MathDB
Expressing n as a+b from the infinite sets A and B

Source: Baltic Way 1997

January 28, 2011
combinatorics proposedcombinatorics

Problem Statement

a) Prove the existence of two infinite sets AA and BB, not necessarily disjoint, of non-negative integers such that each non-negative integer nn is uniquely representable in the form n=a+bn=a+b with aA,bBa\in A,b\in B. b) Prove that for each such pair (A,B)(A,B), either AA or BB contains only multiples of some integer k>1k>1.