MathNet 2011 Question 59

Let $x$, $y$, $z$ be positive real numbers such that $x^3 + y^3 + z^3 = 1$. Prove that
$$
x^2 + y^2 + z^2 \gt x^5 + y^5 + z^5 + 2x^2y^2z^2(x + y + z).
$$