Seeing the regularity in the equations, and the difference (16) between 20 and 36, and 36 and 52, led me to think about powers of 2, and quickly got x=y=z=4 as an answer.

Then, I decided to write y as a function of x as per the first equation; substitute in the second, and write z as a function of x, and finally substitute in the third, that became an enormous polynomial in x... which I didn't want to solve!