Question 147
(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 $\frac{1}{9}$.
(posed by Qian Zhanwang)
(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 $\frac{1}{9}$.
(posed by Qian Zhanwang)