RO-NMO 2008 Problem 102

Let $a,b,c$ be 3 complex numbers such that $a|bc| + b|ca| + c|ab| = 0.$ Prove that $|(a-b)(b-c)(c-a)| \geq 3\sqrt 3 |abc|.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.