Problem

Source: III International Festival of Young Mathematicians Sozopol 2012, Theme for 10-12 grade

Tags: algebra



If $a$, $b$, and $c$ are positive numbers, determine the least possible value of the following expression: $\frac{1}{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}-\frac{2}{\frac{a}{c}+\frac{c}{b}+\frac{b}{a}}$.