{"status": "success", "data": {"description_md": "Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\\mathcal{P}$ and $\\mathcal{Q}$. The intersection of planes $\\mathcal{P}$ and $\\mathcal{Q}$ is the line $\\ell$. The distance from line $\\ell$ to the point where the sphere with radius $13$ is tangent to plane $\\mathcal{P}$ is $\\tfrac{m}{n}$, 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": "

Two spheres with radii 36 and one sphere with radius 13 are each externally tangent to the other two spheres and to two different planes \\mathcal{P} and \\mathcal{Q}. The intersection of planes \\mathcal{P} and \\mathcal{Q} is the line \\ell. The distance from line \\ell to the point where the sphere with radius 13 is tangent to plane \\mathcal{P} is \\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": 5, "problem_name": "2021 AIME II Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/21_aime_II_p11", "prev": "/problem/21_aime_II_p09"}}