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\begin{center}
${\bf 37\th}$ \textbf{United States of America Mathematical
Olympiad}
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\begin{center}
\textbf{Day I \hspace{6mm} 12:30 PM -- 5 PM EDT}
\end{center}

\begin{center}
\textbf{April 29, 2008}
\end{center}

\bigskip

\begin{enumerate}

\item
 Prove that for each positive integer $n$, there are pairwise
relatively prime integers $k_0,k_1,\ldots, \linebreak k_n$, all
strictly greater than $1$, such that $k_0k_1\cdots k_n-1$ is the
product of two consecutive integers.





\vspace*{.3in}

\item %FENG3
  Let $ABC$ be an  acute, scalene triangle, and let $M,\,N$,
and $P$ be the midpoints of $\overline{BC},\,\overline{CA}$, and
$\overline{AB}$, respectively. Let the perpendicular bisectors of
$\overline{AB}$ and $\overline{AC}$ intersect ray $AM$ in points
$D$ and $E$ respectively, and let lines $BD$ and $CE$ intersect in
point  $F$, inside of  triangle $ABC$.   Prove that points
$A,\,N,\,F$, and $P$ all lie on one circle.

\vspace*{.3in}

\item %CAR3
 Let $n$ be a positive integer. Denote by $S_n$ the set of
points $(x,y)$ with integer coordinates such that
\[ |x| + \left|y+\frac{1}{2}\right|  < n. \]
A \emph{path} is a sequence of distinct points
$(x_1,y_1),(x_2,y_2), \dots,(x_\ell,y_\ell)$ in $S_n$ such that,
for $i=2,\dots,\ell$, the distance between $(x_i,y_i)$ and
$(x_{i-1},y_{i-1})$ is 1 (in other words, the points $(x_i,y_i)$
and $(x_{i-1},y_{i-1})$ are neighbors in the lattice of points
with integer coordinates).

Prove that the points in $S_n$ cannot be partitioned into fewer
than $n$ paths (a partition of $S_n$ into $m$ paths is a set
$\mathcal P$ of $m$ nonempty paths such that each point in $S_n$
appears in exactly one of the $m$ paths in $\mathcal P$).


\end{enumerate}


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

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