USAJMO 2025 Question 4

Let $n$ be a positive integer and let $a_0\ge a_1\ge\cdots\ge a_n\ge0$ be integers. Prove that $$\sum_{i=0}^n i\binom{a_i}{2}\le\frac12\binom{a_0+a_1+\cdots+a_n}{2}.$$

Proposed by Hung-Hsun Yu.

Manual essay; 1 editorial point, not official contest scoring.