Nemotron-IMO-Bench 2026 A27

Let $n\geq 4$ and let $a_1\geq a_2\geq\ldots\geq a_n$ be real numbers such that $a_1=a_2$ and $a_{n-1}=a_n$. Prove that $$n\sum_{i=1}^n a_ia_{i+1}\geq (\sum_{i=1}^n a_i)^2,$$ where $a_{n+1}=a_1$.