Ta có:
\(\left(a-1\right)\left(b-1\right)\left(c-1\right)=abc-\left(ab+bc+ca\right)+\left(a+b+c\right)-1\)
\(=abc-\left(ab+bc+ca\right)+5\ge0\)
\(\Rightarrow abc\ge ab+bc+ca-5\)(1)
\(\left(a-3\right)\left(b-3\right)\left(c-3\right)=abc-3\left(ab+bc+ca\right)+9\left(a+b+c\right)-27\)
\(=abc-3\left(ab+bc+ca\right)+27\le0\)
\(\Rightarrow abc\le3\left(ab+bc+ca\right)-27\)(2)
(1)(2) suy ra \(ab+bc+ca-5\le3\left(ab+bc+ca\right)-27\)
\(\Leftrightarrow ab+bc+ca\ge11\).
\(6^2=\left(a+b+c\right)^2=a^2+b^2+c^2+2\left(ab+bc+ca\right)\)
\(\Leftrightarrow a^2+b^2+c^2=36-2\left(ab+bc+ca\right)\le36-2.11=14\)
Dấu \(=\)khi \(\hept{\begin{cases}\left(a-1\right)\left(b-1\right)\left(c-1\right)=0\\\left(a-3\right)\left(b-3\right)\left(c-3\right)=0\\a+b+c=6\end{cases}}\Leftrightarrow\hept{\begin{cases}a=1\\b=2\\c=3\end{cases}}\)và các hoán vị.
