This could well be too general a question, but I'd be interested in solutions to special cases too. Say you have some finite set of positive real numbers $x_i$, when is it the case that $\sum_i x_i > \prod_i x_i$? And when are they equal?

The special case that prompted this was an argument about whether any number is equal to the sum of its prime factors.

Any references or quick proofs welcome.