\documentclass[12pt]{article}
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\def\calA{\mathcal{A}}
\def\calB{\mathcal{B}}
\def\calF{\mathcal{F}}
\def\calG{\mathcal{G}}
\def\calL{\mathcal{L}}
\def\calO{\mathcal{O}}
\def\calP{\mathcal{P}}
\def\calR{\mathcal{R}}
\def\calS{\mathcal{S}}
\def\calT{\mathcal{T}}
\def\ida{\mathfrak{a}}
\def\idb{\mathfrak{b}}
\def\idm{\mathfrak{m}}
\def\idn{\mathfrak{n}}
\def\idp{\mathfrak{p}}
\def\CC{\mathbb{C}}
\def\FF{\mathbb{F}}
\def\NN{\mathbb{N}}
\def\PP{\mathbb{P}}
\def\QQ{\mathbb{Q}}
\def\RR{\mathbb{R}}
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}
\def\ZZ{\mathbb{Z}}
\def\bov{\mathbf{v}}

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\def\soln{{\bf \noindent Solution:  } \nopagebreak}
\def\first{{\bf \noindent First Solution:  } \nopagebreak}
\def\second{{\bf \noindent Second Solution:  } \nopagebreak}
\def\third{{\bf \noindent \newline Third Solution:  } \nopagebreak}
\def\fourth{{\bf \noindent \newline Fourth Solution:  } \nopagebreak}
\def\fifth{{\bf \noindent \newline Fifth Solution:  } \nopagebreak}
\def\sixth{{\bf \noindent \newline Sixth Solution:  } \nopagebreak}
\def\note{{\bf \noindent \newline Note:  } \nopagebreak}
\def\comment{{\bf \noindent Comment:  } \nopagebreak}
\def\hint{{\bf \noindent \newline Hint:  } \nopagebreak}
\def\source{{\bf \noindent \newline Source:  } \nopagebreak}

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\def\lf{\lfloor}
\def\rf{\rfloor}
\def\lc{\lceil}
\def\rc{\rceil}
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\def\R]{\right]}
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\def\Avec{{(A_{1}, \ldots, A_{n})}}
\def\xvec{{(x_{1}, \ldots, x_{n})}}
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%\newtheorem{lemma}{Lemma}

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\def\sang{\sin \angle}

%\newcounter{felist}
%\def\bfe{\begin{list}{\alph{felist}}{\usecounter{fe}
\def\bi{\begin{itemize}}
\def\ei{\end{itemize}}
\def\dd{\dots}
\def\half{\frac{1}{2}}
\def\sang{\sin \angle}


\def\binom#1#2{{#1 \choose #2}}
\def\bZ{\mathbb{Z}}

\def\head#1{\begin{center} {\bf #1} \end{center}}

\pagestyle{empty}
\begin{document}
%\input epsf

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



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


\begin{center}
{\bf April 30, 2003}
\end{center}

\bigskip


\be
\setcounter{enumi}{3}
\ii %{\bf A\&F1}
Let $ABC$ be a triangle. A circle passing through $A$ and $B$
intersects  segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines
$AB$ and $DE$ intersect at $F$ while lines $BD$ and $CF$ intersect at
$M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.

\vspace{5mm}

\ii %{\bf A\&F2}%T2 and ZF
Let $a, b, c$ be positive real numbers. Prove that
\[
\frac{(2a+b+c)^2}{2a^2+(b+c)^2} + \frac{(2b+c+a)^2}{2b^2+(c+a)^2} +
\frac{(2c+a+b)^2}{2c^2+(a+b)^2} \le 8.
\]

\vspace{5mm}

\ii %{\bf BAR3}
A positive integer is written at each vertex of a regular
hexagon so that the sum of all numbers written is $2003^{2003}$.
Bert makes a sequence of moves of the following form: Bert picks a
vertex and replaces the number written there by the absolute value
of the difference between the numbers written at the two
neighboring vertices. Prove that Bert can always make a sequence
of moves ending at the position with all six numbers equal to
zero.

\ee

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

\end{document}