Mock AIME I 2012 Problems/Problem 8
Problem
Suppose that the complex number satisfies
. If
is the maximum possible value of
,
can be expressed in the form
. Find
.
Solution
We begin by dividing both sides by to obtain
. Now, consider that we may write
with
a positive real number so that
and
some real number. Then,
Since we need
, we must have
, or equivalently,
. By the quadratic equation, this has roots
and to maximize
, we take the larger root
which is clearly maximized when
is minimized. Since
, the maximum value of
will occur where
, so the maximum value of
occurs where
and finally we find that the maximum value of
is
Taking the fourth power, the desired answer is
.