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 $ A_0,A_1,\ldots ,A_{m-1}, $ 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