2008 AMC 12B Problems/Problem 19
Problem 19
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
We need only concern ourselves with the imaginary portions of and (both of which must be 0). These are:
Since appears in both equations, we let it equal 0 to simplify the equations. This yields two single-variable equations. Equation 1 tells us that the imaginary part of must be , and equation two tells us that the real part of must be . Therefore, . There are no restrictions on , so to minimize 's absolute value, we let .
, answer choice B.
See Also
2008 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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