"the problem was solved" is correct and usable in written English. You can use it whenever you want to convey that a given problem has been dealt with successfully. For example, "The team put their heads together and, eventually, the problem was solved.".
The problem was solved, or so most thought.
One talking duck later, the problem was solved.
The problem was solved by moving the bin eight feet away from the neighbor's apartment.
Our team was not going to participate unless the problem was solved properly.
Then, when the insight arrived, Dodge immediately realized that the problem was solved.
Once the problem was solved, though, I saw the potential instantly.
When I did this and cleaned the flue and burner again, the problem was solved.
Thanks to Ludwig my first paper got accepted! The editor wrote me that my manuscript was well-written
Listya Utami K.
PhD Student in Biology, Bandung Institute of Technology, Indonesia