Problem:
Let v and w be distinct, randomly chosen roots of the equation z1997−1=0. Let m/n be the probability that 2+3≤∣v+w∣, where m and n are relatively prime positive integers. Find m+n.
Solution:
Because the 1997 roots of the equation are symmetrically distributed in the complex plane, it is no loss of generality to assume that v=1. Let w=cosθ+isinθ, with −180∘<θ<180∘. It is required to find the probability that
∣1+w∣2=∣(1+cosθ)+isinθ∣2=2+2cosθ≥2+3
which is equivalent to cosθ≥213. Thus ∣θ∣≤30∘. Because w=1, the only possible values of θ are
θ=±1997360∘,±1997720∘,±19971080∘,…,±1997360k∘
where k=⌊1997/12⌋=166. Hence the probability is 2⋅166/1996=83/499, and m+n=83+499=582.