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.