1998 APMO Problems/Problem 3

Problem

Let $a, b, c$ be positive real numbers. Prove that $(1+\frac{a}{b})(1+\frac{b}{c})(1+\frac{c}{a}) \geq 2(1+\frac{a+b+c}{\sqrt[\leftroot{-2}\uproot{2}3]{abc}})$.

Solution

WLOG, assume $abc = 1$. Then \[(a+b+c)(ab+ac+bc-2)-abc \geq 2\] \[(a+b+c)(ab+ac+bc-2) \geq 3\]

Then by AM-GM, \[\frac{a+b+c}{3} \geq \sqrt[\leftroot{-2}\uproot{2}3]{abc}\] \[a+b+c \geq 3\] and \[ab+ac+bc \geq 3\] so \[ab+ac+bc-2 \geq 1\]

And so $(a+b+c)(ab+ac+bc-2) \geq 3$.

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