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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1998 APMO Problems/Problem 3

Problem

Solution