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 $j$.

Let $O$ be the circumcenter and $H$ the orthocenter 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.

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

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 $n$.

Prove that the inequality \[\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9\left(ab+bc+ca\right)\]holds for all positive reals $a$, $b$, $c$.