{"status": "success", "data": {"description_md": "Circles $\\omega_1$ and $\\omega_2$ intersect at two points $P$ and $Q$, and their common tangent line closer to $P$ intersects $\\omega_1$ and $\\omega_2$ at points $A$ and $B$, respectively. The line parallel to line $AB$ that passes through $P$ intersects $\\omega_1$ and $\\omega_2$ for the second time at points $X$ and $Y$, respectively. Suppose $PX = 10, PY = 14,$ and $PQ = 5$. Then the area of trapezoid $XABY$ is $m\\sqrt{n}$ 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": "

Circles \\omega_1 and \\omega_2 intersect at two points P and Q, and their common tangent line closer to P intersects \\omega_1 and \\omega_2 at points A and B, respectively. The line parallel to line AB that passes through P intersects \\omega_1 and \\omega_2 for the second time at points X and Y, respectively. Suppose PX = 10, PY = 14, and PQ = 5. Then the area of trapezoid XABY is 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": 4, "problem_name": "2023 AIME II Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/23_aime_II_p10", "prev": "/problem/23_aime_II_p08"}}