{"status": "success", "data": {"description_md": "A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\\frac m{\\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\\lfloor m+\\sqrt{n}\\rfloor.$ (The notation $\\lfloor x\\rfloor$ denotes the greatest integer that is less than or equal to $x$.)\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>A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let <span class=\"katex--inline\">h</span> be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then <span class=\"katex--inline\">h</span> can be written in the form <span class=\"katex--inline\">\\frac m{\\sqrt{n}},</span> where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are positive integers and <span class=\"katex--inline\">n</span> is not divisible by the square of any prime. Find <span class=\"katex--inline\">\\lfloor m+\\sqrt{n}\\rfloor.</span> (The notation <span class=\"katex--inline\">\\lfloor x\\rfloor</span> denotes the greatest integer that is less than or equal to <span class=\"katex--inline\">x</span>.)</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": 6, "problem_name": "2006 AIME I Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/06_aime_I_p15", "prev": "/problem/06_aime_I_p13"}}