IMO 1984 LL MOR37

Origin: MOR

Problem

Denote by [x] the greatest integer not exceeding x. For all real k > 1, define two sequences: an(k) = [nk] and bn(k) = nk k −1 . If A(k) = {an(k) : n \inN} and B(k) = {bn(k) : n \inN}, prove that A(k) and B(k) form a partition of N if and only if k is irrational.