1984 AIME Problems/Problem 7
Contents
Problem
The function f is defined on the set of integers and satisfies
Find .
Solution 1
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
:
Solution 2
We start by finding values of the function right under 1000 since they require little repetition.
$$ (Error compiling LaTeX. Unknown error_msg)\begin{align*}f(999)=f(f(1004))=f(1001)=998 f(998)=f(f(1003))=f(1000)=997 f(997)=f(f(1002))=f(999)=998 f(996)=f(f(1001))=f(998)=997 \\
Soon we realize the for integers
either equal
or
based on it parity. (If short on time, a guess of 998 or 997 can be taken now.)
If
is even
if
is odd
.
has even parity, so
.
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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