{"status": "success", "data": {"description_md": "In right $\\triangle ABC$ with hypotenuse $\\overline{AB}$, $AC = 12$, $BC = 35$, and $\\overline{CD}$ is the altitude to $\\overline{AB}$. Let $\\omega$ be the circle having $\\overline{CD}$ as a diameter. Let $I$ be a point outside $\\triangle ABC$ such that $\\overline{AI}$ and $\\overline{BI}$ are both tangent to circle $\\omega$. The ratio of the perimeter of $\\triangle ABI$ to the length $AB$ can be expressed in the form $\\displaystyle\\frac{m}{n}$, 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": "

In right \\triangle ABC with hypotenuse \\overline{AB}, AC = 12, BC = 35, and \\overline{CD} is the altitude to \\overline{AB}. Let \\omega be the circle having \\overline{CD} as a diameter. Let I be a point outside \\triangle ABC such that \\overline{AI} and \\overline{BI} are both tangent to circle \\omega. The ratio of the perimeter of \\triangle ABI to the length AB can be expressed in the form \\displaystyle\\frac{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": "2009 AIME I Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/09_aime_I_p13", "prev": "/problem/09_aime_I_p11"}}