The decision to choose one tense of another does not have anything to do with the fact that the problem was solved or not, but with the fact that the problem was something that happened in a given time, or was something that used to happen recurrently.
IfTaking the present as the point the action is referred from, if you say
Miguel tuvo un problema
you are stating that Miguel had a problem one, a given day. If you say
Miguel tenía un problema
you are stating that Miguel used to have a problem that happened once in a while. Complete sentences for the two cases could be:
Miguel tuvo un problema ayer: su perro se tiró un pedo y casi muere asfixiado por el hedor.
(The problem happened just yesterday.)Miguel tenía un problema con su perro: solía tirarse tales pedos que tenía que tener la ventana siempre abierta.
(The problem used to happen every day, but it is not happening any more.)
In fact, if the problem started in the past but it is still not solved, you can use the present tense:
Miguel tiene un problema: su perro se tiró un pedo ayer y la casa aún huele mal.
Note that so far I have used the tenses taking the present as the point in which the action is referred from. That is, we are talking now abou a problem happened in the past. But if we speak about the problem from the point of view of the past, things change. Then you could refer to the problem as something ongoing:
Miguel tenía un problema: su perro se había tirado un pedo y no sabía cómo ahuyentar el mal olor.
(Miguel's problem is still the present from the narrator's point of view.)