hoangghiep

$LHS = \sum \frac{a^2}{b} = \frac{1}{4} \left(\sum \frac{a^2}{b}+3 \sum \frac{a^2}{b} \right) \geq \frac{1}{4} \left(\sum \frac{a^2}{b}+3\frac{(\sum a)^2}{\sum a} \right)= \frac{1}{4} \left(\sum \frac{a^2}{b} +3\sum a \right) = \frac{1}{4} \sum \left(\frac{a^2}{b}+3b\right) = \frac{1}{4} \sum \left(\frac{a^2+b^2}{b}+2b \right)$ $\geq \frac{1}{4} \sum 2\sqrt{\frac{a^2+b^2}{b}.2b}=\frac{1}{4} \sum 2\sqrt{2(a^2+b^2)} = \sum \frac{\sqrt{a^2+b^2}}{\sqrt{2}} = RHS $