MathDB

8

Part of 1999 AIME Problems

Problems(1)

Supporting triples

Source:

12/18/2005
Let T\mathcal{T} be the set of ordered triples (x,y,z)(x,y,z) of nonnegative real numbers that lie in the plane x+y+z=1.x+y+z=1. Let us say that (x,y,z)(x,y,z) supports (a,b,c)(a,b,c) when exactly two of the following are true: x≥a,y≥b,z≥c.x\ge a, y\ge b, z\ge c. Let S\mathcal{S} consist of those triples in T\mathcal{T} that support (12,13,16).\left(\frac 12,\frac 13,\frac 16\right). The area of S\mathcal{S} divided by the area of T\mathcal{T} is m/n,m/n, where mm and nn are relatively prime positive integers, find m+n.m+n.
Supportgeometryanalytic geometryparallelograminequalitiesAMCAIME