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$.

Prove that the equation \[ x^{2008}+ 2008!= 21^{y}\] doesn't have solutions in integers.

Click for solution Assume $ x$ and $ y$ are integers which satisfy the given equation. It is easy to see that $ x$ and $ y$ must be positive. Note that $ x$ is divisible by 21. Let $ x=21n$. Hence the given equation becomes \[ (21n)^{2008}+2008!=21^y \] Thus $ y>2008$. Note that both $ (21n)^{2008}$ and $ 21^y$ are divisible by $ 21^{2008}$, but $ 2008!$ is not divisible by $ 21^{2008}$, which is a contradiction. Therefore, there are no solutions in integers.

Let $ ABC$ a triangle and $ X$, $ Y$ and $ Z$ points at the segments $ BC$, $ AC$ and $ AB$, respectively.Let $ A'$, $ B'$ and $ C'$ the circuncenters of triangles $ AZY$,$ BXZ$,$ CYX$, respectively.Prove that $ 4(A'B'C')\geq(ABC)$ with equality if and only if $ AA'$, $ BB'$ and $ CC'$ are concurrents. Note: $ (XYZ)$ denotes the area of $ XYZ$

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.