MathNet 2020 — Math Olympiad Vault — Part 01 Question 33

Suppose $x$, $y$, and $z$ are real numbers that satisfy $x + y + z \gt 0$, $x y + y z + z x \gt 0$ and $x y z \gt 0$. Prove that $x$, $y$, and $z$ must all be positive.