Problem

Source: 2007 Bulgarian Autumn Math Competition, Problem 12.3

Tags: Symmetric inequality, three variable inequality, algebra, inequalities



Find all real numbers $r$, such that the inequality \[r(ab+bc+ca)+(3-r)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 9\]holds for any real $a,b,c>0$.