Không mất tính tổng quát, giả sử \(a\le b\le c\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le\frac{3}{a}\)
\(\Rightarrow\frac{3}{a}\ge\frac{4}{5}\Rightarrow a\le\frac{15}{4}\Rightarrow a< 4\)
Mặt khác \(\frac{1}{a}< \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{4}{5}\Rightarrow a>\frac{5}{4}\Rightarrow a>1\)
\(\Rightarrow1< a< 4\Rightarrow\left[{}\begin{matrix}a=2\\a=3\end{matrix}\right.\)
- Với \(a=2\Rightarrow\frac{1}{b}+\frac{1}{c}=\frac{3}{10}\)
\(\Rightarrow\frac{3}{10}\le\frac{2}{b}\Rightarrow b\le\frac{20}{3}\Rightarrow b< 7\)
\(\frac{1}{b}< \frac{1}{b}+\frac{1}{c}=\frac{3}{10}\Rightarrow b>\frac{10}{3}\Rightarrow b>3\)
\(\Rightarrow3< b< 7\Rightarrow b=\left\{4;5;6\right\}\Rightarrow c=\left\{20;10;\frac{2}{15}\left(l\right)\right\}\)
- Với \(a=3\Rightarrow...\) xét tương tự bên trên
