2021 Fall AMC 12B Problems/Problem 21
Revision as of 01:25, 24 November 2021 by Lopkiloinm (talk | contribs) (Created page with "==Problem== For real numbers <math>x</math>, let <cmath>P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)</cmath> where <math>i = \sqrt{-1}</math>. For how many...")
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 $\boxed{(A) 0}