Question 10
$n$ 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 $k \gt 8$. 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 $2/113$.