## គណិតវិទ្យា​អូឡាំពិច​អាស៊ី​ប៉ាស៊ីភិច ២០០៤

Time allowed: 4 hours
No calculators are to be used
Each question is worth 7 points

Problem 1.
Determine all finite nonempty sets $\mathbb{S}$ of positive integers satisfying $\displaystyle \frac{i + j}{(i, j)}$ is an element of $\mathbb{S}$ for all $i, j \in \mathbb{S}$; where $(i, j)$ is the greatest common divisor of $i$ and $j$.

Problem 2.
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.

Problem 3.
Let a set $\mathbb{S}$ of $2004$ points in the plane be given, no three of which are collinear. Let $\mathbb{L}$ 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 $\mathbb{S}$ with at most two colours, such that for any points $p, q$ of $\mathbb{S}$, the number of lines in $\mathbb{L}$ which separate $p$ from $q$ is odd if and only if $p$ and $q$ have the same colour.

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

Problem 4.
For a real number $x$, let $\lfloor x \rfloor$ stand for the largest integer that is less than or equal to $x$. Prove that $\lfloor \frac{(n-1)!}{n(n+1)} \rfloor$ is even for every positive integer $n$.

សំនួរ​ទី​៥.
ចូរ​បង្ហាញ​ថា $(a^2 + 2)(b^2 + 2)(c^2 + 2) \geq 9(ab + bc + ca)$ ចំពោះ​គ្រប់​ចំនួន​ពិត $a; b; c > 0$.