Question 178
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
$$
\sum_{i=0}^{n} i \binom{a_i}{2} \le \frac{1}{2} \binom{a_0 + a_1 + \dots + a_n}{2}.
$$
$$
\sum_{i=0}^{n} i \binom{a_i}{2} \le \frac{1}{2} \binom{a_0 + a_1 + \dots + a_n}{2}.
$$