<s>[INST] Prove for any positive real numbers $a,b,c$ $\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2} \geq \frac{a+b+c}{3}$ Since the problem sheets says I should use Cauchy-Schwarz inequality, I used
$\frac{{a_1}^2}{x_1}+\frac{{a_2}^2}{x_2}+\frac{{a_3}^2}{x_3}$ 
$\geq \frac{(a_1+a_2+a_3)^2}{x_1+x_2+x_3}$
I first multiplied each term by $a,b,c$ to get a perfect square on top like
$\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}$ 
$=\frac{a^4}{a(a^2+ab+b^2)}+\frac{b^4}{b(b^2+bc+c^2)}+\frac{c^4}{c(c^2+ca+a^2)}$
$\hspace{120pt}\geq \frac{(a^2+b^2+c^2)^2}{a^3+b^3+c^3+ab(a+b)+bc(b+c)+ca(c+a)} $
But I am still stuck for few hours. This is not a homework, but is a set of problems to prepare for AMC/USAMO.
(Note: I started off with a general question and got closed as off topic. I have another genuine Math Question now, and I am just trying to see if math.se is going to be useful) 
 [/INST] We need to prove that 
$$\sum_{cyc}\left(\frac{a^3}{a^2+ab+b^2}-\frac{a}{3}\right)\geq0$$ or
$$\sum_{cyc}\frac{2a^3-a^2b-ab^2}{a^2+ab+b^2}\geq0$$ or
$$\sum_{cyc}\left(\frac{a(a-b)(2a+b)}{a^2+ab+b^2}-(a-b)\right)\geq0$$ or
$$\sum_{cyc}\frac{(a-b)^2(a+b)}{a^2+ab+b^2}\geq0.$$
Done!
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