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.
What is the minimum number of points necessary to mark inside a convex \(n\)-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?