{"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": "<p>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$.</p>&#10;<hr><p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "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"}}