Question 70
Let $a$, $b$, $c$ be distinct positive integers and let $k$ be a positive integer such that
$$
ab + bc + ca \ge 3k^2 - 1.
$$
Prove that $\frac{1}{3}(a^3 + b^3 + c^3) \ge abc + 3k$.
$$
ab + bc + ca \ge 3k^2 - 1.
$$
Prove that $\frac{1}{3}(a^3 + b^3 + c^3) \ge abc + 3k$.