1984 AIME Problems/Problem 8
Contents
Problem
The equation has complex roots with argument
between
and
in the complex plane. Determine the degree measure of
.
Solution 1
We shall introduce another factor to make the equation easier to solve. If is a root of
, then
. The polynomial
has all of its roots with absolute value
and argument of the form
for integer
(the ninth degree roots of unity). Now we simply need to find the root within the desired range that satisfies our original equation
.
This reduces to either
or
. But
can't be
because if
, then
. (When we multiplied by
at the beginning, we introduced some extraneous solutions, and the solution with
was one of them.) This leaves
.
Solution 2
The substitution simplifies the equation to
. Applying the quadratic formula gives roots
, which have arguments of
and
respectively.
We can write them as
and
.
So we can use De Moivre's theorem (which I would suggest looking at if you never heard of it before) to find the fractional roots of the expressions above!
For
we have
and
Similarly for
, we have
and
The only argument out of all these roots that fits the description is
~ blueballoon
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |