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If $a,b,c\ge0:a+b+c=3,$ prove $\sqrt{a+b+b^2}+\sqrt{b+c+c^2}+\sqrt{c+a+a^2}\ge 3\sqrt{3}.$
Alternative proof.
By using Cauchy-Schwarz and AM-GM $$\sqrt{a+b+b^2}\ge \frac{a+2b}{\sqrt{a+b+1}}\ge 2\sqrt{3}\cdot\frac{a+2b}{a+b+4}.$$It implies $$L.H.S\ge2\sqrt{3}\cdot\left(\frac{a+2b}{a+b+4}+\...
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