{"status": "success", "data": {"description_md": "Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\\overline{BC}, \\overline{CA},$ and $\\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\\overline{DE}, \\overline{EF},$ and $\\overline{FD},$ respectively, with the property that $P$ is on $\\overline{CQ}, Q$ is on $\\overline{AR},$ and $R$ is on $\\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a+b\\sqrt{c},$ where $a, b$ and $c$ are integers, and $c$ is not divisible by the square of any prime. What is $a^{2}+b^{2}+c^{2}$?\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": "

Let ABC be equilateral, and D, E, and F be the midpoints of \\overline{BC}, \\overline{CA}, and \\overline{AB}, respectively. There exist points P, Q, and R on \\overline{DE}, \\overline{EF}, and \\overline{FD}, respectively, with the property that P is on \\overline{CQ}, Q is on \\overline{AR}, and R is on \\overline{BP}. The ratio of the area of triangle ABC to the area of triangle PQR is a+b\\sqrt{c}, where a, b and c are integers, and c is not divisible by the square of any prime. What is a^{2}+b^{2}+c^{2}?

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": "1998 AIME Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/98_aime_p13", "prev": "/problem/98_aime_p11"}}