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\begin{center}
{\bf 32nd International Mathematical Olympiad}\\[.1in]
{\bf First Day \rule[.05in]{.25in}{.01in} July 17, 1991} \\
{\bf Time Limit: 4}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
\end{center}
\begin{enumerate}
\item
Given a triangle $\, ABC, \,$ let $\, I \,$ be the center of its
inscribed
circle. The internal bisectors of the angles $\, A,B,C \,$ meet
the
opposite sides in $\, A',B',C'\, $ respectively. Prove that
\[
\frac{1}{4} \; < \; \frac{AI \cdot BI \cdot CI}{AA' \cdot BB'
\cdot
CC'} \; \leq \; \frac{8}{27}.
\]
\item
Let $\, n > 6 \,$ be an integer and $\, a_1, a_2, \ldots, a_k \,$
be all the
natural numbers less than $n$ and relatively prime to $n$.
If
\[
a_2 - a_1 = a_3 - a_2 = \cdots = a_k - a_{k-1} > 0,
\]
prove that $\, n \,$ must be either a prime number or a power of
$\, 2$.
\item
Let $S = \{1,2,3,\ldots,280\}$. Find the smallest integer $n$
such that each $n$-element subset of $S$ contains five numbers
which are pairwise relatively prime.
\end{enumerate}
\begin{center}
{\bf Second Day \rule[.05in]{.25in}{.01in} July 18, 1991} \\
{\bf Time Limit: 4}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
\end{center}
\begin{enumerate}
\setcounter{enumi}{3}
\item
Suppose $\, G \,$ is a connected graph with $\, k \,$ edges.
Prove that it
is
possible to label the edges $\, 1,2,\ldots, k \,$
in such a way that at
each vertex which belongs to two or more edges, the greatest
common divisor of the integers labeling those edges is equal to
1.
[A {\em graph\/} consists of a set of points, called {\em
vertices\/},
together with a set of {\em edges\/} joining certain pairs of
distinct vertices. Each pair of vertices $\, u,v \,$ belongs to
at
most
one edge. The graph $G$ is {\em connected\/} if for each pair of
distinct vertices $\, x,y \,$ there is some sequence of vertices
$\, x = v_0, v_1, v_2, \ldots, v_m = y \,$ such that each pair
$\, v_i, v_{i+1} \; (0 \leq i < m)\,$ is joined by an edge of $\,
G$.]
\item
Let $\, ABC \,$ be a triangle and $\, P \,$ an interior point of
$\, ABC \,$. Show
that at least one of the angles $\, \angle PAB, \; \angle PBC,
\; \angle PCA \,$ is less than or equal to $30^\circ$.
\item
An infinite sequence $\, x_0, x_1, x_2, \ldots \, $ of real
numbers is
said to be {\em bounded\/} if there is a constant $\, C \,$ such
that
$\, |x_i| \leq C \,$ for every $\, i \geq 0$.
Given any real number $\, a > 1, \,$ construct a bounded infinite
sequence
$x_0, x_1, x_2, \ldots \, $ such that
\[
|x_i - x_j| |i - j|^a \geq 1
\]
for every pair of distinct nonnegative integers $i,j$.
\end{enumerate}
\end{document}