Question 8
Let $T$ 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 $S$ consist of those triples in $T$ that support $\left(\frac{1}{2}, \frac{1}{3}, \frac{1}{6}\right).$ The area of $S$ divided by the area of $T$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$