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\title{USAMO 2021/1}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 65}
\maketitle
\section*{Problem}
Rectangles $BCC_1B_2$, $CAA_1C_2$, and $ABB_1A_2$ are erected
outside an acute triangle $ABC$. Suppose that
\[ \angle BC_1C + \angle CA_1A + \angle AB_1B = 180^\circ. \]
Prove that lines $B_1C_2$, $C_1A_2$, and $A_1B_2$ are concurrent.
\section*{Video}
\href{https://www.youtube.com/watch?v=9WNgDETHOlI&t=890}{\texttt{https://youtu.be/9WNgDETHOlI}}
\newpage
\section*{Solution}
The angle condition implies the circumcircles of the three
rectangles concur at a single point $P$.
Then $\dang C P B_2 = \dang C P A_1 = 90\dg$,
hence $P$ lies on $A_1 B_2$ etc., so we're done.
\begin{remark*}
As one might guess from the two-sentence solution,
the entire difficulty of the problem
is getting the characterization of the concurrence point.
\end{remark*}
\end{document}