We write
\begin{align*}
\frac{(1 + 5z)(4z + 3x)(5x + 6y)(y + 18)}{xyz} &= \frac{4}{5} \cdot \frac{(1 + 5z)(5z + \frac{15}{4} x)(5x + 6y)(y + 18)}{xyz} \\
&= \frac{4}{5} \cdot \frac{4}{3} \cdot \frac{(1 + 5z)(5z + \frac{15}{4} x)(\frac{15}{4} z + \frac{9}{2} y)(y + 18)}{xyz} \\
&= \frac{4}{5} \cdot \frac{4}{3} \cdot \frac{2}{9} \cdot \frac{(1 + 5z)(5z + \frac{15}{4} x)(\frac{15}{4} x + \frac{9}{2} y)(\frac{9}{2} y + 81)}{xyz} \\
&= \frac{32}{135} \cdot \frac{(1 + 5z)(5z + \frac{15}{4} x)(\frac{15}{4} x + \frac{9}{2} y)(\frac{9}{2} y + 81)}{xyz}.
\end{align*}Let $a = 5z,$ $b = \frac{15}{4} x,$ and $c = \frac{9}{2} y,$ so $z = \frac{1}{5} a,$ $x = \frac{4}{15} b,$ and $y = \frac{2}{9} c.$  Then
\begin{align*}
\frac{32}{135} \cdot \frac{(1 + 5z)(5z + \frac{15}{4} x)(\frac{15}{4} x + \frac{9}{2} y)(\frac{9}{2} y + 81)}{xyz} &= \frac{32}{135} \cdot \frac{(1 + a)(a + b)(b + c)(c + 81)}{\frac{4}{15} b \cdot \frac{2}{9} c \cdot \frac{1}{5} a} \\
&= 20 \cdot \frac{(1 + a)(a + b)(b + c)(c + 81)}{abc} \\
&= 20 \cdot (1 + a) \left( 1 + \frac{b}{a} \right) \left( 1 + \frac{c}{b} \right) \left( 1 + \frac{81}{c} \right).
\end{align*}By AM-GM,
\begin{align*}
1 + a &= 1 + \frac{a}{3} + \frac{a}{3} + \frac{a}{3} \ge 4 \sqrt[4]{\left( \frac{a}{3} \right)^3}, \\
1 + \frac{b}{a} &= 1 + \frac{b}{3a} + \frac{b}{3a} + \frac{b}{3a} \ge 4 \sqrt[4]{\left( \frac{b}{3a} \right)^3}, \\
1 + \frac{c}{b} &= 1 + \frac{c}{3b} + \frac{c}{3b} + \frac{c}{3b} \ge 4 \sqrt[4]{\left( \frac{c}{3b} \right)^3}, \\
1 + \frac{81}{c} &= 1 + \frac{27}{c} + \frac{27}{c} + \frac{27}{c} \ge 4 \sqrt[4]{\left( \frac{27}{c} \right)^3},
\end{align*}so
\begin{align*}
20 \cdot (1 + a) \left( 1 + \frac{b}{a} \right) \left( 1 + \frac{c}{b} \right) \left( 1 + \frac{81}{c} \right) &\ge 20 \cdot 256 \sqrt[4]{\left( \frac{a}{3} \right)^3 \cdot \left( \frac{b}{3a} \right)^3 \cdot \left( \frac{c}{3b} \right)^3 \cdot \left( \frac{27}{c} \right)^3} \\
&= 5120.
\end{align*}Equality occurs when
\[1 = \frac{a}{3} = \frac{b}{3a} = \frac{c}{3b} = \frac{27}{c},\]or $a = 3,$ $b = 9,$ and $c = 27,$ which means $x = \frac{12}{5},$ $y = 6,$ and $z = \frac{3}{5}.$  Therefore, the minimum value is $5120.$
다음과 같이 작성합니다.
\begin{align*}
\frac{(1 + 5z)(4z + 3x)(5x + 6y)(y + 18)}{xyz} &= \frac{4}{5} \cdot \frac{(1 + 5z)(5z + \frac{15}{4} x)(5x + 6y)(y + 18)}{xyz} \\
&= \frac{4}{5} \cdot \frac{4}{3} \cdot \frac{(1 + 5z)(5z + \frac{15}{4} x)(\frac{15}{4} z + \frac{9}{2} y)(y + 18)}{xyz} \\
&= \frac{4}{5} \cdot \frac{4}{3} \cdot \frac{2}{9} \cdot \frac{(1 + 5z)(5z + \frac{15}{4} x)(\frac{15}{4} x + \frac{9}{2} y)(\frac{9}{2} y + 81)}{xyz} \\
&= \frac{32}{135} \cdot \frac{(1 + 5z)(5z + \frac{15}{4} x)(\frac{15}{4} x + \frac{9}{2} y)(\frac{9}{2} y + 81)}{xyz}.
\end{align*}$a = 5z,$b = \frac{15}{4} x,$ c = \frac{9}{2} y,$이므로 $z = \frac{1}{5} a,$ $x = \frac{4}{15} b,$ $y = \frac{2}{9} c.$ 그러면 다음과 같이 됩니다.
\begin{align*}
\frac{32}{135} \cdot \frac{(1 + 5z)(5z + \frac{15}{4} x)(\frac{15}{4} x + \frac{9}{2} y)(\frac{9}{2} y + 81)}{xyz} &= \frac{32}{135} \cdot \frac{(1 + a)(a + b)(b + c)(c + 81)}{\frac{4}{15} b \cdot \frac{2}{9} c \cdot \frac{1}{5} a} \\
&= 20 \cdot \frac{(1 + a)(a + b)(b + c)(c + 81)}{abc} \\
&= 20 \cdot (1 + a) \left( 1 + \frac{b}{a} \right) \left( 1 + \frac{c}{b} \right) \left( 1 + \frac{81}{c} \right).
\end{align*}By AM-GM,
\begin{align*}
1 + a &= 1 + \frac{a}{3} + \frac{a}{3} + \frac{a}{3} \ge 4 \sqrt[4]{\left( \frac{a}{3} \right)^3}, \\
1 + \frac{b}{a} &= 1 + \frac{b}{3a} + \frac{b}{3a} + \frac{b}{3a} \ge 4 \sqrt[4]{\left( \frac{b}{3a} \right)^3}, \\
1 + \frac{c}{b} &= 1 + \frac{c}{3b} + \frac{c}{3b} + \frac{c}{3b} \ge 4 \sqrt[4]{\left( \frac{c}{3b} \right)^3}, \\
1 + \frac{81}{c} &= 1 + \frac{27}{c} + \frac{27}{c} + \frac{27}{c} \ge 4 \sqrt[4]{\left( \frac{27}{c} \right)^3},
\end{align*}so
\begin{align*}
20 \cdot (1 + a) \left( 1 + \frac{b}{a} \right) \left( 1 + \frac{c}{b} \right) \left( 1 + \frac{81}{c} \right) &\ge 20 \cdot 256 \sqrt[4]{\left( \frac{a}{3} \right)^3 \cdot \left( \frac{b}{3a} \right)^3 \cdot \left( \frac{c}{3b} \right)^3 \cdot \left( \frac{27}{c} \right)^3} \\
&= 5120.
\end{align*}평등은 다음과 같은 경우에 발생합니다.
\1 = \frac{a}{3} = \frac{b}{3a} = \frac{c}{3b} = \frac{27}{c},\]또는 $a = 3,$ $b = 9,$ 및 $c = 27,$ 즉, $x = \frac{12}{5},$ $y = 6,$ 및 $z = \frac{3}{5}.$ 따라서 최소값은 $5120.$입니다.