In the case the referent is specific. and so if an article is used it should be definite. But
The Hilbert twenty-fourth problem
is incorrect. (as FumbleFingers says in a comment, this is because a "number" adjective should come before the noun in normal use.) It could be rephrased as
The twenty-fourth Hilbert problem
Hilbert's twenty-fourth ...
Even though I don't know the specifics of your case, I would suggest this:
The tensile tests were performed at a XX strain rate for the sample with the XX mm gauge part.
The first indefinite article could be used since there may be infinite instances of such "XX strain rate".
That sample is indeed a specific one.
That "XX mm gauge part" ...
If there is more than one document current in a particular context, meaning, for example, that there are documents that have not expired, you can certainly refer to any one of them as "a current document".
You can even use it in a general sense:
A current document will have a date in the upper right hand corner that is not earlier than today's ...