KöMaL 2021 A809

Let the lengths of the sides of triangle $ABC$ be denoted by $a$, $b$ and $c$ using the standard notations. Let $S$ denote the centroid of triangle $ABC$. Prove that for an arbitrary point $P$ in the plane of the triangle the following inequality is true: $a\cdot PA^3+b\cdot PB^3+c\cdot PC^3\ge 3abc\cdot PS.$