Mock AIME 6 2006-2007 Problems/Problem 7
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
~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.