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Hi Cezar my friend, it is along time I haven't met you. Here is my solution: Let I be incenter use 2\sin\frac{A}{2}\cos\frac{A}{2}=\sin A=\frac{a}{2R},\sin\frac{A}{2}=\frac{r}{IA} , it is equivalent to \sum\frac{4RIA}{ra}\ge 4\sqrt{3}\frac{R}{r}\Leftrightarrow \sum\frac{IA}{a}\ge\sqrt{3} Which is tr...