2021 Fall AMC 12B Problems/Problem 21
Problem
For real numbers , let
where
. For how many values of
with
does
Solution
Let . Now
.
and
so there is a real number
between
and
. The other
's must be complex conjugates since all coefficients of the polynomial are real. The magnitude of those complex
's squared is
which is greater than
. If
is real number then
must have magnitude of
, but all the solutions for
do not have magnitude of
, so the answer is
~lopkiloinm