China Mathematical Competition

As $\sin x$ is a convex function for $x\in (0,\frac{\pi }{6})$, we have $\frac{3}{\pi }x<\sin x<x$. Then $$\sin \frac{\pi }{13}<\frac{\pi }{13}<\frac{1}{4},\sin \frac{\pi }{12}>\frac{3}{\pi }\times \frac{\pi }{12}=\frac{1}{4},$$ $$\sin \frac{\pi }{10}<\frac{\pi }{10}<\frac{1}{3},\sin \frac{\pi }{9}>\frac{3}{\pi }\times \frac{\pi }{9}=\frac{1}{3},$$ that is $$\sin \frac{\pi }{13}<\frac{1}{4}<\sin \frac{\pi }{12}<\sin \frac{\pi }{11}<\sin \frac{\pi }{10}<\frac{1}{3}<\sin \frac{\pi }{9}.$$ Therefore, all the possible values of positive integers $n$ are $10,11,12$, and their sum is 33. The answer is 33.