2016 IMO Shortlist A1

Let $a, b$ and $c$ be positive real numbers such that $\min \{a b, b c, c a\} \geq 1$. Prove that $$\sqrt[3]{\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)} \leq\left(\frac{a+b+c}{3}\right)^{2}+1$$

令 $a、b$ 和 $c$ 为正实数，使得 $\min \{a b, b c, c a\} \geq 1$。证明 $$\sqrt[3]{\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)} \leq\left(\frac{a+b+c}{3}\right)^{2}+1$$