\documentclass[11pt]{scrartcl}
\usepackage{evan}

\begin{document}
\title{Twitch 072.1}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 72}
\maketitle

\section*{Problem}
Each point of a three-dimensional space is colored
with one of two colors such that
whenever an isosceles triangle $ABC$ with $AB = AC$
has vertices of the same color $c$ it follows that
the midpoint of $BC$ also is colored with $c$.
Prove that there exists a perpendicular square prism
with all vertices of equal color.

\section*{Video}
\href{https://www.youtube.com/watch?v=7Qjr5CwzQDU&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/7Qjr5CwzQDU}}



\newpage

\section*{Solution}
Assume for contradiction that this is not the case,
and call the colors red and green.
We proceed in several steps.
\begin{claim*}
  The midpoint of any two points of the same color is the same color.
\end{claim*}
\begin{proof}
  Assume for contradiction we found two green points $B$ and $C$
  whose midpoint is red.
  Then every point on the perpendicular bisecting plane
  $\mathcal P$ of $B$ and $C$ is necessarily also red.

  Now, take any red point $X$ not on $\mathcal P$
  (it clearly exists or we'd be done).
  Then every point on the plane halfway between $X$ and $\mathcal P$
  red would be red too;
  since if the midpoint of $XP$ was green for $P \in \mathcal P$,
  then we can complete it to a red isosceles triangle $XPQ$
  for some $Q \in \mathcal P$.

  Since we have two parallel entirely red planes, contradiction.
\end{proof}

So, assume henceforth the claim.
\begin{claim*}
  In any two-coloring of the following set,
  there is a monochromatic isosceles right triangle:
  \begin{center}
  \begin{asy}
    dotfactor *= 2;
    size(3cm);
    dot( (0,0) );
    dot( (1,1) );
    dot( (1,-1) );
    dot( (-1,1) );
    dot( (-1,-1) );
  \end{asy}
  \end{center}
\end{claim*}
\begin{proof}
  Exhaustive check.
  (Note that if two opposite vertices are the same color,
  the midpoint shares that color too.)
\end{proof}

Take $33$ parallel stacks of that set.
Then you can find two whose colorings are the same.
On the other hand, given a monochromatic right isosceles triangle,
taking all three midpoints gives a monochromatic square.
The end.

\end{document}
