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

\begin{document}
\title{Twitch 123.4}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 123}
\maketitle

\section*{Problem}
Find the real numbers $x$ and $y$ that minimize the sum
\[ \sqrt{2(x^2 + y^2)} + \sqrt{2(x^2 + y^2) + 2(x - y) + 1}
  + \sqrt{2(x^2 + y^2) + 2(x + y) + 1}. \]

\section*{Video}
\href{https://www.youtube.com/watch?v=Qvq4pTg-C8o&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/Qvq4pTg-C8o}}



\newpage

\section*{Solution}
Let $S$ be sum. Then
\begin{align*}
\frac{S}{\sqrt2} &= \sqrt{x^2+y^2} \\
&+ \sqrt{(x+1/2)^2+(y-1/2)^2} \\
&+ \sqrt{(x+1/2)^2+(y+1/2)^2}
\end{align*}
Let
\begin{align*}
A &= (0,0) \\
B &= (-1/2, 1/2) \\
C &= (-1/2, -1/2).
\end{align*}
Want to minimize $PA+PB+PC$ where $P \in \RR^2$,
so take $P$ to be the Fermat point:
\[ P = \left( -\half + \frac{1}{2\sqrt3}, 0 \right). \]

\end{document}
