One of my favorite Putnam problems due to a slick solution.
$R$ is at $(3, 4)$ on the cartesian plane. To try to confuse $R$, the devious $S$ decides to rotate $R$ about the point $(1, 0)$ by $36^\circ$. $S$ then rotates $R$ by $36^\circ$ about the point $(2, 0)$, then $36^\circ$ about the point $(3, 0)$, then $(4, 0)$, etc., until finally rotating her $36^\circ$ about the point $(10, 0)$. Where does $R$ end exactly and why?
(Edit) Additional hint: Narmer and xnor have the correct solution below, but there is still a clever proof it works that no one has gotten. If you're curious, it involves only very basic geometry, and doesn't require much more than
putting a regular polygon in the right starting location.