Nemotron-IMO-Bench 2026 A30

Let $m,n$ be nonnegative integers and let $a_0,\ldots, a_m, b_0,\ldots, b_n$ nonnegative real numbers. Let $c_k:=\max_{i+j=k}a_ib_j.$ Prove that $$\frac 1{m+n+1}\sum_{k=0}^{m+n}c_k\ge\frac 1{(m+1)(n+1)}\sum_{i=0}^{m}a_i\sum_{j=0}^{n}b_j.$$