GULF 2012 Problem 4

Fawzi cuts a spherical cheese completely into (at least three) slices of equal thickness. He starts at one end, making successive parallel cuts, working through the cheese until the slicing is complete. The discs exposed by the first two cuts have integral areas.

list Prove that all the discs that he cuts have integral areas. (ii) Prove that the original sphere had integral surface area if, and only if, the area of the second disc that he exposes is even.[/list]