No, they can't solve a problem. They can quickly calculate results for a specific input set, if the problem is already solved beforehand. But they cannot tackle a new problem on their own. No computer has ever solved a previously unsolved problem. They don't have problem solving skills, all they can do is compute.
Given 2 points on a randomly generated map of any size, the shortest path between them is a previously unsolved problem. Let's go ahead and say it is some ridiculous size like 1 light year with 500 billion random objects of random sizes thrown in so that you can't say that somewhere someone has solved it before. Here is a link to explain how a computer is going to solve that problem:
http://theory.stanford.edu/~amitp/GameProgramming/AStarComparison.html
