INMO '92 Question 9:
Find $n$ such that in a regular $n$-gon $A_1A_2 ...A_n$ we have $$\frac{1}{A_1A_2}=\frac{1}{A_1A_3}+\frac{1}{A_1A_4}$$
I tried the following
Assume it is inscribed in a circle.
Then length of chord is $2\sin(\theta)$ where $\theta$ is half the angle subtended at the center between consecutive points. So, $\theta=\frac{180^\circ}{n}$.
Then we get $$\csc(\theta)=\csc(2\theta)+\csc(3\theta)$$ Not sure quite how to proceed from there- using double and triple angle formulae doesn't seem to work