MathDB

Complex roots.

Source:

November 27, 2005

algebrapolynomialAMCAIMEnumber theoryrelatively primegeometric series

Problem Statement

The polynomial

P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}

has 34 complex roots of the form

z_k=r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34

, with

0<a_1\le a_2\le a_3\le\cdots\le a_{34}<1

and

r_k>0

. Given that

a_1+a_2+a_3+a_4+a_5=m/n

, where

m

and

n

are relatively prime positive integers, find

m+n

.

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