3 edited tag
2 Wanted to give an additional hint for a solution.

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

# 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?