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Partition of the positive rationals into disjoint subsets

Source: IMO Shortlist 1993, India 4

March 15, 2006
modular arithmeticnumber theoryrelatively primepartitionalgebraIMO Shortlist

Problem Statement

a) Show that the set Q+ \mathbb{Q}^{ + } of all positive rationals can be partitioned into three disjoint subsets. A,B,C A,B,C satisfying the following conditions: BA=B;&B2=C;&BC=A; BA = B; \& B^2 = C; \& BC = A; where HK HK stands for the set {hk:h∈H,k∈K} \{hk: h \in H, k \in K\} for any two subsets H,K H, K of Q+ \mathbb{Q}^{ + } and H2 H^2 stands for HH. HH. b) Show that all positive rational cubes are in A A for such a partition of Q+. \mathbb{Q}^{ + }. c) Find such a partition Q+=A∪B∪C \mathbb{Q}^{ + } = A \cup B \cup C with the property that for no positive integer n≤34, n \leq 34, both n n and n+1 n + 1 are in A, A, that is, min{n∈N:n∈A,n+1∈A}>34. \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34.
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