Points on a line are colored in 2 colours. Prove that it is always possible to find three points of the same color with one being the midpoint.
I took the 2 colours to be red and blue. I approached this problem in the following way:
I tried to prove it by contradiction by trying to make a configuration where the midpoint of two points of the same colour can never be the same colour. Take a line segment from the line such that the endpoints are both red. Then consider the midpoint to be blue as the condition would be satisfied anyway if it was red. Then from the 2 endpoints keep going closer to the midpoint and we get a pair of points in each case.
Since the midpoint is blue I have to force the pair of points to be both red or a pair of red and blue. I tried drawing this out and could not deduce anything of use from this.
I want to know how to approach this problem and what would be the thought process in this kind of case, rather than a direct solution.
Suppose otherwise.
Assume the line is the real number line with points labeled by their values.
Let $x,y$ be distinct points with the same color, say color $A$.
Then the points $2x-y$ and $2y-x$ must both have the other color.
Now what can you say about the color of the point ${\Large{\frac{x+y}{2}}}$?