Difference between revisions of "1984 AIME Problems/Problem 8"
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== Problem == | == Problem == | ||
− | The equation <math> | + | 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 thet complex plane. Determine the degree measure of <math>\theta</math>. |
== Solution == | == Solution == | ||
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* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
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+ | [[Category:Trigonometry Problems]] |
Revision as of 08:29, 18 October 2007
Problem
The equation has complex roots with argument between and in thet 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 |