USAJMO 2025 Problem 4

Let $n$ be a positive integer, and let $a_0, a_1,\dots, a_n$ be nonnegative integers such that $a_0\ge a_1\ge \dots\ge a_n.$ Prove that $$\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.$$ Note: $\binom{k}{2}=\frac{k(k-1)}{2}$ for all nonnegative integers $k$ .