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min of (a^{-n}+b)/(1-a)+(b^{-n}+c)/(1-b)+(c^{-n}+a)/(1-c) if a+b+c=1, a,b,c>0

Source: Balkan BMO Shortlist 2016 A4

July 30, 2019
algebrainequalitiesminimumthree variable inequality

Problem Statement

The positive real numbers a,b,ca, b, c satisfy the equality a+b+c=1a + b + c = 1. For every natural number nn find the minimal possible value of the expression E=a−n+b1−a+b−n+c1−b+c−n+a1−cE=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}
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