{"status": "success", "data": {"description_md": "In $\\triangle ABC$ with side lengths $AB=13$, $BC=14$, and $CA=15$, let $M$ be the midpoint of $\\overline{BC}$. Let $P$ be the point on the circumcircle of $\\triangle ABC$ such that $M$ is on $\\overline{AP}$. There exists a unique point $Q$ on segment $\\overline{AM}$ such that $\\angle PBQ = \\angle PCQ$. Then $AQ$ can be written as $\\frac{m}{\\sqrt{n}}$, 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 with side lengths AB=13, BC=14, and CA=15, let M be the midpoint of \\overline{BC}. Let P be the point on the circumcircle of \\triangle ABC such that M is on \\overline{AP}. There exists a unique point Q on segment \\overline{AM} such that \\angle PBQ = \\angle PCQ. Then AQ can be written as \\frac{m}{\\sqrt{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": 5, "problem_name": "2023 AIME II Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/23_aime_II_p13", "prev": "/problem/23_aime_II_p11"}}