MathDB

n

numbers are written down along the circumference. Their sum equals to zero, and one of them equals

1

.
a) Prove that there are two neighbours with their difference not less than

n/4

.
b) Prove that there is a number that differs from the arithmetic mean of its two neighbours not less than on

8/(n^2)

.
c) Try to improve the previous estimation, i.e what number can be used instead of

8

?
d) Prove that for

n=30

there is a number that differs from the arithmetic mean of its two neighbours not less than on

2/113

, give an example of such

30

numbers along the circumference, that not a single number differs from the arithmetic mean of its two neighbours more than on

2/113

232