APMO 2004

Determine all finite nonempty sets $S$ of positive integers satisfying

$$\frac{i+j}{(i, j)} \text { is an element of } S \text { for all } i, j \text { in } S \text {, }$$

where $(i, j)$ is the greatest common divisor of $i$ and $j$.

Let $O$ be the circumcentre and $H$ the orthocentre of an acute triangle $A B C$. Prove that the area of one of the triangles $\mathrm{AOH}, \mathrm{BOH}$ 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 $\mathcal{L}$ 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 $\mathcal{L}$ 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

$$\left\lfloor\frac{(n-1)!}{n(n+1)}\right\rfloor$$

is even for every positive integer $n$.

Prove that

$$\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9(a b+b c+c a)$$

for all real numbers $a, b, c>0$.