Generalized Inequality!

By Rearrangement Inequality, we have $\sum_{cyc} \frac{a_{1}}{a_{2}+a_{3}} \geq \frac{a_{1}}{a_{1}+a_{n}}$ and $\sum_{cyc} \frac{a_{1}}{a_{2}+a_{3}} \geq \frac{a_{n}}{a_{1}+a_{n}}$ adding them up, we have $\sum_{cyc} \frac{a_{1}}{a_{2}+a_{3}} \geq \frac{n}{2}$ and hence, we have $m=\frac{n}{2}$ , so $\frac{m}{n}=\boxed{2}$