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\begin{document}
\centerline{\large \bf New Zealand Mathematical Olympiad Camp}
\bigskip
\medskip
\centerline{\large \bf Christchurch, 1998}
\bigskip
\medskip
\centerline{\Large \bf Problem Set 1}
\vspace{1cm}
\centerline{\Large \bf Problems}
\vspace{1cm}
\begin{enumerate}
\item The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ are
perpendicular. Through the midpoints of $AB$ and $AD$ the lines
perpendicular to the
corresponding opposite sides $CD$ and $BC$ are drawn. Prove that these
lines
intersect at a point on the line $AC$.
\item Prove that, for each positive integer $n$, there exists an integer
$k$
which can be expressed as the sum of two squares of integers and such
that
${n\le k