Mathematics and artificial intelligence (AI) have had a symbiotic relationship since Allen Turing dreamed of taking Hilbert’s tenth problem into the realm of computation that would blur the distinction between human and machine reasoning. Every aspect of AI has mathematical roots, and here have been some bilateral developments. For example, efforts to improve computational logic led to new results in mathematical logic, itself. Intelligent tutors have improved, particularly with modern programming paradigms, and some contribute to mathematics education. I see this as a part of mathematics because I view mathematics as not just a collection of facts, but as a process mathematical reasoning. Once this is accepted, mathematics and artificial intelligence interleave throughout each of their branches.
As indicated in the title, problem solving involves two activities: modeling, sometimes called abstraction of a problem, and theorem proving. Both involve logical and analogical reasoning. Research into these has spawned several subdisciplines that have very similar goals and strategies, but which have different names, or have evolved by professional communities that tend not to overlap.