Nemotron-IMO-Bench 2026 N36

Let $n>3$ be an integer and let $A,B$ be subsets of $\{1,\ldots, n\}$ such that $ab+1$ is a perfect square for all $a\in A$ and $b\in B$. Prove that $$\min (|A|, |B|)\leq 1+\lfloor\log_{2+\sqrt{3}}n\rfloor.$$