IMO 1968 SL 19

Origin: ITA

Problem

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 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 $\frac{1}{5}$ apart?

Solution

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 $20$.