1984 AIME Problems/Problem 7
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 :
Note that we should also be suspicious if our answer is - it is a -digit number, and we were not asked to, say, divide our number by .
Solution 2
We start by finding values of the function right under since they require iteration of the function.
Soon we realize the for integers either equal or based on it parity. (If short on time, a guess of or can be taken now.) If is even if is odd . has even parity, so . The result may be rigorously shown through induction.
Solution 3
Assume that is to be performed times. Then we have In order to find , we want to know the smallest value of Because then From which we'll get a numerical value for .
Notice that the value of we expect to find is basically the smallest such that after is performed times and then is performed back times, the result is greater than or equal to .
In this case, the value of for is , because Thus
~ Nafer
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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