Problem #2144


For each $x$ in $[0,1]$, define

$\begin{cases}  f(x) = 2x, \qquad\qquad \mathrm{if} \quad 0 \leq x \leq \frac{1}{2};\\  f(x) = 2-2x, \qquad \mathrm{if} \quad \frac{1}{2} < x \leq 1.  \end{cases}$

Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. For how many values of $x$ in $[0,1]$ is $f^{[2005]}(x) = \frac {1}{2}$?

$(\mathrm {A}) \ 0 \qquad (\mathrm {B}) \ 2005 \qquad (\mathrm {C})\ 4010 \qquad (\mathrm {D}) \ 2005^2 \qquad (\mathrm {E})\ 2^{2005}$

This problem is copyrighted by the American Mathematics Competitions.

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