Let $a, b$ and $c$ be positive real numbers such that min $\{ab, bc, ca\} \ge 1$. Prove that $$\sqrt[3]{(a^2 + 1)(b^2 + 1)(c^2 + 1)} \le (\frac{a+b+c}{3} )^2 + 1 $$
Source: 2017 Saudi Arabia IMO TST IV p1
Tags: algebra, inequalities
Let $a, b$ and $c$ be positive real numbers such that min $\{ab, bc, ca\} \ge 1$. Prove that $$\sqrt[3]{(a^2 + 1)(b^2 + 1)(c^2 + 1)} \le (\frac{a+b+c}{3} )^2 + 1 $$