MathNet 2003 — Part 01 Question 44

Let $x$, $y$, $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1$. Try that
$$
x^2 y + x y^2 + z^2 y + z^2 x + 2 x y z \le \frac{1}{3}
$$