I don't believe such thing to exist.
It is uncommon, but it's really not all that different from "I don't want this thing to exist." (different verb, different adjective) You can even let both these adjectives act as pronouns, and omit the "thing." "I don't want this to exist." "I don't want such to exist."
I believe him to have passed the exam. I want / expect him to have passed the exam. (i.e., no problem)
I believe him to pass the exam. This is a logical problem. You can't believe something that hasn't happened yet. I want / expect him to pass the exam.
- A.