Difference between revisions of "2021 Fall AMC 12B Problems/Problem 21"
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− | Let <math>a=\cos(x)+i\sin(x)</math>. Now <math>P(a)=1+a-a^2+a^3</math>. <math>P(-1)=-2</math> and <math>P(0)=1</math> so there is a real number <math>a_1</math> between <math>-1</math> and <math>0</math>. The other <math>a</math>'s must be complex conjugates since all coefficients of the polynomial are real. The magnitude of those complex <math>a</math>'s squared is <math>\frac{1}{a_1}</math> which is greater than <math>1</math>. If <math>x</math> is real number then <math>a</math> must have magnitude of <math>1</math>, but all the solutions for <math>a</math> do not have magnitude of <math>1</math>, so the answer is <math>\boxed{(A) 0}</math> ~lopkiloinm | + | Let <math>a=\cos(x)+i\sin(x)</math>. Now <math>P(a)=1+a-a^2+a^3</math>. <math>P(-1)=-2</math> and <math>P(0)=1</math> so there is a real number <math>a_1</math> between <math>-1</math> and <math>0</math>. The other <math>a</math>'s must be complex conjugates since all coefficients of the polynomial are real. The magnitude of those complex <math>a</math>'s squared is <math>\frac{1}{a_1}</math> which is greater than <math>1</math>. If <math>x</math> is real number then <math>a</math> must have magnitude of <math>1</math>, but all the solutions for <math>a</math> do not have magnitude of <math>1</math>, so the answer is <math>\boxed{\textbf{(A)}\ 0 }</math> ~lopkiloinm |
Revision as of 02:08, 24 November 2021
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