We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.
James furiously cuts a rectangular sheet of paper with scissors. Every second he cuts a random piece by an unsystematic rectilinear cut into two parts.
a) Find the mathematical expectation of the number of sides of a polygon (made from a piece of paper) that James randomly picks up after an hour of such work.
b) Solve the same problem if at first the piece of paper had the form of an arbitrary polygon.