CR-MO 2006

Let $f$ be a function that satisfies : $$f(x)+2f\left(\frac{x+\frac{2001}2}{x-1}\right) = 4014-x.$$ Find $f(2004)$ .

Let $n$ be a positive integer, and let $p$ be a prime, such that $n>p$ . Prove that : $$\binom np \equiv \left\lfloor\frac{n}{p}\right\rfloor \ \pmod p.$$