MathDB

(a) For given complex numbers

a_1,a_2,a_3,a_4

, we define a function

P:\mathbb C\to\mathbb C

P(z)=z^5+a_4z^4+a_3z^3+a_2z^2+a_1z

. Let

w_k=e^{2ki\pi/5}

, where

k=0,\ldots,4

. Prove that

P(w_0)+P(w_1)+P(w_2)+P(w_3)+P(w_4)=5.

(b) Let

A_1,A_2,A_3,A_4,A_5

be five points in the plane. A pentagon is inscribed in the circle with center

A_1

and radius

R

. Prove that there is a vertex

S

of the pentagon for which

SA_1\cdot SA_2\cdot SA_3\cdot SA_4\cdot SA_5\ge R^5.

Problem 5