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strange long inequality

Source: Iran 3rd round 2011-algebra exam-p4

September 7, 2011
inequalitiesfunctionLaTeXCauchy Inequalityinequalities proposed

Problem Statement

For positive real numbers a,ba,b and cc we have a+b+c=3a+b+c=3. Prove a1+(b+c)2+b1+(a+c)2+c1+(a+b)2≤3(a2+b2+c2)a2+b2+c2+12abc\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}.
proposed by Mohammad Ahmadi
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