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${\bf 37\th}$ \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 30, 2008}
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\item
 Let
$\mathcal{P}$ be a convex polygon with $n$ sides, $n \ge 3$. Any
set of $n-3$ diagonals of $\mathcal{P}$ that do not intersect in
the interior of the polygon determine a {\em triangulation} of
$\mathcal{P}$ into $n-2$ triangles. If $\mathcal{P}$ is regular
and there is a triangulation of $\mathcal{P}$ consisting of only
isosceles triangles, find all the possible values of $n$.





\vspace*{.3in}


\item  Three nonnegative real numbers $r_1$, $r_2$, $r_3$ are
written on a blackboard. These numbers have the property that
there exist integers $a_1$, $a_2$, $a_3$, not all zero, satisfying
$a_1r_1+a_2r_2+a_3r_3=0$. We are permitted to perform the
following operation: find two numbers $x$, $y$ on the blackboard
with $x\le y$, then erase $y$ and write $y-x$ in its place. Prove
that after a finite number of such operations, we can end up with
at least one $0$ on the blackboard.

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\item At a certain mathematical conference, every pair of
mathematicians are either friends or strangers. At mealtime, every
participant eats in one of two large dining rooms. Each
mathematician insists upon eating in a room which contains an even
number of his or her friends. Prove that the number of ways that
the mathematicians may be split between the two rooms is a power
of two (i.e., is of the form $2^k$ for some positive integer $k$).


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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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