Question 166
Let $n \ge 3$ be a positive integer. Consider in the complex plane a regular polygon $P_1P_2\dots P_n$ inscribed in the unit circle, and let $A$ be the set of affixes of the vertices of the polygon. Let $a_1, a_2, \dots, a_{n-1}$ be complex numbers with the property:
$$
|z^{n-1} + a_1z^{n-2} + a_2z^{n-3} + \dots + a_{n-2}z + a_{n-1}| = 1, \text{ for any } z \in A.
$$
Show that $a_1 = a_2 = \dots = a_{n-1} = 0$.
$$
|z^{n-1} + a_1z^{n-2} + a_2z^{n-3} + \dots + a_{n-2}z + a_{n-1}| = 1, \text{ for any } z \in A.
$$
Show that $a_1 = a_2 = \dots = a_{n-1} = 0$.