Question 162
Fix integers $n \ge 2$ and $1 \le m \le n-1$. Let $a_0$, $a_1$, $\dots$, $a_n$ be non-negative real numbers satisfying $a_0 + a_1 + \dots + a_n = 1$. Prove that, if $\sum_{k=0}^n a_k x^k \lt x^m$ for some $0 \lt x \lt 1$, then $\sum_{k=0}^{m-1} (m-k)a_k \lt \sum_{k=m+1}^n (k-m)a_k$.
The Problem Selection Committee
The Problem Selection Committee