{"status": "success", "data": {"description_md": "Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ 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 constructed from $M_i$ towards the interior of the octagon such that $R_1 \\perp R_3$, $R_3 \\perp R_5$, $R_5 \\perp R_7$, and $R_7 \\perp R_1$. 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 $\\cos 2 \\angle A_3 M_3 B_1$ can be written in the form $m - \\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

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, and 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 constructed from M_i towards the interior of the octagon such that R_1 \\perp R_3, R_3 \\perp R_5, R_5 \\perp R_7, and 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 can be written in the form m - \\sqrt{n}, where m and n are positive integers. Find m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "2011 AIME I Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/11_aime_I_p15", "prev": "/problem/11_aime_I_p13"}}