{"status": "success", "data": {"description_md": "In $\\triangle ABC$, $AB=10$, $\\angle A=30^\\circ$, and $\\angle C=45^\\circ$. Let $H,D$, and $M$ be points on line $\\overline{BC}$ such that $\\overline{AH}\\perp\\overline{BC}$, $\\angle BAD=\\angle CAD$, and $BM=CM$. Point $N$ is the midpoint of segment $\\overline{HM}$, and point $P$ is on ray $AD$ such that $\\overline{PN}\\perp\\overline{BC}$. Then $AP^2=\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime 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": "

In \\triangle ABC, AB=10, \\angle A=30^\\circ, and \\angle C=45^\\circ. Let H,D, and M be points on line \\overline{BC} such that \\overline{AH}\\perp\\overline{BC}, \\angle BAD=\\angle CAD, and BM=CM. Point N is the midpoint of segment \\overline{HM}, and point P is on ray AD such that \\overline{PN}\\perp\\overline{BC}. Then AP^2=\\tfrac{m}{n}, where m and n are relatively prime 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": "2014 AIME II Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/14_aime_II_p15", "prev": "/problem/14_aime_II_p13"}}