US-TST 2000 Problem 1

Let $a, b, c$ be nonnegative real numbers. Prove that $$\frac{a+b+c}{3} - \sqrt[3]{abc} \leq \max\{(\sqrt{a} - \sqrt{b})^2, (\sqrt{b} - \sqrt{c})^2, (\sqrt{c} - \sqrt{a})^2\}.$$