HMMT-Nov 2009 guts, Problem 1

If $f(x)=x /(x+1)$, what is $f(f(f(f(2009))))$ ?

Solution: $f(f(x))=\frac{(x /(x+1))}{(x /(x+1))+1}=x / 2 x+1, f(f(f(f(x))))=x / 4 x+1=\frac{2009}{8037}$ $\fbox{\frac{2009}{8037}}$.

Answer: \frac{2009}{8037}