Mock AIME 3 Pre 2005 Problems/Problem 4

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Problem

$\zeta_1, \zeta_2,$ and $\zeta_3$ are complex numbers such that

$\begin{align*}\zeta_1+\zeta_2+\zeta_3&=1\\ \zeta_1^2+\zeta_2^2+\zeta_3^2&=3\\ \zeta_1^3+\zeta_2^3+\zeta_3^3&=7\end{align*}$ (Error compiling LaTeX. Unknown error_msg)

Compute $\zeta_1^{7} + \zeta_2^{7} + \zeta_3^{7}$.

Solution

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