Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$

Question

I've tried getting everything on the left side and transforming it into something squared so that I can prove it's bigger or equals to 0 but I've been unsuccessful.

Answer 1

For positives $a$, $b$ and $c$ by C-S we obtain: $$\sum_{cyc}\frac{a^2}{b}\geq\frac{\left(\sum\limits_{cyc}a\right)^2}{\sum\limits_{cyc}b}=\frac{(a+b+c)^2}{a+b+c}=a+b+c.$$