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sum (a+b)/(a^2+ab+b^2) <=2 if 1/a+1/b+1/c =3 for a,b,c>0

Source: 2020 Greek JBMO TST p2

November 14, 2020
algebrainequalities

Problem Statement

Let a,b,ca,b,c be positive real numbers such that 1a+1b+1c=3\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3. Prove that a+ba2+ab+b2+b+cb2+bc+c2+c+ac2+ca+a2≤2\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2 When is the equality valid?
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