RU-ALLRUSSIAN 1976 Problem 13

n real numbers are written around a circle. One of the numbers is 1 and the sum of the numbers is 0. Show that there are two adjacent numbers whose difference is at least $n / 4$ . Show that there is a number which differs from the arithmetic mean of its two neighbours by at least $8 / n^2$ . Improve this result to some $k / n^2$ with $k > 8$ . Show that for $n = 30$ , we can take $k = 1800 / 113$ . Give an example of 30 numbers such that no number differs from the arithmetic mean of its two neighbours by more than $2 / 113$ .