3B. Let $z$ and $w$ be complex numbers such that $|z|=|w|=|z-w|$. Calculate the value of $\left(\frac{Z}{w}\right)^{300}$.
Solution. If $x=\frac{z}{w}$, then

$$
|x|=\left|\frac{Z}{w}\right|=\frac{|z|}{|w|}=1,|x-1|=\left|\frac{Z}{w}-1\right|=\frac{|z-w|}{|w|}=1
$$

From

$$
1=|x-1|^{2}=(x-1)(\overline{x-1})
$$

we get $x \bar{x}-x-\bar{x}+1=1$. Since $1=|x|^{2}=x \bar{x}$, it follows that $x-1+\bar{x}=0$. If we multiply by $x(x \neq 0)$, we will get $x^{2}-x+1=0$, and from there, multiplying by $x+1$, $(x \neq-1)$, we will get $x^{3}+1=0$, or $x^{3}=-1$. Finally, $\left(\frac{z}{w}\right)^{300}=\left(x^{3}\right)^{100}=1$.