Nemotron-IMO-Bench 2026 C16

Let $m\geq n\geq 2022$ be integers and let $a_1,\ldots, a_n$ and $b_1,\ldots, b_n$ be real numbers. Prove that there are at most $3n\sqrt{m\log n}$ pairs $(i,j)$ with $1\leq i,j\leq n$ and $$|a_i+b_j-ij|\leq m.$$