3060.51 – Red and Blue Point Line Segment

Suppose every point in a plane is colored red or colored blue. Show that there exists some line segment in the plane such that its end points and its midpoint are all the same color.

Solution

Pick two points $x$ and $y$ that are, say, red. We propose to work with the line $xy$ (see the figure). On this line, locate points $M, N,$ and $P$ such that $Mx = xy = yP$ and $xN = Ny$.

Now, by cases:

1. If $N$ is red, we’re done: $xNy$.
2. If $N$ is blue, then look at $M$.
3. If $M$ is red, we’re done: $Mxy$.
4. If $M$ is blue, then look at $P$. 1. If $P$ is red, then we’re done: $xyP$. 2. If $P$ is blue, then we’re still done: $MNP$.

Lovely problem, eh? (Only if you like proof by cases!). Note: there are several more problems about this red and blue plane, all with the 3060 Stella number.