KöMaL 2024 B5370

Let $k$ be a positive integer, and assume that real numbers $a_1$, $\ldots$, $a_k$ satisfy ${\sum_{i=1}^k (k-i+1) a_i=0}$. Prove that there exists positive integer $m\le k$ for which $2m\sum_{i=1}^{m} a_i\le \sum_{i=1}^m i a_i$.