<s>[INST] Show $ \frac {a^3}{a^2+b^2+c^2-bc}+\frac {b^3}{a^2+b^2+c^2-ca} +\frac {c^3}{a^2+b^2+c^2-ab} \le \frac {(a+b+c)^2 (a^4+b^4+c^4)}{6abc(ab+bc+ca)} $ If $a,b,c$ are positive real numbers then how do we prove that $ \dfrac {a^3}{a^2+b^2+c^2-bc}+\dfrac {b^3}{a^2+b^2+c^2-ca} +\dfrac {c^3}{a^2+b^2+c^2-ab} \le \dfrac {(a+b+c)^2 (a^4+b^4+c^4)}{6abc(ab+bc+ca)} $ 
Please help. Thanks 
 [/INST] Noting $ab+bc+ca \le a^2+b^2+c^2$ and $a^4+b^4+c^4 \ge \frac13(a^2+b^2+c^2)^2$, it is sufficient to show:
$$\sum_{cyc} \frac{a^3}{a^2+b^2+c^2-bc} \le \frac{(a+b+c)^2(a^2+b^2+c^2)}{18 abc}$$
Now $a^2+(b^2+c^2-bc) \ge a^2+bc$. So it is enough to show
$$ \frac{(a+b+c)^2(a^2+b^2+c^2)}{18 abc} \ge \sum_{cyc} \frac{a^3}{a^2+bc} = (a+b+c)-abc\sum_{cyc}\frac1{a^2+bc}$$
where we have used $\dfrac{a^3}{a^2+bc} = a - \dfrac{abc}{a^2+bc}$.
This can be written as a quadratic in $\frac{a+b+c}{abc}$:
$$ (a^2+b^2+c^2)\left(\frac{a+b+c}{abc}\right)^2-18\left(\frac{a+b+c}{abc}\right)+18\sum_{cyc}\frac1{a^2+bc} \ge 0$$
and the discriminant condition is then:
$$(a^2+b^2+c^2)\sum_{cyc}\frac1{a^2+bc} \ge \tfrac92$$
which follows readily from Cauchy-Schwarz inequality.
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