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

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

\begin{center}
\textbf{April 28, 2004}
\end{center}

\bigskip

\begin{enumerate}
\setcounter{enumi}{3}

\item %WOO1
Alice and Bob play a game on a 6 by 6 grid.  On his or her turn, a
player chooses a rational number not yet appearing in the grid and
writes it in an empty square of the grid.  Alice goes first and then
the players alternate.  When all squares have numbers written in them,
in each row, the square with the greatest number in that row is
colored black.  Alice wins if she can then draw a line from the top of
the grid to the bottom of the grid that stays in black squares, and
Bob wins if she can't.  (If two squares share a vertex, Alice can draw
a line from one to the other that stays in those two squares.)  Find,
with proof, a winning strategy for one of the players.

\vspace{5mm}

\item %AND2
Let $a, b$ and $c$ be positive real numbers. Prove that
\[
(a^5-a^2+3)(b^5-b^2+3)(c^5-c^2+3) \ge (a+b+c)^3.
\]

\vspace{5mm}

\item %FEN1
A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the
center of $\omega$. Suppose that
\[
(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2.
\]
Prove that $ABCD$ is an isosceles trapezoid.

\end{enumerate}

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

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