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\begin{center}
{\bf 34nd International Mathematical Olympiad}\\[.1in]
{\bf First Day \rule[.05in]{.25in}{.01in} July 18, 1993} \\
{\bf Time Limit: 4}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
\end{center}
\begin{enumerate}
\item
Let $f(x) = x^n + 5x^{n-1} + 3$, where $n > 1$ is an integer. Prove that
$f(x)$ cannot be expressed as the product of two nonconstant polynomials
with integer coefficients.
\item
Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB =
\angle ACB + \pi/2$ and $AC \cdot BD = AD \cdot BC$.
\begin{itemize}
\item[(a)] Calculate the ratio $(AB \cdot CD)/(AC \cdot BD)$.
\item[(b)] Prove that the tangents at $C$ to the circumcircles of
$\triangle ACD$ and $\triangle BCD$ are perpendicular.
\end{itemize}
\item
On an infinite chessboard, a game is played as follows. At the start,
$n^2$ pieces are arranged on the chessboard in an $n$ by $n$ block of
adjoining squares, one piece in each square. A move in the game is a jump
in a horizontal or vertical direction over an adjacent occupied square to
an unoccupied square immediately beyond. The piece which has been jumped
over is removed.
Find those values of $n$ for which the game can end with only one piece
remaining on the board.
\end{enumerate}
\begin{center}
{\bf Second Day \rule[.05in]{.25in}{.01in} July 19, 1993} \\
{\bf Time Limit: 4}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
\end{center}
\begin{enumerate}
\setcounter{enumi}{3}
\item
For three points $P,Q,R$ in the plane, we define $m(PQR)$ as the minimum
length of the three altitudes of $\triangle PQR$. (If the points are
collinear, we set $m(PQR) = 0$.)
Prove that for points $A,B,C,X$ in the plane,
\[
m(ABC) \leq m(ABX) + m(AXC) + m(XBC).
\]
\item
Does there exist a function $f: \mathbf{N} \to \mathbf{N}$ such that
$f(1) = 2$, $f(f(n)) = f(n) + n$ for all $n \in \mathbf{N}$, and $f(n) <
f(n+1)$ for all $n \in \mathbf{N}$?
\item
There are $n$ lamps $L_0, \dots, L_{n-1}$ in a circle ($n>1$), where we
denote $L_{n+k} = L_k$. (A lamp at all times is either on or off.)
Perform steps $s_0, s_1, \dots$ as follows: at step $s_i$, if $L_{i-1}$
is lit, switch $L_i$ from on to off or vice versa, otherwise do nothing.
Initially all lamps are on. Show that:
\begin{itemize}
\item[(a)] There is a positive integer $M(n)$ such that after $M(n)$
steps all the lamps are on again;
\item[(b)] If $n = 2^k$, we can take $M(n) = n^2 - 1$;
\item[(c)] If $n = 2^k + 1$, we can take $M(n) = n^2 - n + 1$.
\end{itemize}
\end{enumerate}
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