Nemotron-IMO-Bench 2026 N48

Let $n>1$ be an integer. Prove that $2n-1$ is a prime number if and only if for any $n$ pairwise distinct positive integers $a_1, a_2, \ldots , a_n$ there exist $i, j \in \{1, 2, \ldots , n\}$ such that $$\frac{a_i+a_j}{\gcd(a_i,a_j)} \geq 2n-1.$$