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2 Wanted to give an additional hint for a solution.
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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.

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?

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.

1
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Do a barrel roll! (i.e. a Euclidean plane rotation puzzle)

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?