{"status": "success", "data": {"description_md": "Let $ ABCD$ be an isosceles trapezoid with $ \\overline{AD}||\\overline{BC}$ whose angle at the longer base $ \\overline{AD}$ is $ \\dfrac{\\pi}{3}$. The diagonals have length $ 10\\sqrt {21}$, and point $ E$ is at distances $ 10\\sqrt {7}$ and $ 30\\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \\overline{AD}$. The distance $ EF$ can be expressed in the form $ 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": "<p>Let $ ABCD$ be an isosceles trapezoid with $ \\overline{AD}||\\overline{BC}$ whose angle at the longer base $ \\overline{AD}$ is $ \\dfrac{\\pi}{3}$. The diagonals have length $ 10\\sqrt {21}$, and point $ E$ is at distances $ 10\\sqrt {7}$ and $ 30\\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \\overline{AD}$. The distance $ EF$ can be expressed in the form $ m\\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. 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": "2008 AIME I Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/08_aime_I_p11", "prev": "/problem/08_aime_I_p09"}}