Do statements giving background about a problem require proper citation?
I describe the problem in almost exactly the same way as some earlier papers.


If your description of the problem is based on the description in another paper, then you have to cite that paper. If your description is not based on another paper, but is such a standard thing, obvious to any expert in the field, that you have naturally written it in a similar way to other papers, then the following applies:

Statements like 


  this is a problem that has challenged mathematicians for over half a century 


or


  the computability of the matrix, given only partial spectral data quickly becomes intractable in higher dimensions


can benefit from citation(s). Not necessarily to avoid plagiarism - if you don't include citations, you're probably not misleading anyone into thinking it's an original contribution - but to support the statements with evidence.

Statements giving background about a problem often benefit from citations  because without evidence, the reader might not be convinced of the truth of the statement. (Or, even if we know our audience will agree with the statement, we sometimes cite evidence to show that we know the evidence.)

To some degree, this depends on conventions in your field, though. 


In some fields, statements like the ones above would typically be supported by citations to a handful of papers that either directly support the statement with evidence, or show that other experts in the field agree with this statement and have made similar statements themselves. 
In other fields, the standard is to only cite things you really use, and in that case you probably wouldn't cite anything in this situation. 


You can look for some high-quality published papers on topics similar to yours to get a sense of what the convention is in your field. If in doubt: cite.