Math Gold Medalist

Lor

2023 AIME II 

Problem 8

Let $\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$ where $i = \sqrt{-1}.$ Find the value of the product\[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).\]

de Moivre’s theorem

nth Root of Unity

Gauss Idea (Rainbow Idea)

w is 7th root of unity

1+w+w^2+…+w^6=0

Apply Gauss idea on multiplication of the problem.

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   Solution