We are given a fixed point on a circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0,1,2,\ldots$ from it we obtain points with abscissas $n = 0,1,2,\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $1/5$ apart?
I have a few questions about this. Firstly, what does the question mean by abscissa? Are they talking about the abscissa of convergence? Secondly I don't really get the distance thing going $0,1,2,\ldots.$ Below is the official solution:
Why don't they ever go more than $+1$ distance (except for going across the fixed point)? Finally, since we are worried about actual distance not curved, how can they be sure this is in fact the first such pair of points?