Number Theory

$\frac{1}{x} - \frac{1}{xy} - \frac{1}{xyz} = \frac{1}{3}$

Multiply by x on both sides

$1 - \frac{1}{y} - \frac{1}{yz} = \frac{x}{3}$

x < 3

if x = 2:

$\frac{1}{y}+\frac{1}{yz} = \frac{1}{3}$

$1 + \frac{1}{z} = \frac{y}{3}$

Meaning z = 3, which cannot work in x > y > z

if x = 1:

$\frac{1}{y}+\frac{1}{yz} = \frac{2}{3}$

$1 + \frac{1}{z} = \frac{2y}{3}$

Also meaning z = 3, which doesn't work

Therefore, there is no solution.