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\begin{document}
\begin{center}
${\bf 37}^{\mbox{\bf th}}$ {\bf International
Mathematical Olympiad} \\[.1in]
{\bf Mumbai, India} \\ [.05in]
{\bf Day I \hspace{.25in} 9 a.m. - 1:30 p.m.}\\[.05in]
{\bf July 10, 1996}
\end{center}
\vspace*{.3in}
\begin{enumerate}
\item %% IMO 1
We are given a positive integer $r$ and a rectangular board $ABCD$
with dimensions $|AB| = 20, |BC| = 12$. The rectangle is divided into
a grid of $20 \times 12$ unit squares. The following moves are
permitted on the board: one can move from one square to another only
if the distance between the centers of the two squares is $\sqrt{r}$.
The task is to find a sequence of moves leading from the square
with $A$ as a vertex to the square with $B$ as a vertex.
\begin{enumerate}
\item[(a)]
Show that the task cannot be done if $r$ is divisible by 2 or 3.
\item[(b)]
Prove that the task is possible when $r=73$.
\item[(c)]
Can the task be done when $r=97$?
\end{enumerate}
\item %% IMO2
Let $P$ be a point inside triangle $ABC$ such that
\[
\angle APB - \angle ACB = \angle APC - \angle ABC.
\]
Let $D, E$ be the incenters of triangles $APB, APC$, respectively. Show
that $AP, BD, CE$ meet at a point.
\item %% IMO3
Let $S$ denote the set of nonnegative integers. Find all functions $f$
from $S$ to itself such that
\[
f(m + f(n)) = f(f(m)) + f(n) \qquad \forall m, n \in S.
\]
\end{enumerate}
\pagebreak %% DAY 2
\begin{center}
${\bf 37}^{\mbox{\bf th}}$ {\bf International
Mathematical Olympiad} \\[.1in]
{\bf Mumbai, India} \\ [.05in]
{\bf Day II \hspace{.25in} 9 a.m. - 1:30 p.m.}\\[.05in]
{\bf July 11, 1996}
\end{center}
\vspace*{.3in}
\begin{enumerate}
\setcounter{enumi}{3}
\item %% IMO4
The positive integers $a$ and $b$ are such that the numbers $15a +
16b$ and $16a - 15b$ are both squares of positive integers. What is
the least possible value that can be taken on by the smaller of these
two squares?
\item %% IMO5
Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BC$
is parallel to $EF$, and $CD$ is parallel to $FA$. Let $R_A, R_C, R_E$
denote the circumradii of triangles $FAB, BCD, DEF$, respectively, and
let $P$ denote the perimeter of the hexagon. Prove that
\[
R_A + R_C + R_E \geq \frac{P}{2}.
\]
\item %% IMO6
Let $p,q,n$ be three positive integers with $p+q < n$. Let $(x_{0},
x_{1}, \dots, x_{n})$ be an $(n+1)$-tuple of integers satisfying the
following conditions:
\begin{enumerate}
\item[(a)] $x_{0} = x_{n} = 0$.
\item[(b)] For each $i$ with $1 \leq i \leq n$, either $x_{i} -
x_{i-1} = p$ or $x_{i} - x_{i-1} = -q$.
\end{enumerate}
Show that there exist indices $i < j$ with $(i,j)
\neq (0, n)$, such that $x_{i} = x_{j}$.
\end{enumerate}
\end{document}