{"status": "success", "data": {"description_md": "Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\\sqrt{3}$, and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$?\n\n$\\textbf{(A)} ~47\\qquad\\textbf{(B)} ~52\\qquad\\textbf{(C)} ~55\\qquad\\textbf{(D)} ~58\\qquad\\textbf{(E)} ~63$", "description_html": "

Let ABCDEF be an equiangular hexagon. The lines AB, CD, and EF determine a triangle with area 192\\sqrt{3} , and the lines BC, DE, and FA determine a triangle with area 324\\sqrt{3} . The perimeter of hexagon ABCDEF can be expressed as m +n\\sqrt{p} , where m, n, and p are positive integers and p is not divisible by the square of any prime. What is m + n + p ?

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\\textbf{(A)} ~47\\qquad\\textbf{(B)} ~52\\qquad\\textbf{(C)} ~55\\qquad\\textbf{(D)} ~58\\qquad\\textbf{(E)} ~63

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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": 4, "problem_name": "2021 AMC 10A Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc10A_p22", "prev": "/problem/21_amc10A_p20"}}