Find all $ f: Q^ + \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) = f(x)$ and $ (x + 1)f(x - 1) = xf(x)$ for all rational numbers that are bigger than 1.

Quadrilateral $ ABCD$ has an inscribed circle which centered at $ O$ with radius $ r$. $ AB$ intersects $ CD$ at $ P$; $ AD$ intersects $ BC$ at $ Q$ and the diagonals $ AC$ and $ BD$ intersects each other at $ K$. If the distance from $ O$ to the line $ PQ$ is $ k$, prove that $ OK\cdot\ k = r^2$.

Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.

For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 + p + Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p - 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?

In a triangle $ ABC$ incircle touches the sides $ AB$, $ AC$ and $ BC$ at $ C_1$, $ B_1$ and $ A_1$ respectively. Prove that $ \sqrt {\frac {AB_1}{AB}} + \sqrt {\frac {BC_1}{BC}} + \sqrt {\frac {CA_1}{CA}}\leq\frac {3}{\sqrt {2}}$ is true.

In a class of $ n\geq 4$ some students are friends. In this class any $ n - 1$ students can be seated in a round table such that every student is sitting next to a friend of him in both sides, but $ n$ students can not be seated in that way. Prove that the minimum value of $ n$ is $ 10$.