2012 AMC 12A Problem 25

Let f(x)=|2\{x\}-1| where \{x\} denotes the fractional part of x. The number n is the smallest positive integer such that the equation nf(xf(x))=x has at least 2012 real solutions. What is n? Note: the fractional part of x is a real number y=\{x\} such that 0\le y<1 and x-y is an integer.

\textbf{(A)}\ 30\qquad\textbf{(B)}\ 31\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 62\qquad\textbf{(E)}\ 64

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.