<s>[INST] How prove this inequality $\sum_{cyc}\sqrt{\frac{2b+2c}{a}-1}\ge 3\sqrt{3}$ 
Let $a,b,c>0 ,2b+2c-a\ge 0,2c+2a-b\ge 0,2a+2b-c\ge 0$ show that
  $$\sqrt{\dfrac{2b+2c}{a}-1}+\sqrt{\dfrac{2c+2a}{b}-1}+\sqrt{\dfrac{2a+2b}{c}-1}\ge 3\sqrt{3}$$

I try use AM-GM and Cauchy-Schwarz inequality and from here I don't see what to do
 [/INST] By AM-GM
$$\sum_{cyc}\sqrt{\frac{2b+2c}{a}-1}=\sum_{cyc}\frac{2\sqrt3(2b+2c-a)}{2\sqrt{3a(2b+2c-a)}}\geq$$
$$\geq\sum_{cyc}\frac{2\sqrt3(2b+2c-a)}{3a+2b+2c-a}=\sum_{cyc}\frac{2\sqrt3(2b+2c-a)}{2(a+b+c)}=3\sqrt3$$
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