<s>[INST] Inequality. $(a^2+bc)(b^2+ca)(c^2+ab) \geq abc(a+b)(b+c)(c+a)$ Let $a,b,c$ be three real positive(strictly) numbers. Prove that: 
$$(a^2+bc)(b^2+ca)(c^2+ab) \geq abc(a+b)(b+c)(c+a).$$
I tried : 
$$abc\left(a+\frac{bc}{a}\right)\left(b+\frac{ca}{b}\right)\left(c+\frac{ab}{c}\right)\geq abc(a+b)(b+c)(c+a) $$
and now I want to try to prove that for example $$a+\frac{bc}{a} \geq a+b$$
but I don't know if is is a good idea. 
Thanks:) 
 [/INST] $$LHS-RHS=abc\sum_{sym} a(a-b)(a-c)+\frac{1}{2}(ab+bc+ca)\left[c^2(a-b)^2+a^2(b-c)^2+b^2(c-a)^2\right]\ge 0$$
True by Schur 3 deg
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