Question 119
We call a sequence $a_0, \ldots, a_{1389}$ of real numbers *concave* if for every $0 \lt i \lt 1389$ we have $a_i \ge \frac{a_{i-1} + a_{i+1}}{2}$. Find the maximum number $c$ such that for every *concave* sequence of nonnegative numbers, we have
$$
\sum_{i=0}^{1389} i a_i^2 \ge c \sum_{i=0}^{1389} a_i^2
$$
$$
\sum_{i=0}^{1389} i a_i^2 \ge c \sum_{i=0}^{1389} a_i^2
$$