3060.52 – Red and Blue Point Triangle

Suppose every point in a plane is colored either red or blue. Show that there is an triangle somewhere in the plane whose vertices are all the same color.

Solution

  1. Find two points, A and B, that are the same color, say blue. Then make a triangular array of points as shown in the figure below.

  2. If C or D is blue, we're done. So assume C is red, and D is red.

  3. If E is red, we're done, △\triangle CDE is red. If E is blue, then we're done, △\triangle ABE is blue.

3060_52_solution_54f02de5d7.png