\documentclass{article}
\usepackage{amssymb}
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\title{March 2002 Mock Exam}
\date{A four hour examination}
\author{Hungarian MO Round 3 for non-specialist classes}
\begin{document}
\maketitle

\begin{enumerate}
\item The escribed circle to the side $AB$ of
$\Delta ABC$ touches the lines $AB$, $AC$ (produced) at $P, Q$
respectively. The escribed circle to the side $BC$ touches the
lines $AC, AB$ at $U, X$ respectively. Prove that the intersection
of $PQ$ and $UX$ is as far from $AB$ as from $BC$.
\item For which natural numbers $n$, if any, is there 
an $n$-gon containing exactly $n^2 - 30n + 236$ acute angles?
\item Let $n > 1$ be a fixed integer. Find $x_1, \ldots, x_n$
real numbers so that $\sum_i x_i = 2(n-1)$, $\sum (x_i-1)^2 = n$
and $x_n$ is maximized subject to these conditions.
\end{enumerate}
\end{document}