ISL 1988 Problem 78

It is proposed to partition a set of positive integers into two disjoint subsets $A$ and $B$ subject to the conditions

i.) 1 is in $A$ ii.) no two distinct members of $A$ have a sum of the form $2^k + 2, k = 0,1,2, \ldots;$ and iii.) no two distinct members of B have a sum of that form.

Show that this partitioning can be carried out in unique manner and determine the subsets to which 1987, 1988 and 1989 belong.