Problem

Source: Romanian TST for 2019 IMO

Tags: set theory, number theory



Let be two natural numbers m,n, and m pairwise disjoint sets of natural numbers A0,A1,,Am1, each having n elements, such that no element of A_{i\pmod m} is divisible by an element of A_{i+1\pmod m} , for any natural number i. Determine the number of ordered pairs (a,b)\in\bigcup_{0\le j < m} A_j\times\bigcup_{0\le j < m} A_j such that a|b and such that \{ a,b \}\not\in A_k, for any k\in\{ 0,1,\ldots ,m-1 \} . Radu Bumbăcea