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Supporting triples

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December 18, 2005

Supportgeometryanalytic geometryparallelograminequalitiesAMCAIME

Problem Statement

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.

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