CAPS 1996 Problem 3

The base of a regular quadrilateral pyramid $\pi$ is a square with side length $2a$ and its lateral edge has length a $\sqrt{17}$ . Let $M$ be a point inside the pyramid. Consider the five pyramids which are similar to $\pi$ , whose top vertex is at $M$ and whose bases lie in the planes of the faces of $\pi$ . Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of $\pi$ , and find for which $M$ equality holds.