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VMO 2018 P4

Source: Vietnam MO 2018 1st day 4th problem

January 11, 2018
analytic geometryinequalities

Problem Statement

On the Cartesian plane the curve (C)(C) has equation x2=y3x^2=y^3. A line dd varies on the plane such that dd always cut (C)(C) at three distinct points with xx-coordinates x1, x2, x3x_1,\, x_2,\, x_3. a. Prove that the following quantity is a constant: x1x2x323+x2x3x123+x3x1x223.\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: x12x2x33+x22x3x13+x32x3x13<−154.\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}.
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