Question 50
Let $P$ be an interior point to an equilateral triangle of altitude $1$. If $x$, $y$, $z$ are the distances from $P$ to the sides of the triangle, then prove that:
$$
x^2 + y^2 + z^2 \geq x^3 + y^3 + z^3 + 6xyz.
$$
$$
x^2 + y^2 + z^2 \geq x^3 + y^3 + z^3 + 6xyz.
$$