BY-MO 2007 Problem 8

Let $(m,n)$ be a pair of positive integers. (a) Prove that the set of all positive integers can be partitioned into four pairwise disjoint nonempty subsets such that none of them has two numbers with absolute value of their difference equal to either $m$ , $n$ , or $m+n$ . (b) Find all pairs $(m,n)$ such that the set of all positive integers can not be partitioned into three pairwise disjoint nonempty subsets satisfying the above condition.