Question 101
Let $A$ be a set of pairwise distinct positive integers. A division of $A$ into two disjoint non-empty subsets $A_1$ and $A_2$ is good, if the least common multiple of the elements of $A_1$ is the same as the greatest common divisor of the elements of $A_2$. Show that for a set with $3n + 2$ elements the maximal possible number of good divisions is equal to $2n$. (A simplification of the problem C3 from the IMO Shortlist 2015.)