\documentclass[11pt]{scrartcl}
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\begin{document}
\title{Singapore 2017/J2/2}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 20}
\maketitle

\section*{Problem}
Let $n$ be a positive integer and $a_1$, $a_2$, \dots, $a_{2n}$
are $2n$ distinct integers.
Given the equation
\[ \left\lvert x - a_1 \right\rvert
  \left\lvert x - a_2 \right\rvert
  \dots
  \left\lvert x-a_{2n} \right\rvert
  = (n!)^2
\]
has an integer solution, determine its value in terms of the $a_i$.





\newpage

\section*{Solution}
The left-hand side has $2n$ different nonzero numbers.
So the absolute values are at least $n!^2$;
hence it follows that the
$a_i$ are a shift of $(-n, -(n-1), \dots, -1, 1, \dots, n)$.

One of several valid functions is to take
$x$ to be the average of the $a_i$.

\end{document}
