19. (ITA 5) We are given a fixed point on the 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 abscisas $n=0,1,2, \ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $1 / 5$ apart?
19. We shall denote by $d_{n}$ the shortest curved distance from the initial point to the $n$th point in the positive direction. The sequence $d_{n}$ goes as follows: $0,1,2,3,4,5,6,0.72,1.72, \ldots, 5.72,0.43,1.43, \ldots, 5.43,0.15=d_{19}$. Hence the required number of points is 20 .