ELMO SL 2014 A6

Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$ . Prove that $$(a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}.$$Proposed by Sammy Luo