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\begin{document}

\begin{center}
${\bf 33\rd}$ \textbf{United States of America Mathematical Olympiad}
\end{center}

\begin{center}
\textbf{Day I \hspace{6mm} 12:30 PM -- 5 PM EDT}
\end{center}

\begin{center}
\textbf{April 27, 2004}
\end{center}

\bigskip

\begin{enumerate}

\item %AND1
Let $ABCD$ be a quadrilateral circumscribed about a circle, whose
interior and exterior angles are at least $60\dg$. Prove that
\[
\frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|.
\]
When does equality hold?

\vspace{5mm}

\item %KED1
Suppose $a_1, \dots, a_n$ are integers
whose greatest common divisor is 1. Let $S$ be a set of integers with the
following properties.
\begin{enumerate}
\item[(a)] For $i=1, \dots, n$, $a_i \in S$.
\item[(b)] For $i,j = 1, \dots, n$ (not necessarily distinct), 
$a_i - a_j \in S$.
\item[(c)] For any integers $x,y \in S$, if $x+y \in S$, then $x-y \in S$.
\end{enumerate}
Prove that $S$ must be equal to the set of all integers.

\vspace{5mm}

\item %LIU1
For what real values of $k>0$ is it possible to dissect a $1 \times k$
rectangle into two similar, but noncongruent, polygons?

\end{enumerate}

\vfill
{\small
\begin{center}
Copyright \copyright\ \ Committee on the American Mathematics
Competitions,\\
Mathematical Association of America
\end{center}
}

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