If all three are integers, then their product must be an integer. Their product = 3+ xyz +3/xyz + 1/(xyz*xyz). It looks like their product can only be an integer if xyz = 1, but I haven't proved this yet (I'm on vacation). Given this start, can anybody prove that xyz must equal 1?
If we can prove this, then 1/yz = x/xyz = x, so 2x must be an integer and it's denominator must be 1 or 2. Same for y and z.
Then, if none of the denominators are 2, the only solution will be (1,1,1). (because xyz = 1). If one of the denominators = 2, the only solution is (2,1,1/2). If two of the denominators are 2, the only solution is (4,1/2,1/2). And there are no solutions where all three denominators equal 2.
Outline proof (spoiler)
