14

Part of 2016 ASDAN Math Tournament

Problems(2)

Source:

In the diagram to the right, squares are drawn on the side of the triangle with side lengths

5

6

7

as shown below. The corners of adjacent squares are then connected. What is the area of the resulting hexagon?

Source:

x,y,z

are positive real numbers that satisfy \begin{align*} x+y+z&=xyz\\ \frac{x^2}{16(1+x^2)}=\frac{y^2}{25(1+y^2)}&=\frac{z^2}{36(1+z^2)}. \end{align*} Compute

\frac{x^2(1+x^2)^2}{x^2(1+z^2)^2}.