Question 57
Let $n \ge 3$ be a positive integer.
a) Prove that there exist $z_1, z_2, \dots, z_n \in \mathbb{C}$ such that
$$
\frac{z_1}{z_2} + \frac{z_2}{z_3} + \dots + \frac{z_{n-1}}{z_n} + \frac{z_n}{z_1} = n i.
$$
b) For which $n$ there exist the complex numbers $z_1, z_2, \dots, z_n$, having the same absolute value, such that
$$
\frac{z_1}{z_2} + \frac{z_2}{z_3} + \dots + \frac{z_{n-1}}{z_n} + \frac{z_n}{z_1} = n i?
$$
a) Prove that there exist $z_1, z_2, \dots, z_n \in \mathbb{C}$ such that
$$
\frac{z_1}{z_2} + \frac{z_2}{z_3} + \dots + \frac{z_{n-1}}{z_n} + \frac{z_n}{z_1} = n i.
$$
b) For which $n$ there exist the complex numbers $z_1, z_2, \dots, z_n$, having the same absolute value, such that
$$
\frac{z_1}{z_2} + \frac{z_2}{z_3} + \dots + \frac{z_{n-1}}{z_n} + \frac{z_n}{z_1} = n i?
$$