Question 23
Let $n$ be a positive integer, $a_1, a_2, \dots, a_{2n+1}$ be $2n+1$ positive real numbers. For $k = 1, 2, \dots, 2n+1$, define
$$
b_k = \max_{0 \le m \le n} \left( \frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right),
$$
where the subscript of $a_i$ is taken modulo $2n+1$. Prove: the number of subscripts $k$ satisfying that $b_k \ge 1$ does not exceed $2 \sum_{i=1}^{2n+1} a_i$.
$$
b_k = \max_{0 \le m \le n} \left( \frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right),
$$
where the subscript of $a_i$ is taken modulo $2n+1$. Prove: the number of subscripts $k$ satisfying that $b_k \ge 1$ does not exceed $2 \sum_{i=1}^{2n+1} a_i$.