2012 AIME I Problems/Problem 6
Problem 6
The complex numbers and
satisfy
and the imaginary part of
is
, for relatively prime positive integers
and
with
Find
Solution
Substituting the first equation into the second, we find that and thus
So
must be a
nd root of unity, and thus the imaginary part of
will be
for some
with
But note that
is prime and
by the conditions of the problem so the denominator in the argument of this value will always be
and thus
See also
2012 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |