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Found problems: 1

2006 Balkan MO, 1
Let $ a$, $ b$, $ c$ be positive real numbers. Prove the inequality \[ \frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}. \]
inequalities theorems inequalities rearrangement inequality AM-GM inequality Cauchy-Schwarz inequality algebra