KöMaL 2005 B3829

Let $a_1$, $a_2$, ..., $a_n$ denote positive numbers. Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+\ldots+\frac{a_{n-1}^2}{a_{n-1}+a_n}+ \frac{a_n^2}{a_n+a_1}\ge\frac{1}{2}(a_1+a_2+\ldots+a_n).$