Let $a, b$ and $c$ be positive real numbers satisfying $$\frac{1}{a + 2019}+\frac{1}{b + 2019}+\frac{1}{c + 2019}=\frac{1}{2019}.$$Prove that $abc \ge 4038^3$.
Source: New Zealand NZMOC Camp Selection Problems 2018 p5
Tags: algebra, inequalities
Let $a, b$ and $c$ be positive real numbers satisfying $$\frac{1}{a + 2019}+\frac{1}{b + 2019}+\frac{1}{c + 2019}=\frac{1}{2019}.$$Prove that $abc \ge 4038^3$.