CGMO 2003 Problem 4

(1) Prove that there exist five nonnegative real numbers $a, b, c, d$ and $e$ with their sum equal to 1 such that for any arrangement of these numbers around a circle, there are always two neighboring numbers with their product not less than $\frac{1}{9}.$

(2) Prove that for any five nonnegative real numbers with their sum equal to 1 , it is always possible to arrange them around a circle such that there are two neighboring numbers with their product not greater than $\frac{1}{9}.$