APMO 2013 Problem 4

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying: (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs either to $A$ or to $B$, then $i+a$ belongs to $A$ or $i-b$ belongs to $B$.

Prove that $a|A|=b|B|$. (Here $|X|$ denotes the number of elements in the set $X$.)