Nemotron-IMO-Bench 2026 C23

Let $1<a_1<a_2<\ldots < a_n < 2a_1$ be integers and let $m$ be the number of distinct prime factors of $a_1a_2\cdots a_n$. Prove that $$(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.$$