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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)

See Also

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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