RMM 2012 Problem 3

Each positive integer number is coloured red or blue. A function $f$ from the set of positive integer numbers into itself has the following two properties: (a) if $x \leq y$, then $f(x) \leq f(y)$; and (b) if $x, y$ and $z$ are all (not necessarily distinct) positive integer numbers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$.

Prove that there exists a positive number $a$ such that $f(x) \leq a x$ for all positive integer numbers $x$.