IMO 1992 LL IND25

Origin: IND

Problem

(a) Show that the set N of all natural numbers can be parti- tioned into three disjoint subsets A, B, and C satisfying the following conditions: A2 = A, B2 = C, C2 = B, AB = B, AC = C, BC = A, where HK stands for {hk | h \inH, k \inK} for any two subsets H, K of N, and H2 denotes HH. (b) Show that for every such partition of N, min{n \inN | n \inA and n + 1 \inA} is less than or equal to 77.