Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 7"
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But to get how many more '''distinct''' complex roots, we must subtract the complex roots of <math>P_{1004}</math> that can be found in <math>Q_{1003}</math> | But to get how many more '''distinct''' complex roots, we must subtract the complex roots of <math>P_{1004}</math> that can be found in <math>Q_{1003}</math> | ||
| − | complex roots of <math>P_{1004}:\; x=e^{\frac{1}{1005}2\pi i},e^{\frac{2}{1005}2\pi i},\cdots,e^{\frac{1004}{1005}2\pi i}</math> | + | complex roots of <math>P_{1004}:\; x=e^{\frac{1}{1005}2\pi i},e^{\frac{2}{1005}2\pi i},e^{\frac{3}{1005}2\pi i},\cdots,e^{\frac{1004}{1005}2\pi i}</math> for a total of <math>1004</math> complex roots. |
Revision as of 19:21, 26 November 2023
Problem
Let 
and 
for all integers 
. How many more distinct complex roots does 
have than 
?
Solution
The roots of 
will be in the form 
for 
with the only real solution when 
is odd and 
and the rest are complex.
Therefore, each will have distinct complex roots when is even and 
distinct complex roots when is odd.
The roots of 
will be all of the roots of 
which will include several repeated roots.
To get how many more complex roots does have than that will be the number of complex roots of 
.
But to get how many more distinct complex roots, we must subtract the complex roots of that can be found in
complex roots of 
for a total of 
complex roots.
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
