VMO 2018 Problem 4

On the Cartesian plane the curve $(C)$ has equation $x^2=y^3$ . A line $d$ varies on the plane such that $d$ always cut $(C)$ at three distinct points with $x$ -coordinates $x_1, x_2, x_3$ . a. Prove that the following quantity is a constant: $$\sqrt[3]{\frac{x_1x_2}{x_3^2}}+\sqrt[3]{\frac{x_2x_3}{x_1^2}}+\sqrt[3]{\frac{x_3x_1}{x_2^2}}.$$ b. Prove the following inequality: $$\sqrt[3]{\frac{x_1^2}{x_2x_3}}+\sqrt[3]{\frac{x_2^2}{x_3x_1}}+\sqrt[3]{\frac{x_3^2}{x_3x_1}}<-\frac{15}{4}.$$