I believe x=(10^k+1)*p^2/q^2 is the general formula. Here we need
1) (10^k+1) mod q^2 = 0
2) 0.1
2) is needed to make x a k digit number, and I chose p = 10. You can also choose p = 4 to 9.
The key is how to find k and q. It is only easy for q = 11.