Finite number of points are colored red and blue in a plane so that the distance between 2 points with different colors is smaller than 1.
Prove that there is a circle with diameter of $\sqrt{2}$ so that there are only points of the same color outside of the circle.
I thought about inducting on the number points, but I couldn't figure out the inductive step.
Some things I have found out:
If you consider one red point all blue points must lie inside the circle centered at the red point with radius 1. If there are multiple red points the blue points must lie inside the intersection of these circles. Furthermore if there are points of both colors then all of the points can be covered by a circle with radius 2.
Some progress: I was thinking about inducting and this is my line of thought. First of all we assume that there are points of both colors (otherwise it is trivial). Then we try to induct on $x:=$ the number of red points. The cases $x=1$ and $x=2$ are trivial.
Let's look at the case of $x=3$: Because of the condition that blue points have to exist the red points have to form a triangle, such that no side is longer than 2 (we can actually get something stronger but I can't precisely find out what).
@Hugo604 I don't really know what competition you are talking about. I found this problem on aops and thought it was a nice problem. If you could provide a link to the problems of the German homework problem I would delete the problem myself.