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3

Part of 2019 India Regional Mathematical Olympiad

Problems(2)

Cyclic Non-homogeneous in 3 variables

Source: RMO 2019 P3

10/20/2019
Let a,b,ca,b,c be positive real numbers such that a+b+c=1a+b+c=1. Prove that aa2+b3+c3+bb2+a3+c3+cc2+a3+b3≤15abc\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\leq\frac{1}{5abc}
InequalityRMOinequalities
System of Equations

Source: RMO 2019 Paper 2 P3

11/10/2019
Find all triples of non-negative real numbers (a,b,c)(a,b,c) which satisfy the following set of equations a2+ab=ca^2+ab=c b2+bc=ab^2+bc=a c2+ca=bc^2+ca=b
indiaRMOalgebraInequalitysystem of equations