2023 AMC 12A Problems/Problem 16
Problem
Consider the set of complex numbers satisfying . The maximum value of the imaginary part of can be written in the form , where and are relatively prime positive integers. What is ?
Solution
First, substitute in .
Let and
We are trying to maximize , so we'll turn the equation into a quadratic to solve for in terms of .
We want to maximize , due to the fact that is always negatively contributing to 's value, that means we want to minimize .
Due to the trivial inequality:
If we plug 's minimum value in, we get that 's maximum value is
Then and
- CherryBerry
See Also
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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All AMC 12 Problems and Solutions |
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