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Part of 2021 Serbia JBMO TSTs

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Serbia JBMO TST 2021 problem 1

Source: Serbia JBMO TST 2021

1/3/2022
Prove that for positive real numbers a,b,ca, b, c the following inequality holds: \begin{align*} \frac{a}{9bc+1}+\frac{b}{9ca+1}+\frac{c}{9ab+1}\geq \frac{a+b+c}{1+(a+b+c)^2} \end{align*} When does equality occur?
algebrainequalities