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\begin{center}
${\bf 28^{th}}$ {\bf United States of America Mathematical Olympiad}
\end{center}

\begin{center}
{\bf  Part  I \hspace{ 6mm} 9 a.m. -- 12 noon}
\end{center}

\begin{center}
{\bf April 27, 1999}
\end{center}

\bigskip
\begin{itemize}

\item[1.]
Some checkers placed on an $n \times n$ checkerboard satisfy the following
conditions:
\begin{enumerate}
\item[(a)] every square that does not contain a checker shares a side
with one that does;
\item[(b)] given any pair of squares that contain checkers, there is
a sequence of squares containing checkers, starting and ending with
the given squares, such that every two consecutive squares of the
sequence share a side.
\end{enumerate}
Prove that at least $(n^{2}-2)/3$ checkers have been placed on the
board.

\item[2.]
Let $ABCD$ be a cyclic quadrilateral. Prove that
\[
|AB - CD| + |AD - BC| \geq 2|AC - BD|.
\]

\item[3.]
Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible
by $p$, such that
\[
\{ra/p\} + \{rb/p\} + \{rc/p\} + \{rd/p\} = 2
\]
for any integer $r$ not divisible by $p$.
Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$
are divisible by $p$.
(Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.)

\end{itemize}

\vfill
%\vspace{110 mm}

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

\newpage

\begin{center}
${\bf 28^{th}}$ {\bf United States of America Mathematical Olympiad}
\end{center}



\begin{center}
{\bf  Part  II \hspace{ 6mm} 1 p.m. -- 4 p.m.}
\end{center}


\begin{center}
{\bf April 27, 1999}
\end{center}

\bigskip

\begin{itemize}

\item[4.]
Let $a_{1}, a_{2}, \dots, a_{n}$ ($n > 3$) be real numbers such that
\[
a_{1} + a_{2} + \cdots + a_{n} \geq n \qquad \mbox{and} \qquad
a_{1}^{2} + a_{2}^{2} +
\cdots + a_{n}^{2} \geq n^{2}.
\]
Prove that $\max(a_{1}, a_{2}, \dots,
a_{n}) \geq 2$.

\item[5.]
The Y2K Game is played on a
$1 \times 2000$ grid as follows.
Two players in turn write either an S or
an O in an empty square. The first player who produces three
consecutive boxes that spell SOS wins. If all boxes are filled without
producing SOS then the game is a draw. Prove that the second player
has a winning strategy.

\item[6.]
Let $ABCD$ be an isosceles trapezoid with $AB \parallel CD$. The inscribed
circle $\omega$ of triangle $BCD$ meets $CD$ at $E$. Let $F$ be a point on
the (internal) angle bisector of $\angle DAC$ such that $EF \perp CD$. Let the
circumscribed circle of triangle $ACF$ meet line $CD$ at $C$ and $G$. Prove
that the triangle $AFG$ is isosceles.

\end{itemize}

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



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