It's the Arnold's known problem.

gobathegreat wrote: Prove the inequality: $$\frac{A+a+B+b}{A+a+B+b+c+r}+\frac{B+b+C+c}{B+b+C+c+a+r}>\frac{C+c+A+a}{C+c+A+a+b+r}$$where $A$, $B$, $C$, $a$, $b$, $c$ and $r$ are positive real numbers https://artofproblemsolving.com/community/c6h402918p2244716 https://artofproblemsolving.com/community/c6h554998p3224587