{"status": "success", "data": {"description_md": "Let $ \\overline{MN}$ be a diameter of a circle with diameter $ 1$. Let $ A$ and $ B$ be points on one of the semicircular arcs determined by $ \\overline{MN}$ such that $ A$ is the midpoint of the semicircle and $ MB=\\frac35$. Point $ C$ lies on the other semicircular arc. Let $ d$ be the length of the line segment whose endpoints are the intersections of diameter $ \\overline{MN}$ with the chords $ \\overline{AC}$ and $ \\overline{BC}$. The largest possible value of $ d$ can be written in the form $ r-s\\sqrt{t}$, where $ r$, $ s$, and $ t$ are positive integers and $ t$ is not divisible by the square of any prime. Find $ r+s+t$.\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 $ \\overline{MN}$ be a diameter of a circle with diameter $ 1$. Let $ A$ and $ B$ be points on one of the semicircular arcs determined by $ \\overline{MN}$ such that $ A$ is the midpoint of the semicircle and $ MB=\\frac35$. Point $ C$ lies on the other semicircular arc. Let $ d$ be the length of the line segment whose endpoints are the intersections of diameter $ \\overline{MN}$ with the chords $ \\overline{AC}$ and $ \\overline{BC}$. The largest possible value of $ d$ can be written in the form $ r-s\\sqrt{t}$, where $ r$, $ s$, and $ t$ are positive integers and $ t$ is not divisible by the square of any prime. Find $ r+s+t$.

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": "2009 AIME II Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/09_aime_II_p14"}}