Middle European Mathematical Olympiad 2010 Problem T-2

For each integer $n \geq 2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$, we have $$\frac{a_1^2 + \cdots + a_n^2}{n} \geq \left(\frac{a_1 + \cdots + a_n}{n}\right)^2 + C_n \cdot (a_1 - a_n)^2.$$