2014 AMC 12A Problems/Problem 24
Problem
Let , and for , let . For how many values of is ?
Solution 1
1. Draw the graph of by dividing the domain into three parts.
2. Look at the recursive rule. Take absolute of the previous function and down by 1 to get the next function.
3. Count the x intercepts of the each function and find the pattern.
The pattern turns out to be solutions,for x interval:[1,99], the function gain only one extra solution after because there is no summit on the graph any more, and the answer is thus . (Revised by Flamedragon & Jason,C)
Solution 2
First, notice that the recursion and the definition of require that for all such that , if , then is even. Now, we can do case work on to find which values of (such that ) make even. The answer comes out to be all the even values of in the range , in the domain . So, the answer is or .
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2014amc12a/383
~ dolphin7
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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