Once again I have come across an olympiad-type problem which probably requires some sort of insight even though it looks simple. The question is as follows:
Let $a$, $b$ and $c$ be positive real numbers. Prove that:
$(a+b)(b+c)(c+a)$ $\geqslant$ $8(a+b-c)(b+c-a)(c+a-b)$
I have tried to multiply out the LHS but unfortunately it didn't get me much...
I found that if one of $a$, $b$ or $c$ is greater than or equal to the sum of the other two, then the inequality is trivially true, since LHS is positive while RHS isn't.
Would there be a quick and easy formula or known inequality that I could use to make this problem simpler? Or is this just a 'bash-and-solve' type question?
Any help, comments or edits are greatly appreciated! Thanks! :)
This question appeared in the South African Mathematics Olympiad in 2008.