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sum cyc 1/(1+x^3)=3/2

Source: Romanian District Olympiad 2022 - Grade 9 - Problem 2

March 27, 2022
Inequalitycyclic inequalityinequalitiesalgebra

Problem Statement

a)a) Prove that 2x3−3x2+1≥0, (∀)x≥0.2x^3-3x^2+1\geq 0,~(\forall)x\geq0. b)b) Let x,y,z≥0x,y,z\geq 0 such that 21+x3+21+y3+21+z3=3.\frac{2}{1+x^3}+\frac{2}{1+y^3}+\frac{2}{1+z^3}=3. Prove that 1−x1−x+x2+1−y1−y+y2+1−z1−z+z2≥0.\frac{1-x}{1-x+x^2}+\frac{1-y}{1-y+y^2}+\frac{1-z}{1-z+z^2}\geq 0.
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