2023 AMC 12B Problems/Problem 22
Contents
Problem
A real-valued function has the property that for all real numbers and Which one of the following cannot be the value of
Solution 1
Substituting we get Substituting we find This gives Plugging in implies , so answer choice is impossible.
~AtharvNaphade
Solution 2
First, we set and . Thus, the equation given in the problem becomes
Thus, or 1.
Case 1: .
We set . Thus, the equation given in the problem becomes
Thus, for all .
Case 2: .
We set . Thus, the equation given in the problem becomes
Thus, for any ,
Therefore, an infeasible value of is
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 3
The relationship looks suspiciously like a product-to-sum identity. In fact, which is basically the relation. So we know that is a valid solution to the function. However, if we define where is arbitrary, the above relation should still hold for so any value in can be reached, so choices and are incorrect.
In addition, using the similar formula for hyperbolic cosine, we know The range of is so choice is incorrect.
Therefore, the remaining answer is choice
~kxiang
Video Solution 1 by OmegaLearn
Video Solution 2
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2023 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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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