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${\bf 34\th}$ \textbf{United States of America Mathematical
Olympiad}
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\begin{center}
\textbf{Day I \hspace{6mm} 12:30 PM -- 5 PM EDT}
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\textbf{April 19, 2005}
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\bigskip

\begin{enumerate}

\item %FENG1
Determine all composite positive integers $n$ for which it is
possible to arrange all divisors of $n$ that are greater than 1 in
a circle so that no two adjacent divisors are relatively prime.


\vspace{5mm}

\item %GELCA3
 Prove that the
system
\begin{align*} x^6+x^3+x^3y+y & = 147^{157} \\[.1in]
          x^3+x^3y+y^2+y+z^9 & = 157^{147}
          \end{align*}
has no solutions in integers $x$, $y$, and $z$.


\vspace{5mm}

\item %FENG3
Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two
points on side $BC$. Construct point $C_1$ in such a way that
convex quadrilateral $APBC_1$ is cyclic, $QC_1
\parallel CA$, and $C_1$ and $Q$ lie on opposite sides of line
$AB$. Construct point $B_1$ in such a way that convex
quadrilateral $APCB_1$ is cyclic, $QB_1 \parallel BA$, and $B_1$
and $Q$  lie on opposite sides of line $AC$.  Prove that points
$B_1, C_1,P$, and $Q$ lie on a circle.

\end{enumerate}

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

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