Answer to Question #266500 in Discrete Mathematics for Vishnu

Question

Each point on a straight line is colored either red or blue. Prove that we can

find three points of the same color such that one is the midpoint of the other two.

Answer 1

Let us consider two points A and B of the same color.

Let O be the midpoint of the points A and B;

A’ be a point such that B is a midpoint of the points A and A’;

B’ be a point such that A is a midpoint of the points B and B’.

If one of these points (O, A’, B’) has the same color as points A and B, then we’ve found three points of the same color such that one is the midpoint of the other two.

Otherwise points O, A’, B’ have the same color (if A and B are blue, then O, A’, B’ are red).

So, the points O, A’, B’ are the same color and O is midpoint of A’ and B’.