{"status": "success", "data": {"description_md": "In $\\triangle ABC$, $AB = 3$, $BC = 4$, and $CA = 5$. Circle $\\omega$ intersects $\\overline{AB}$ at $E$ and $B$, $\\overline{BC}$ at $B$ and $D$, and $\\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\\tfrac{DG}{EG} = \\tfrac{3}{4}$, length $DE=\\tfrac{a\\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.\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": "

In \\triangle ABC, AB = 3, BC = 4, and CA = 5. Circle \\omega intersects \\overline{AB} at E and B, \\overline{BC} at B and D, and \\overline{AC} at F and G. Given that EF=DF and \\tfrac{DG}{EG} = \\tfrac{3}{4}, length DE=\\tfrac{a\\sqrt{b}}{c}, where a and c are relatively prime positive integers, and b is a positive integer not divisible by the square of any prime. Find a+b+c.

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": 6, "problem_name": "2014 AIME I Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/14_aime_I_p14"}}