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.