Partitioning can be carried out in unique manner
Source: IMO LongList 1988, United Kingdom 3, Problem 78 of ILL
November 9, 2005
combinatorics unsolvedcombinatorics
Problem Statement
It is proposed to partition a set of positive integers into two disjoint subsets and subject to the conditions
i.) 1 is in
ii.) no two distinct members of have a sum of the form 2^k \plus{} 2, k \equal{} 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.