2008 AMC 12B Problems/Problem 19
Contents
Problem
A function is defined by for all complex numbers , where and are complex numbers and . Suppose that and are both real. What is the smallest possible value of ?
Solution 1:
We need only concern ourselves with the imaginary portions of and (both of which must be 0). These are:
Let and then we know and Therefore which reaches its minimum when by the Trivial Inequality. Thus, the answer is
Solution 2:
Since and are both real we get,
Solving, we get , can be anything, to minimize the value we set , so then the answer is . Thus, the answer is
By: Quaratinium
See Also
2008 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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All AMC 12 Problems and Solutions |
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