Task B-2.1. (20 points) Determine all complex numbers $z$ such that the moduli of the numbers $z, \frac{1}{z}, 1-z$ are equal.
Solution. According to the condition of the problem, $|z|=\left|\frac{1}{z}\right|=|1-z|$

From the equality $|z|=\left|\frac{1}{z}\right|$, we get $|z|=1$, and then from $|z|=|1-z|$ it follows that $|1-z|=1$.

If we denote $z=x+i y$, we have $x^{2}+y^{2}=1$ and $(1-x)^{2}+y^{2}=1$.

(5 points)

By subtracting these equations, we get $x=\frac{1}{2}$.

Furthermore, $y^{2}=1-x^{2}=\frac{3}{4}$, from which $y_{1,2}= \pm \frac{\sqrt{3}}{2}$.

(10 points)

The sought complex numbers are $z_{1}=\frac{1}{2}+\frac{\sqrt{3}}{2} i$ and $z_{2}=\frac{1}{2}-\frac{\sqrt{3}}{2} i$.

(5 points)