Problem

Source: ELMO 2013/2, by Evan Chen; also Shortlist A5

Tags: inequalities, logarithms, function, calculus, derivative, inequalities unsolved, 3-variable inequality



Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. Proposed by Evan Chen