Nemotron-IMO-Bench 2026 C20

Let $n>1$ be an integer. Let $A,B$ be subsets of the set $D$ of positive divisors of $n$ such that $a\nmid b$ and $b\nmid a$ for any $a\in A$ and $b\in B$. Prove that $$\sqrt{|A|}+\sqrt{|B|}\leq \sqrt{|D|}.$$