Difference between revisions of "1984 AIME Problems/Problem 7"
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== Solution == | == Solution == | ||
| − | Define <math>f^{h} = f(f(\cdots f(f(x))\cdots))</math>, where the function <math>f</math> is performed <math>h</math> times. We find that <math> f(84) = f(f(89) = f^2(89) = f^3(94) = \ldots f^{y}(1004)</math>. <math>1004 = 84 + 5(y - 1) \Longrightarrow y = 185</math>. So we now need to reduce <math>f^{185}(1004)</math>. | + | Define <math>f^{h} = f(f(\cdots f(f(x))\cdots))</math>, where the function <math>f</math> is performed <math>h</math> times. We find that <math> f(84) = f(f(89)) = f^2(89) = f^3(94) = \ldots f^{y}(1004)</math>. <math>1004 = 84 + 5(y - 1) \Longrightarrow y = 185</math>. So we now need to reduce <math>f^{185}(1004)</math>. |
Let’s write out a couple more iterations of this function: | Let’s write out a couple more iterations of this function: | ||
Revision as of 11:18, 5 February 2011
Problem
The function f is defined on the set of integers and satisfies 
Find 
.
Solution
Define 
, where the function 
is performed 
times. We find that 
. 
. So we now need to reduce 
.
Let’s write out a couple more iterations of this function:
So this function reiterates with a period of 2 for 
. It might be tempting at first to assume that 
is the answer; however, that is not true since the solution occurs slightly before that. Start at 
:
![\[f^{3}(1004)=f^{2}(1001)=f(998)=f^{2}(1003)=f(1000)=\boxed{997}\]](http://latex.artofproblemsolving.com/f/b/8/fb80f9ca09d81d49b8754df2ae578bf487bb2f1b.png)
See also
| 1984 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
