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?

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(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.