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\begin{document}
%\input epsf

\begin{center}
${\bf 43\rd}$ {\bf IMO Team Selection Test} \\[.1in]
{\bf Lincoln, Nebraska} \\ [.05in]
{\bf Day I \hspace{.25in} 8:30 a.m. - 1:00 p.m.}\\[.05in]
{\bf June 21, 2002}
\end{center}

\vspace*{.3in}

\be
\ii [1.] %EMM
%Titu
Let $ABC$ be a triangle. Prove that
\[
\sin\frac{3A}{2} + \sin\frac{3B}{2} + \sin\frac{3C}{2} \le
\cos\frac{A-B}{2} + \cos\frac{B-C}{2} + \cos\frac{C-A}{2}.
\]


\ii [2.]
%Kiran MMH
Let $p$ be a prime number greater than 5. For any integer $x$,
define
\[
f_p(x) = \sum_{k=1}^{p-1} \frac{1}{(px+k)^2}.
\]
Prove that for all positive integers $x$ and $y$ the numerator of
$f_p(x)-f_p(y)$, when written in lowest terms, is divisible by
$p^3$.

\ii [3.] %MMH
%Zuming and Zhongtao
Let $n$ be an integer greater than 2, and $P_1, P_2, \cdots , P_n$
distinct points in the plane. Let $\mathcal S$ denote the union of
all segments $P_1P_2, P_2P_3, \dots , P_{n-1}P_{n}$. Determine if
it is always possible to find points $A$ and $B$ in $\mathcal S$
such that $P_1P_n \parallel AB$ (segment $AB$ can lie on line
$P_1P_n$) and $P_1P_n = kAB$, where (1) $k = 2.5$; (2) $k = 3$.

\ee

\vfill

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

\begin{center}
${\bf 43\rd}$ {\bf IMO Team Selection Test} \\[.1in]
{\bf Lincoln, Nebraska} \\ [.05in]
{\bf Day II \hspace{.25in} 8:30 a.m. - 1:00 p.m.}\\[.05in]
{\bf June 22, 2002}
\end{center}

\vspace*{.3in}

\be
\ii [4.]
%Kiran MMM
Let $n$ be a positive integer and let $S$ be a set of $2^n+1$
elements. Let $f$ be a function from the set of two-element
subsets of $S$ to $\{0, \dots, 2^{n-1}-1\}$. Assume that for any
elements $x, y, z$ of $S$, one of $f(\{x,y\}), f(\{y,z\}), f(\{z,
x\})$ is equal to the sum of the other two. Show that there exist
$a, b, c$ in $S$ such that $f(\{a,b\}), f(\{b,c\}), f(\{c,a\})$
are all equal to 0.


\ii [5.]
%Zuming MMH
Consider the family of  nonisoceles triangles $ABC$ satisfying the
property $AC^2 + BC^2 = 2 AB^2$. Points $M$ and $D$ lie on side
$AB$ such that $AM = BM$ and $\ang ACD = \ang BCD$. Point $E$ is
in the plane such that $D$ is the incenter of triangle $CEM$.
Prove that exactly one of the ratios
\[
\frac{CE}{EM}, \quad \frac{EM}{MC}, \quad \frac{MC}{CE}
\]
is constant.

\ii [6.]
%Titu MMH
Find in explicit form all ordered pairs of positive integers $(m,
n)$ such that $mn-1$ divides $m^2 + n^2$.

\ee

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