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${\bf 36\th}$ \textbf{United States of America Mathematical
Olympiad}
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\textbf{Day II \hspace{6mm} 12:30 PM -- 5 PM EDT}
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\textbf{April 25, 2007}
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\item %BAR1
An {\em animal} with $n$ {\em cells} is a connected figure
consisting of  $n$ equal-sized square cells.\footnote {Animals are
also called {\em polyominoes}. They can be defined inductively.
Two cells are {\em adjacent} if they share a complete edge. A
single cell is an animal, and given an animal with $n$-cells, one
with $n+1$ cells is obtained by adjoining a new cell by making it
adjacent to one or more existing cells.} The figure below shows an
8-cell animal.
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A {\em dinosaur} is an animal with at least 2007 cells. It is said
to be {\em primitive} if its cells cannot be partitioned into two
or more dinosaurs. Find with proof the maximum number of cells in
a primitive dinosaur.





\vspace*{.3in}


\item %AND2
Prove that for every nonnegative integer $n$, the number $7^{7^n}
+ 1$ is the product of at least $2n+3$ (not necessarily distinct)
primes.


\vspace*{.3in}

\item %KED3
\noindent Let $ABC$ be an acute triangle with $\omega, \Omega,$
and $R$ being its incircle, circumcircle, and circumradius,
respectively. Circle $\omega_A$ is tangent internally to $\Omega$
at $A$ and tangent externally to $\omega$. Circle $\Omega_A$ is
tangent internally to $\Omega$ at $A$ and tangent internally to
$\omega$. Let $P_A$ and $Q_A$ denote the centers of $\omega_A$ and
$\Omega_A$, respectively. Define points $P_B, Q_B, P_C, Q_C$
analogously. Prove that
\[
8P_AQ_A \cdot P_BQ_B \cdot P_C Q_C \leq R^3,
\]
with equality if and only if triangle $ABC$ is equilateral.

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 \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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