ISL 1966 Problem 43

Given $5$ points in a plane, no three of them being collinear. Each two of these $5$ points are joined with a segment, and every of these segments is painted either red or blue; assume that there is no triangle whose sides are segments of equal color.

a.) Show that:

(1) Among the four segments originating at any of the $5$ points, two are red and two are blue.

(2) The red segments form a closed way passing through all $5$ given points. (Similarly for the blue segments.)

b.) Give a plan how to paint the segments either red or blue in order to have the condition (no triangle with equally colored sides) satisfied.