FI-MO 2012 Problem 5

The [i]Collatz's function[i] is a mapping $f:\mathbb{Z}_+\to\mathbb{Z}_+$ satisfying $$f(x)=\begin{cases} 3x+1,& \text{as }x\text{ is odd} x/2, & \text{as }x\text{ is even.} \end{cases}$$ In addition, let us define the notation $f^1=f$ and inductively $f^{k+1}=f\circ f^k,$ or to say in another words, $f^k(x)=\underbrace{f(\ldots (f}_{k\text{ times}}(x)\ldots ).$ Prove that there is an $x\in\mathbb{Z}_+$ satisfying $$f^{40}(x)> 2012x.$$