Problem

Source: ELMO 2014 Shortlist A6, by Sammy Luo

Tags: inequalities, inequalities proposed



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