{"status": "success", "data": {"description_md": "For each $x$ in $[0,1]$, define\n\n$\\begin{cases}
f(x) = 2x, \\qquad\\qquad \\mathrm{if} \\quad 0 \\leq x \\leq \\frac{1}{2};\\\\
f(x) = 2-2x, \\qquad \\mathrm{if} \\quad \\frac{1}{2} < x \\leq 1.
\\end{cases}$
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 $f^{[2005]}(x) = \\frac {1}{2}$?\n\n$(\\mathrm {A}) \\ 0 \\qquad (\\mathrm {B}) \\ 2005 \\qquad (\\mathrm {C})\\ 4010 \\qquad (\\mathrm {D}) \\ 2005^2 \\qquad (\\mathrm {E})\\ 2^{2005}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

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}

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

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2005 AMC 12A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/05_amc12A_p21", "prev": "/problem/05_amc12A_p19"}}