Nemotron-IMO-Bench 2026 G12

Let $ABC$ be a triangle with side-lengths $a=BC$, $b=CA$ and $c=AB$, and centroid $G$. Prove that for any point $P$ in the same plane as the triangle $$a\cdot PA^3+b\cdot PB^3+c\cdot PC^3\geq 3abc\cdot PG.$$