UZ-MO 2011

Prove that $\forall n\in\mathbb{N}$ , $\exists a,b,c\in$ $\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3)$ such that $n=\frac{ab}{c}$ .

Does existes a function $f:N->N$ and for all positeve integer n $f(f(n)+2011)=f(n)+f(f(n))$