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Which system of equalities implies $a=b=c$

Source: INMO 2016 Problem 2

January 17, 2016

inequalitiesinequalities proposedalgebraalgebra proposed

Problem Statement

For positive real numbers

a,b,c

which of the following statements necessarily implies

a=b=c

: (I)

a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)

, (II)

a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)

? Justify your answer.