<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] $
\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)}
\geq \frac{(a^2+b^2+c^2)^2}{a^3+b^3+c^3+ab(a+b)+bc(b+c)+ca(c+a)} \tag{1}
$
Remember
$\frac{{a_1}^2}{x_1}+\frac{{a_2}^2}{x_2}+\frac{{a_3}^2}{x_3} \geq \frac{(a_1+a_2+a_3)^2}{x_1+x_2+x_3} \tag{2}$
$
a^3+b^3+c^3+ab(a+b)+bc(b+c)+ca(c+a) $
$= a^3+b^3+c^3+a^{2}b+ab^2+b^{2}c+bc^2+c^{2}a+ca^2$
$= (a+b+c)(a^2+b^2+c^2) \tag{3}$
From $(1) $ and $(3)$
$
\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2} $
$\geq \frac{a^2+b^2+c^2}{a+b+c}$
$\geq\frac{a^2}{a+b+c}+\frac{b^2}{a+b+c}+\frac{c^2}{a+b+c}$
Now applying $(2)$ again to the expression on the right
$
\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}$ 
$\geq \frac{a^2}{a+b+c}+\frac{b^2}{a+b+c}+\frac{c^2}{a+b+c}$
$\geq \frac{(a+b+c)^2}{3(a+b+c)} = \frac{a+b+c}{3}$
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