\documentstyle[12pt]{article}
\setlength{\baselineskip}{0.25in}
\setlength{\leftmargin}{0.0in}
\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{0.0in}
\setlength{\textheight}{9in}
\setlength{\headheight}{0.0in}
\setlength{\headsep}{0.0in}
\setlength{\oddsidemargin}{0.0in}
\def\RR{{\mathbf{R}}}
\newenvironment{List}{%
\begin{list}{}{\setlength{\labelwidth}{.15in}
\setlength{\leftmargin}{.55in}
\setlength{\rightmargin}{.25in}
\setlength{\topsep}{0pt}
\setlength{\partopsep}{0pt}
}}{\end{list}}
\begin{document}
\begin{center}
${\bf 33}^{\mbox{\bf rd}}$
{\bf International Mathematical Olympiad} \\[.1in]
{\bf First Day - Moscow - July 15, 1992} \\
{\bf Time Limit: 4}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
\end{center}
\begin{enumerate}
\item
Find all integers $\, a,b,c \,$ with $\, 1 < a < b < c \,$ such
that
\[
(a-1)(b-1)(c-1) \hspace{.2in} \mbox{is a divisor of $\, abc - 1$.}
\]
\item
Let $\, \RR \,$ denote the set of all real numbers. Find all
functions $\, f: \RR \rightarrow \RR \,$ such that
\[
f \left( x^2 + f(y) \right) = y + \left( f(x) \right)^2
\hspace{.2in} \mbox{for all $\, x,y \in R$.}
\]
\item
Consider nine points in space, no four of which are coplanar.
Each
pair of points is joined by an edge (that is, a line segment)
and each edge is either colored blue or red or left uncolored.
Find the smallest value of $\, n \,$ such that whenever exactly
$\, n \,$ edges are colored, the set of colored edges necessarily
contains a triangle all of whose edges have the same color.
\end{enumerate}
\begin{center}
${\bf 33}^{\mbox{\bf rd}}$
{\bf International Mathematical Olympiad} \\[.1in]
{\bf Second Day - Moscow - July 15, 1992} \\
{\bf Time Limit: 4}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
\end{center}
\begin{enumerate}
\setcounter{enumi}{3}
\item
In the plane let $\, C \,$ be a circle, $\, L \,$ a line tangent
to the circle $\, C, \,$ and $\, M \,$ a point on $\, L$.
Find the locus of all points $\, P \,$ with the following
property: there exists two points $\, Q, R \,$ on $\, L \,$
such that $\, M \,$ is the midpoint of $\, QR \,$ and $\, C \,$
is the inscribed circle of triangle $\, PQR$.
\item
Let $\, S \,$ be a finite set of points in three-dimensional
space. Let $\, S_x, \, S_y, \, S_z \,$ be the sets consisting of
the orthogonal projections of the points of $\, S \,$ onto the
$yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that
\[
|S|^2 \leq |S_x| \cdot |S_y| \cdot |S_z|,
\]
where $\, |A| \,$ denotes the number of elements in the finite
set $\, |A|$. (Note: The orthogonal projection of a point onto
a plane is the foot of the perpendicular from that point to
the plane.)
\item
For each positive integer $\, n, \; S(n) \,$ is defined to be
the greatest integer such that, for every positive integer
$\, k \leq S(n), \; n^2\,$ can be written as the sum of $\, k \,$
positive squares.
\begin{List}
\item[(a)]
Prove that $\, S(n) \leq n^2 - 14 \,$ for each $\, n \geq 4$.
\item[(b)]
Find an integer $\, n \,$ such that $\, S(n) = n^2 - 14$.
\item[(c)]
Prove that there are infintely many integers $\, n \,$ such that
$\, S(n) = n^2 - 14$.
\end{List}
\end{enumerate}
\end{document}