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For ##a,b,c \geq 0##, prove that ##(a+b)(b+c)(c+a) \geq 8abc##

(https://www.math.nyu.edu/~bellova/putnam/putnam09_6.pdf)

I started off by expanding the brackets and doing some algebraic rearranging, but I don't think I've got anywhere.

Should I be using induction? Or is this an algebraic problem?

Thanks in advance