KöMaL 2011 B4340

Prove that the following inequality is true for all positive numbers $a_1$,$a_2$,...,$a_n$. $\left(\frac{a_1}{a_2+\dots+a_n}\right)^2+\left(\frac{a_2}{a_3+\dots+a_1}\right)^2+\ldots + \left(\frac{a_n}{a_1+\dots+a_{n-1}}\right)^2\ge \frac{n}{{(n-1)}^2}.$