Difference between revisions of "1984 AIME Problems/Problem 8"
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− | From above, you notice that <math>z^6+z^3+1 = r^9-1 | + | From above, you notice that <math>z^6+z^3+1 = </math>\frac {r^9-1}{r^3-1<math>}</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}$. |
== See also == | == See also == |
Revision as of 10:09, 14 March 2010
Problem
The equation has complex roots with argument between and in the complex plane. Determine the degree measure of .
Solution
If is a root of , then . The polynomial has all of its roots with absolute value and argument of the form for integer .
This reduces to either or . But can't be because if , then and , a contradiction. This leaves .
Also,
From above, you notice that \frac {r^9-1}{r^3-1$}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)\boxed{\theta=160}$.
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 |