<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] We have$:$ $$ \left( {a}^{2}+bc \right)  \left( ac+{b}^{2} \right)  \left( ab+{c}^{
2} \right) -abc \left( a+b \right)  \left( b+c \right)  \left( c+a
 \right) $$
$$=ac \left( {b}^{2}+{c}^{2} \right)  \left( a-b \right) ^{2}+ab \left( {a}^{2}+{c}^
{2} \right)  \left( b-c \right) ^{2} +bc \left( {
a}^{2}+{b}^{2} \right)  \left( a-c \right) ^{2}\geqq 0$$
Done.
</s>