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 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 .
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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All AIME Problems and Solutions |