2022 SSMO Speed Round Problems/Problem 7

Problem

Let $A_1=(1, 0)$ . Define $A_{n+1}$ as the image of $A_n$ under a rotation of either $45^{\circ}$ , $90^{\circ}$ , or $135^{\circ}$ clockwise about the origin, with each choice having a $\frac{1}{3}$ chance of being selected. Find the expected value of the smallest positive integer $n>1$ such that $A_n$ coincides with $A_1$ .

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2022 SSMO Speed Round Problems/Problem 7

Problem

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