Problem

Source: 2019 ELMO Shortlist A1

Tags: algebra, Inequality, inequalities



Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$ Proposed by Milan Haiman