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sum a/(a+b^2) <= 1/4 (sum 1/a) when a + b + c = 1, a,b,c>0

Source: 2016 Grand Duchy of Lithuania, Mathematical Contest p1 (Baltic Way TST)

October 3, 2020
inequalitiesalgebra

Problem Statement

Let a,ba, b and cc be positive real numbers such that a+b+c=1a + b + c = 1. Prove that aa+b2+bb+c2+cc+a2≤14(1a+1b+1c)\frac{a}{a+b^2}+\frac{b}{b+c^2}+\frac{c}{c+a^2} \le \frac{1}{4} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)
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