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{\Large\bf XVI Asian Pacific Mathematics Olympiad \\
March 2004}
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Time allowed: 4 hours

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No calculators are to be used

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Each question is worth 7 points

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{\large\bf Problem 1}. 

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Determine all finite nonempty sets $S$ of positive integers satisfying
$${i+j\over (i,j)}\qquad\mbox{is an element of $S$ for all $i,j$ in $S$},$$
where $(i,j)$ is the greatest common divisor of $i$ and $j$.

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{\large\bf Problem 2}. 

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Let $O$ be the circumcentre and $H$ the orthocentre of an acute triangle 
$ABC$. Prove that the area of one of the triangles $AOH$, $BOH$ and $COH$ 
is equal to the sum of the areas of the other two.

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{\large\bf Problem 3}. 

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Let a set $S$ of 2004 points in the plane be given, no three of which are 
collinear. Let ${\cal L}$ denote the set of all lines (extended indefinitely 
in both directions) determined by pairs of points from the set. Show that it 
is possible to colour the points of $S$ with at most two colours, such that 
for any points $p,q$ of $S$, the number of lines in ${\cal L}$ which separate 
$p$ from $q$ is odd if and only if $p$ and $q$ have the same colour.

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{\it Note}: A line $\ell$ {\it separates} two points $p$ and $q$ if $p$ 
and $q$ lie on opposite sides of $\ell$ with neither point on $\ell$.

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{\large\bf Problem 4}. 

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For a real number $x$, let $\lfloor x\rfloor$ stand for the largest integer 
that is less than or equal to $x$. Prove that 
$$\left\lfloor{(n-1)!\over n(n+1)}\right\rfloor$$
is even for every positive integer $n$.

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{\large\bf Problem 5}. 

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Prove that
$$(a^2+2)(b^2+2)(c^2+2)\ge 9(ab+bc+ca)$$
for all real numbers $a,b,c>0$.

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