'Solve' means to produce the correct answer to a question, 'resolve' means to settle a dispute or bring some matter to a close.
'Problem' has two common meanings. Usually, it refers to a difficult situation of some kind which has undesirable consequences. My toothache is a problem for me, hunger is a problem for the world. The dentist can solve or resolve my toothache problem, but he can't do much about the hunger problem. Here 'solve' and 'resolve' are essentially saying the same: he 'answers' the question of my toothache, he brings the matter of my toothache to a close.
'Problem' is also used in mathematics to mean, essentially, a question, e.g. "At the end of each class, the teacher would write a series of problems on the board, our homework was to solve them." The verb 'resolve' can't be used here.