal) the numbers $a-b$ and $a +$ b have different colours. Prove that $2011$ is coloured white.