# Difference between revisions of "1984 AIME Problems/Problem 8"

Scorpius119 (talk | contribs) (solution added) |
|||

Line 3: | Line 3: | ||

== Solution == | == Solution == | ||

− | + | If <math>r</math> is a root of <math>z^6+z^3+1</math>, then <math>0=(r^3-1)(r^6+r^3+1)=r^9-1</math>. The polynomial <math>x^9-1</math> has all of its roots with absolute value 1 and argument of the form <math>40m^\circ</math> for integer <math>m</math>. | |

+ | |||

+ | |||

+ | This reduces <math>\theta</math> to either 120 or 160. But <math>\theta</math> can't be 120 because if <math>r=\cos 120^\circ +i\sin 120^\circ </math>, then <math>r^3=1</math> and <math>r^6+r^3+1=3</math>, a contradiction. This leaves <math>\theta=160</math>. | ||

== See also == | == See also == | ||

* [[1984 AIME Problems/Problem 7 | Previous problem]] | * [[1984 AIME Problems/Problem 7 | Previous problem]] | ||

* [[1984 AIME Problems/Problem 9 | Next problem]] | * [[1984 AIME Problems/Problem 9 | Next problem]] | ||

* [[1984 AIME Problems]] | * [[1984 AIME Problems]] |

## Revision as of 20:27, 26 March 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 .