OIM 2008

The integers from 1 to $2008^2$ are written on each square of a $2008 \times 2008$ board. For every row and column the difference between the maximum and minimum numbers is computed. Let $S$ be the sum of these 4016 numbers. Find the greatest possible value of $S$ .

Given a triangle $ABC$ , let $r$ be the external bisector of $\angle ABC$ . $P$ and $Q$ are the feet of the perpendiculars from $A$ and $C$ to $r$ . If $CP \cap BA = M$ and $AQ \cap BC=N$ , show that $MN$ , $r$ and $AC$ concur.

Let $P(x) = x^3 + mx + n$ be an integer polynomial satisfying that if $P(x) - P(y)$ is divisible by 107, then $x - y$ is divisible by 107 as well, where $x$ and $y$ are integers. Prove that 107 divides $m$ .

Biribol is a game played between two teams of 4 people each (teams are not fixed). Find all the possible values of $n$ for which it is possible to arrange a tournament with $n$ players in such a way that every couple of people plays a match in opposite teams exactly once.