Nhân vế với vế của giả thiết:
\(\left(a+\dfrac{1}{b}\right)\left(b+\dfrac{1}{c}\right)\left(c+\dfrac{1}{a}\right)=\dfrac{28}{3}\)
\(\Leftrightarrow\left(ab+\dfrac{1}{bc}+\dfrac{a}{c}+1\right)\left(c+\dfrac{1}{a}\right)=\dfrac{28}{3}\)
\(\Leftrightarrow abc+\dfrac{1}{abc}+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+a+b+c=\dfrac{28}{3}\) (1)
Cộng vế với vế giả thiết:
\(\Rightarrow a+b+c+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=4+1+\dfrac{7}{3}=\dfrac{22}{3}\) (2)
(1);(2) \(\Rightarrow abc+\dfrac{1}{abc}+\dfrac{22}{3}=\dfrac{28}{3}\)
\(\Rightarrow abc+\dfrac{1}{abc}=2\)
\(\Rightarrow\left(abc\right)^2-2\left(abc\right)+1=0\)
\(\Rightarrow\left(abc-1\right)^2=0\)
\(\Rightarrow abc=1\)
