Number Choosing Contest.

I think so!

Now for some guessing!

103

127

The smallest integer within the range of [100, 200] that does not break rule 1.

32

59

103: breaks YZ & 1, 3 and 4.
127: breaks YZ & 1, 2, 3 and 4.
???: breaks XY & 2, 3 and 5.
32: breaks YZ & 4 and 5.
59: breaks YZ & 1, 2, 3 and 4.

34, 36, 37, 40, 104

34: breaks YZ & 2 and 4.
36: breaks YZ & 1, 2, 3 and 5.
37: breaks YZ & 1, 2, 3, 4 and 5.
40: breaks YZ & 2, 3 and 5.
104: breaks YZ & 2, 3 and 5.