2007 AMC 12A Problems/Problem 11

Problem

The function f is defined on the set of integers and satisfies $f(n)= \begin{cases} n-3 & \mbox{if }n\ge 1000 \\ f(f(n+5)) & \mbox{if }n<1000 \end{cases}$

Find $\displaystyle f(84)$ .

Solution

Define $\displaystyle f^{h}(x) = f(f(\cdots f(f(x))\cdots))$ , where the function $f$ is performed $h$ times. We find that $f(84) = f(f(89) = f^2(89) = f^3(94) = \ldots f^{y}(1004)$ . $\displaystyle 1004 = 84 + 5(y - 1) \Longrightarrow y = 185$ . So we now need to reduce $\displaystyle f^{185}(1004)$ .

Let’s write out a couple more iterations of this function:

$\displaystyle f^{185}(1004) = f^{184}(1001) = f^{183}(998)$
$= f^{184}(1003) = f^{183}(1000) = f^{182}(997)$
$= f^{183}(1002) = f^{182}(999) = f^{183}(1004)$

So this function reiterates with a period of 2 for $x$ . It might be tempting at first to assume that $f(1004) = 999$ is the answer; however, that is not true since the solution occurs slightly before that. Start at $f^3(1004)$ :

$f^{3}(1004) = f^{2}(1001) = f(998)$
$= f^{2}(1003) = f(1000) = 997$

2007 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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2007 AMC 12A Problems/Problem 11

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