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\begin{center}
${\bf 24}^{\mbox{\bf th}}$ {\bf United States of America
Mathematical Olympiad} \\[.1in]
{\bf April 27, 1995}\\
{\bf Time Limit: 3}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
\end{center}
\begin{enumerate}
\item %AND3
Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is
defined as follows: $\, a_0 = 0, $ $a_1 = 1, \, \ldots,
\, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$
is the least positive integer that does not form an arithmetic
sequence of length $\, p \,$ with any of the preceding terms.
Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained
by writing $\, n \,$ in base $\, p-1 \,$ and reading the result
in base $\, p$.
\item %PNN1
A calculator is broken so that the only keys that still work are
the $\, \sin, \; \cos, $ $\tan, \; \sin^{-1}, \; \cos^{-1}, \,$
and $\, \tan^{-1} \,$ buttons. The display initially shows 0.
Given any positive rational number $\, q, \,$ show that pressing
some finite sequence of buttons will yield $\, q$. Assume that
the calculator does real number calculations with infinite
precision. All functions are in terms of radians.
\item %HUD2
Given a nonisosceles, nonright triangle $\, ABC, \,$ let
$\, O \,$ denote the center of its circumscribed circle, and let
$\, A_1, \, B_1, \,$ and $\, C_1 \,$ be the midpoints of
sides $\, BC, \, CA, \,$ and $\, AB, \,$ respectively.
Point $\, A_2 \,$ is located on the ray
$\, OA_1 \,$ so that $\, \Delta OAA_1 \,$ is similar
to $\, \Delta OA_2A$. Points $\, B_2 \,$
and $\, C_2 \,$ on rays $\, OB_1 \,$ and $\, OC_1, \,$
respectively, are defined similarly.
Prove that lines $\, AA_2, \, BB_2, \,$
and $\, CC_2 \,$ are concurrent, i.e. these three lines intersect
at a point.
\item %PNN3
Suppose $\, q_0, \, q_1, \, q_2, \ldots \; \,$ is an infinite
sequence of integers satisfying the following two conditions:
\begin{List}
\item[(i)] $\, m-n \,$ divides $\, q_m - q_n \,$ for
$\, m > n \geq 0,$
\item[(ii)] there is a polynomial $\, P \,$ such that
$\, |q_n| < P(n) \,$ for all $\, n$.
\end{List}
Prove that there is a polynomial $\, Q \,$ such that
$\, q_n = Q(n) \,$ for all $\, n$.
\item %ROU2
Suppose that in a certain society, each pair of persons can be
classified as either {\em amicable} or {\em hostile}. We shall say
that each member of an amicable pair is a {\em friend} of the
other, and each member of a hostile pair is a {\em foe} of the
other. Suppose that the society has $\, n \,$ persons and $\, q
\,$ amicable pairs, and that for every set of three persons, at
least one pair is hostile. Prove that there is at least one
member of the society whose foes include $\, q(1 - 4q/n^2) \,$ or
fewer amicable pairs.
\end{enumerate}
\vspace*{\fill}
\begin{center}
{\footnotesize Copyright \copyright \hspace{.05in} Committee on
the American
Mathematics Competitions, \\ Mathematical Association of America}
\end{center}
\end{document}