\documentclass[11pt]{scrartcl}
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\begin{document}
\title{JBMO SL 2008 A2}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 71}
\maketitle

\section*{Problem}
Find all real numbers $a,b,c,d$ such that
\begin{align*}
  a+b+c+d &= 20 \\
  ab+bc+cd+da+ac+bd &= 150.
\end{align*}

\section*{Video}
\href{https://www.youtube.com/watch?v=dlrSmf05n-w&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/dlrSmf05n-w}}



\newpage

\section*{Solution}
Note that
\[ a^2+b^2+c^2+d^2 = 20^2 - 2 \cdot 150 = 100 \]
but by Cauchy-Schwarz
\[ 100 \cdot 4 = (a^2+b^2+c^2+d^2)(1+1+1+1) \ge (a+b+c+d)^2 = 400 \]
so equality must hold meaning $a=b=c=d=5$,
and this clearly works.

\end{document}
