Nemotron-IMO-Bench 2026 C12

Let $n>2^k$ and let $1\leq a_1<a_2<\ldots <a_n$ be integers. Prove that $$N:=\prod_{1\leq i<j\leq n} (a_i+a_j)$$ has at least $k+1$ pairwise distinct prime factors.