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

(deleted category)
Line 10: Line 10:
{{AIME box|year=1984|num-b=7|num-a=9}}
{{AIME box|year=1984|num-b=7|num-a=9}}
[[Category:Intermediate Complex Numbers Problems]]
[[Category:Intermediate Trigonometry Problems]]
[[Category:Intermediate Trigonometry Problems]]

Revision as of 21:11, 30 November 2007


The equation $z^6+z^3+1$ has complex roots with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane. Determine the degree measure of $\theta$.


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

This reduces $\theta$ to either $120^{\circ}$ or $160^{\circ}$. But $\theta$ can't be $120^{\circ}$ because if $r=\cos 120^\circ +i\sin 120^\circ$, then $r^3=1$ and $r^6+r^3+1=3$, a contradiction. This leaves $\boxed{\theta=160}$.

See also

1984 AIME (ProblemsAnswer KeyResources)
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
Invalid username
Login to AoPS