{"status": "success", "data": {"description_md": "Let $\\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\\ge a, y\\ge b, z\\ge c.$ Let $\\mathcal{S}$ consist of those triples in $\\mathcal{T}$ that support $\\left(\\frac 12,\\frac 13,\\frac 16\\right).$ The area of $\\mathcal{S}$ divided by the area of $\\mathcal{T}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$\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": "

Let \\mathcal{T} be the set of ordered triples (x,y,z) of nonnegative real numbers that lie in the plane x+y+z=1. Let us say that (x,y,z) supports (a,b,c) when exactly two of the following are true: x\\ge a, y\\ge b, z\\ge c. Let \\mathcal{S} consist of those triples in \\mathcal{T} that support \\left(\\frac 12,\\frac 13,\\frac 16\\right). The area of \\mathcal{S} divided by the area of \\mathcal{T} is m/n, where m and n are relatively prime positive integers, find m+n.

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": 4, "problem_name": "1999 AIME Problem 8", "can_next": true, "can_prev": true, "nxt": "/problem/99_aime_p09", "prev": "/problem/99_aime_p07"}}