A cyclic inequality

The existence of $\lim\limits_{n\to\infty}\gamma_n$, where
$$ \gamma_n=\inf_{a_i>0}\left\{\left[\frac{a_1}{a_2+a_3}+\dots+\frac{a_{n-1}}{a_n+a_1}+\frac{a_n}{a_1+a_2}\right]:\frac{n}2\right\}. $$
is proved, and a simple method of calculating it is derived.