Difference between revisions of "1984 AIME Problems/Problem 8"
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| − | From above, you notice that <math>z^6+z^3+1 = | + | From above, you notice that <math>z^6+z^3+1 = \frac {r^9-1}{r^3-1</math>}<math>. Therefore, the solutions are all of the ninth roots of unity that are not the third roots of unity. After checking, the only angle is </math>\boxed{\theta=160}$. |
== See also == | == See also == | ||
Revision as of 09:09, 14 March 2010
Problem
The equation 
has complex roots with argument 
between 
and 
in the complex plane. Determine the degree measure of .
Solution
If 
is a root of , then 
. The polynomial 
has all of its roots with absolute value 
and argument of the form 
for integer 
.
This reduces to either 
or 
. But can't be because if 
, then 
and 
, a contradiction. This leaves 
.
Also,
From above, you notice that $z^6+z^3+1 = \frac {r^9-1}{r^3-1$ (Error compiling LaTeX. Unknown error_msg)}
\boxed{\theta=160}$.
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 | ||
