1. Determine three complex numbers with a modulus of 1 such that both their sum and their product are equal to 1.
Solution. Let the required numbers be $z_{1}, z_{2}, z_{3}$. Then $z_{1}+z_{2}+z_{3}=1=z_{1} z_{2} z_{3}$ and

$$
z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}=\frac{1}{z_{3}}+\frac{1}{z_{1}}+\frac{1}{z_{2}}=\frac{\overline{z_{3}}}{\left|z_{3}\right|^{2}}+\frac{\overline{z_{1}}}{\left|z_{1}\right|^{2}}+\frac{\overline{z_{2}}}{\left|z_{2}\right|^{2}}=\overline{z_{1}+z_{2}+z_{3}}=1
$$

According to this, from Vieta's formulas, it follows that the required numbers are solutions to the equation

$$
z^{3}-z^{2}+z-1=0
$$

i.e., these are the numbers $1, i$ and $-i$.