Lời giải:
Ta có:\(H=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
\(\Leftrightarrow 18099H=\frac{18099}{a}+\frac{18099}{b}+\frac{18099}{c}\)
\(18099H=\frac{a+b+c}{a}+\frac{a+b+c}{b}+\frac{a+b+c}{c}\)
\(18099H=3+\frac{b}{a}+\frac{c}{a}+\frac{a}{b}+\frac{c}{b}+\frac{a}{c}+\frac{b}{c}\)
Áp dụng BĐT Cô- si ta có:
\(\frac{b}{a}+\frac{c}{a}+\frac{a}{b}+\frac{c}{b}+\frac{a}{c}+\frac{b}{c}\geq 6\sqrt[6]{\frac{b}{a}.\frac{c}{a}.\frac{a}{b}.\frac{c}{b}.\frac{b}{c}.\frac{a}{c}}=6\)
\(\Rightarrow 18099H\geq 3+6=9\)
\(\Rightarrow H\geq \frac{9}{18099}=\frac{1}{2011}\) hay \(H_{\min}=\frac{1}{2011}\)
Dấu bằng xảy ra khi \(a=b=c=6033\)
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