USAMO 1980 Problem 3

A $+B+C$ is an integral multiple of $\pi . x, y, z$ are real numbers. If $x \sin A+y \sin B+z \sin C=x^{2} \sin 2 A+$ $y^{2} \sin 2 B+z^{2} \sin 2 C=0$, show that $x^{n} \sin n A+y^{n} \sin n B+z^{n} \sin n C=0$ for any positive integer $n$.