For all positive real numbers, prove the following inequalities: a) $x^5 + y^5 + z^5 \ge x^4y + y^4z + z^4x$ b) $x^5 + y^5 + z^5 \ge x^2y^2z + y^2z^2x + z^2x^2y$ c) $x^3y^2 +y^3z^2 + z^3x^2 \ge x^2y^2z+y^2z^2x+z^2x^2y$