CMO 1988

Suppose that $S$ is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle (i) whose vertices are all the same colour, and (ii) at least one side of the triangle does not contain a point of the opposite colour.

Let $x_{n + 1} = 4x_n - x_{n - 1}$ , $x_0 = 0$ , $x_1 = 1$ , and $y_{n + 1} = 4y_n - y_{n - 1}$ , $y_0 = 1$ , $y_1 = 2$ . Show that for all $n \ge 0$ that $y_n^2 = 3x_n^2 + 1$ .