First preimage resistance check
Preimage resistance is usually defined not for all the inputs, but for all the outputs, since what you are trying to model is the inability to, given any output $y$, obtain an input $x$ such that $H(x) = y$.

I'm not going to solve the problem for you since you may be able to do it by yourself. Just try to think of the patterns in the output and if this can be related to some input.