Difference between revisions of "1984 AIME Problems/Problem 8"
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== Problem == | == Problem == | ||
| − | The equation <math>z^6+z^3+1</math> has complex roots with argument <math>\theta</math> between <math>90^\circ</math> and <math>180^\circ</math> in | + | The equation <math>z^6+z^3+1</math> has complex roots with argument <math>\theta</math> between <math>90^\circ</math> and <math>180^\circ</math> in the complex plane. Determine the degree measure of <math>\theta</math>. |
== Solution == | == Solution == | ||
Revision as of 11:51, 18 October 2007
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 1 and argument of the form 
for integer 
.
This reduces to either 120 or 160. But can't be 120 because if 
, then 
and 
, a contradiction. This leaves 
.
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 | ||
