IR-TST 2010 Problem 9

Sequence of real numbers $a_0,a_1,\dots,a_{1389}$ are called concave if for each $0<i<1389$ , $a_i\geq\frac{a_{i-1}+a_{i+1}}2$ . Find the largest $c$ such that for every concave sequence of non-negative real numbers: $$\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2$$