I wrote about the book here.
We discuss problems no. ##3, 4, page 21 (solved on page 177) and #38, page 25 (solved on page 185).
Problem #3 "Let $\varepsilon$ be a complex cube root of unity. Prove that
$|z-1|^2+|z-\varepsilon|^2+|z_\varepsilon^2|^2=3(|z|^2+1)"$
Problem #4 "Let $\varepsilon$ be a complex cube root of unity. Prove that
$|z+u|^2+|z+\varepsilon u|^2+|z+\varepsilon^2 u|^2=3(|z|^2+|u|^2)$
whatever $z,\;u\in\mathbb{C}$ are."
Problem #38 "Prove that :
a) $(z-1)^2+(z-\varepsilon)^2+(z-\bar\varepsilon)^2=3z^2$
where $\varepsilon=\cos\frac{2\pi}{3}+\imath \sin\frac{2\pi}{3};$
$\textbf{b)}\;\;\;(z-1)^2+(z-\varepsilon)^2+(z-\varepsilon ^2)^2+\dots+(z-\varepsilon ^{n-1})^2=nz^2$
where $\varepsilon=\cos \frac{2\pi}{n}+\imath \sin \frac{2\pi}{n};$
$$\textbf{c)}\;\;\; \sum_{k=0}^{n-1}|z-\varepsilon ^k|^2=n(|z|^2+1|)."$$
(to be continue)
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