There is no way to copyright someone else's work. Whoever produced the solution owns the copyright.

That is probably incorrect in the United States if it is "Work made for hire".

https://www.copyright.gov/circs/circ09.pdf

It is mostly correct. Unless you sign very carefully worded IP assignment agreement to that effect, even work you get paid for remains under your own copyright. And frankly anyone who would sign that kind of thing on a employment test exercise is... Naive.

I doubt they paid for the interview work.

To be a work for hire, the work must be produced by an employee within the scope of their employment. Consequently, you cannot do work for hire if you have not been hired.

This is not true. If the problem statement itself is a unique creation, and the solution is publicly viewable and gives details on the question itself, the question writer may have the copyright even if you signed no NDA when solving it. Even if not, if a company wants to sue you to try to keep their interviewing IP private (as they see it), they can tie you up in court at a trivial cost to themselves but a life destroying cost to a lone developer.

Source: I once had a dragged out legal dispute with a company that used a GitHub DMCA takedown to tell me to remove a public repo in which I solved their takehome challenge (I had signed no NDA or otherwise any type of agreement of any kind prior to solving it).

If it's not an original question find the original source (a source that is older than the company would be especially nice), and post a solution to the question posed by the original source. The world is a big place, a lot of question are not original.