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:
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 gamma, so to minimize it's absolute value, we let .
, answer choice B.