ELMO SL 2011 A4

In terms of $n\ge2$ , find the largest constant $c$ such that for all nonnegative $a_1,a_2,\ldots,a_n$ satisfying $a_1+a_2+\cdots+a_n=n$ , the following inequality holds: $$\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.$$

Calvin Deng.