Question 173
Let $a$, $b$ and $c$ be non-negative real numbers. Prove that
$$
3(a^2 + b^2 + c^2) \ge (a+b+c)(\sqrt{ab} + \sqrt{bc} + \sqrt{ca}) + (a-b)^2 + (b-c)^2 + (c-a)^2 \ge (a+b+c)^2.
$$
$$
3(a^2 + b^2 + c^2) \ge (a+b+c)(\sqrt{ab} + \sqrt{bc} + \sqrt{ca}) + (a-b)^2 + (b-c)^2 + (c-a)^2 \ge (a+b+c)^2.
$$