Difference between revisions of "1984 AIME Problems/Problem 7"
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== Problem == | == Problem == | ||
− | The function f is defined on the set of | + | The [[function]] f is defined on the [[set]] of [[integer]]s and satisfies |
<math> | <math> | ||
f(n)= | f(n)= | ||
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== Solution == | == Solution == | ||
− | {{solution}} | + | Define <math>\displaystyle 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>\displaystyle 1004 = 84 + 5(y - 1) \Longrightarrow y = 185</math>. So we now need to reduce <math>\displaystyle f^{185}(1004)</math>. |
+ | |||
+ | Let’s write out a couple more iterations of this function: | ||
+ | |||
+ | <div style="text-align:center;"><math>\displaystyle f^{185}(1004) = f^{184}(1001) = f^{183}(998)</math><br /><math>= f^{184}(1003) = f^{183}(1000) = f^{182}(997)</math><br /><math>= f^{183}(1002) = f^{182}(999) = f^{183}(1004)</math></div> | ||
+ | |||
+ | So this function reiterates with a period of 2 for <math>x</math>. It might be tempting at first to assume that <math>f(1004) = 999</math> is the answer; however, that is not true since the solution occurs slightly before that. Start at <math>f^3(1004)</math>: | ||
+ | |||
+ | <div style="text-align:center;"><math>f^{3}(1004) = f^{2}(1001) = f(998)</math><br /><math>= f^{2}(1003) = f(1000) = 997</math></div> | ||
+ | |||
== See also == | == See also == | ||
{{AIME box|year=1984|num-b=6|num-a=8}} | {{AIME box|year=1984|num-b=6|num-a=8}} | ||
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* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | |||
+ | [[Category:Intermediate Algebra Problems]] |
Revision as of 20:31, 10 September 2007
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:
![$\displaystyle f^{185}(1004) = f^{184}(1001) = f^{183}(998)$](http://latex.artofproblemsolving.com/f/0/2/f021a55297891425b24b2d413ab30750ded0c659.png)
![$= f^{184}(1003) = f^{183}(1000) = f^{182}(997)$](http://latex.artofproblemsolving.com/7/a/8/7a80f348127aa35c61a3edbd57cb8fe21925e967.png)
![$= f^{183}(1002) = f^{182}(999) = f^{183}(1004)$](http://latex.artofproblemsolving.com/7/1/1/711b2c1e09a9d998f08affbcd6cfb167592d44cb.png)
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
:
![$f^{3}(1004) = f^{2}(1001) = f(998)$](http://latex.artofproblemsolving.com/c/3/1/c31b2c1b74b0990abf00cc0fd322cb3cdf890656.png)
![$= f^{2}(1003) = f(1000) = 997$](http://latex.artofproblemsolving.com/0/e/0/0e02a85b0dec5f09d8fa3659cdb6a6b4c6cbe496.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 |