2005 AMC 12A Problems/Problem 20

Problem

For each $x$ in $[0,1]$ , define $\begin{array}{clr} f(x) & = 2x, & \text { if } 0 \leq x \leq \frac {1}{2}; \\ f(x) & = 2 - 2x, & \text { if } \frac {1}{2} < x \leq 1. \end{array}$ 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 $f^{[2005]}(x) = \frac {1}{2}$ ? $(\text {A}) \ 0 \qquad (\text {B}) \ 2005 \qquad (\text {C})\ 4010 \qquad (\text {D}) \ 2005^2 \qquad (\text {E})\ 2^{2005}$

Solution 1

For the two functions $f(x)=2x,0\le x\le \frac{1}{2}$ and $f(x)=2-2x,\frac{1}{2}\le x\le 1$ ,as long as $f(x)$ is between $0$ and $1$ , $x$ will be in the right domain, so we don't need to worry about the domain of $x$ .

Also, every time we change $f(x)$ , the expression for the final answer in terms of $x$ will be in a different form(although they'll all satisfy the final equation), so we get a different starting value of $x$ . Every time we have two choices for $f(x$ ) and altogether we have to choose $2005$ times. Thus, $2^{2005}\Rightarrow\boxed{E}$ .

Note: the values of x that satisfy $f^{[n]}(x) = \frac {1}{2}$ are $\frac{1}{2^{n+1}}$ , $\frac{3}{2^{n+1}}$ , $\frac{5}{2^{n+1}}$ , $\cdots$ , $\frac{2^{n+1}-1}{2^{n+1}}$ .

Solution 2

We are given that $f^{[2005]}(x) = \frac {1}{2}$ . Thus, $f(f^{[2004]}(x))=\frac{1}{2}$ . Let $f^{[2004]}(x)$ be equal to $y$ . Thus $f(y)=\frac{1}{2}$ or $y=\frac{1}{4}$ or $\frac{3}{4}$ . Now we know $f^{[2004]}(x)$ is equal to $\frac{1}{4}$ or $\frac{3}{4}$ . Now we know that $f(f^{[2003]}(x))=\frac{1}{4}$ or $\frac{3}{4}$ . Now we solve for $f^{[2003]}(x)$ and let $f^{[2003]}(x)=z$ . Thus $f(z)$ is equal to $\frac{1}{8}$ , $\frac{7}{8}$ , $\frac{5}{8}$ ,and $\frac{3}{8}$ . As we see, $f^{[2005]}(x)$ has 1 solution, $f^{[2004]}(x)$ has 2 solutions, and $f^{[2003]}(x)$ has 4 solutions. Thus for each iteration we double the number of possible solutions. There are 2005 iterations and thus the number of solutions is $2^{2005}$ $\Rightarrow\boxed{E}$ (not rigorous)

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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2005 AMC 12A Problems/Problem 20

Contents

Problem

Solution 1

Solution 2

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