CN-TST 2021 Problem 11

Let $n$ be a positive integer and $a_1,a_2,\ldots a_{2n+1}$ be positive reals. For $k=1,2,\ldots ,2n+1$ , denote $b_k = \max_{0\le m\le n}\left(\frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right)$ , where indices are taken modulo $2n+1$ . Prove that the number of indices $k$ satisfying $b_k\ge 1$ does not exceed $2\sum_{i=1}^{2n+1} a_i$ .