{"status": "success", "data": {"description_md": "Torus $\\mathcal T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane a distance $6$ from the center of the circle. When a sphere of radius $11$ rests inside $\\mathcal T$, it is internally tangent to $\\mathcal T$ along a circle with radius $r_{i}$, and when it rests outside $\\mathcal T$, it is externally tangent along a circle with radius $r_{o}$. The difference $r_{i}-r_{o}=\\tfrac{m}{n}$, 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": "

Torus \\mathcal T is the surface produced by revolving a circle with radius 3 around an axis in the plane a distance 6 from the center of the circle. When a sphere of radius 11 rests inside \\mathcal T, it is internally tangent to \\mathcal T along a circle with radius r_{i}, and when it rests outside \\mathcal T, it is externally tangent along a circle with radius r_{o}. The difference r_{i}-r_{o}=\\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": 4, "problem_name": "2024 AIME II Problem 8", "can_next": true, "can_prev": true, "nxt": "/problem/24_aime_II_p09", "prev": "/problem/24_aime_II_p07"}}