2011 AIME I Problems/Problem 14

Let $A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}A_{7}A_{8}$ be a regular octagon. Let $M_{1},M_{3},M_{5},M_{7}$ be the midpoints of sides $\overline{A_{1}A_{2}},\overline{A_{3}A_{4}},\overline{A_{5}A_{6}}$ , and $\overline{A_{7}A_{8}}$ , respectively. For $i=1,3,5,7$ , ray $R_{i}$ is suspended such that $R_{1}\perp R_{3}, R_{3}\perp R_{5}, R_{5}\perp R_{7}, R_{7}\perp R_{1}$ . Pairs of rays $R_{1}$ and $R_{3}$ , $R_{3}$ and $R_{5}$ , $R_{5}$ and $R_{7}$ , and $R_{7}$ and $R_{1}$ meet at $B_{1},B_{3},B_{5},B_{7}$ , respectively. If $B_{1}B_{3}=A_{1}A_{2}$ , then $\cos{2\angle{A_{3}M_{3}B_{1}}$ (Error compiling LaTeX. Unknown error_msg) can be written as $m-\sqrt{n}$ , where $m$ and $n$ are positive integers. Find $m+n$ .

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2011 AIME I Problems/Problem 14