\[\boxed{\text{854.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]
\[a > 0,\ \ b > 0,\ \ a \neq b\]
\[Допустим\ \ a^{3} + b^{3} > ab(a + b)\]
\[a^{3} + b^{3} - ab(a + b) =\]
\[= (a + b)\left( a^{2} - ab + b^{2} \right) -\]
\[- ab(a + b) =\]
\[= (a + b)\left( a^{2} - ab + b^{2} - ab \right) =\]
\[= (a + b)\left( a^{2} - 2ab + b^{2} \right) =\]
\[= (a + b)(a - b)^{2} > 0 \Longrightarrow a^{3} +\]
\[+ b^{3} > ab(a + b) - верно.\]