Estonian Math Competitions

Consider any circle such that all of the endpoints of the marked rays are inside the circle. Choose a tangent line of the circle that is not parallel to any of the marked rays. Let this tangent line be $l_0$ and let $l_{\alpha }$ be the tangent line we get by rotating $l_0$ counterclockwise by angle $\alpha$ with respect to the center of the circle. For any $\alpha$, let $f(\alpha )$ be the number of marked rays that intersect with $l_{\alpha }$. Since $l_0$ and $l_{\pi }$ are two parallel lines and all of the endpoints of the marked rays are between them, we know from the definition of $l_0$ that $f(0)+f(\pi )=2n$. Without loss of generality, let $f(0)\le n$ and $f(\pi )\ge n$. Because no two rays have the same direction, when $\alpha$ increases continuously $f(\alpha )$ can at any time instance only change by at most one. Thus $f(\alpha )$ ranges over all integral values between $f(0)$ and $f(\pi )$. Hence $f(\alpha )=n$ for some $\alpha$.

The argument above does not use the assumption that the initial points of the marked rays are distinct, whence it solves both parts of the problem.