{"status": "success", "data": {"description_md": "Rectangle $ABCD$ and a semicircle with diameter $AB$ are coplanar and have nonoverlapping interiors. Let $\\mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $\\ell$ meets the semicircle, segment $AB$, and segment $CD$ at distinct points $N$, $U$, and $T$, respectively. Line $\\ell$ divides region $\\mathcal{R}$ into two regions with areas in the ratio $1: 2$. Suppose that $AU = 84$, $AN = 126$, and $UB = 168$. Then $DA$ can be represented as $m\\sqrt {n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. 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": "

Rectangle ABCD and a semicircle with diameter AB are coplanar and have nonoverlapping interiors. Let \\mathcal{R} denote the region enclosed by the semicircle and the rectangle. Line \\ell meets the semicircle, segment AB, and segment CD at distinct points N, U, and T, respectively. Line \\ell divides region \\mathcal{R} into two regions with areas in the ratio 1: 2. Suppose that AU = 84, AN = 126, and UB = 168. Then DA can be represented as m\\sqrt {n}, where m and n are positive integers and n is not divisible by the square of any prime. 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": "2010 AIME I Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/10_aime_I_p14", "prev": "/problem/10_aime_I_p12"}}