Baltic Way 1996 · Problem 1

Solution:

Let $O$ be the circumcentre of the 1996-gon. Consider two diagonals $AB$ and $CD$. There is a rotation around $O$ that takes the point $C$ to $A$ and $D$ to a point $D'$. Clearly the angle of this rotation is a multiple of $2\varphi = 2\pi / 1996$.

The angle $BAD'$ is the inscribed angle on the arc $BD'$, and hence is an integral multiple of $\varphi$, the inscribed angle on the arc between any two adjacent vertices of the 1996-gon. Hence the angle between $AB$ and $CD$ is also an integral multiple of $\varphi$.

Since both $\alpha$ and $\beta$ are integral multiples of $\varphi$, $\alpha / \beta$ is a rational number.