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\centerline{\large \bf New Zealand Mathematical Olympiad Camp}
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\medskip
\centerline{\large \bf Christchurch, 1998}
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\medskip
\centerline{\Large \bf Problem Set 2}
\vspace{1cm}
\centerline{\Large \bf Problems}
\vspace{1cm}
{\bf 1.} Let $ ABCD$ be a convex quadrilateral inscribed in a semicircle
$\Sigma $ of
diameter $AB$. The lines $AC$ and
$BD$ intersect at $E$ and the lines $AD$ and $BC$ at $F$. The line $EF$
intersects the semicircle
$\Sigma $ at
$G$ and the line $AB$ at $H$. Prove that $E$ is the midpoint of line
segment $GH$ if and only if $G$
is the midpoint of line segment $FH$.
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{\bf 2.} Is it possible for the product of five consecutive positive
integers to be a
perfect square?
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{\bf 3.} Let $S$ be the set of all real numbers strictly greater than
$-1$.
Find all
functions $f\colon S\rightarrow S$ satisfying the two conditions:
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(a) $f(x+f(y)+xf(y))=y+f(x)+yf(x)$ for all $x,y\in S$;\par
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(b) $f(x)/x$ is strictly increasing on each of the two intervals
$-1n$ green edges and $q