2004 AIME I Problem 13

The polynomial

$$P(x)=\left(1+x+x^{2}+\ldots+x^{17}\right)^{2}-x^{17}$$

has 34 complex zeros of the form z_{k}=r_{k}\left[\cos \left(2 \pi \alpha_{k}\right)+i \sin \left(2 \pi \alpha_{k}\right)\right], k=1,2,3, \ldots, 34, with 0<\alpha_{1} \leq \alpha_{2} \leq \alpha_{3} \leq \ldots \leq \alpha_{34}<1 and r_{k}>0. Given that \alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}+\alpha_{5}=m / n, where m and n are relatively prime positive integers, find m+n.