1999 AIME Problems/Problem 8

Problem

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

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1999 AIME Problems/Problem 8

Problem

Solution

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