<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] Here's another approach that is less intensive. Show that 
$$ (a^2 + bc) ( b^2 + ca) \geq ab(c+b)(c+a) $$
This is equivalent to 
$$ c (a-b)^2 (a+b) \geq 0 $$
Multiply the 3 cyclic versions of the first inequality, and you get your conclusion. Equality holds if and only if $a=b=c$.

Note: The factorization of the second line is easily guessed, by observing the equality holds when $ c = 0, a = b, a = - b$.
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