Sorry, but I'm going to have to disagree with this response. Not that it's technically incorrect but it doesn't really represent a good answer to OP's question.

Some problems have been proven to be unsolveable. For example In algebra, the Abel–Ruffini theorem states that there is no algebraic solution—that is, solution in radicals

First off a 'math problem' being 'solveable' isn't really well-defined so that's not really something that's been proven. And the fact that the solutions to quintic polynomials may not be expressible in radicals is not really an example of an 'unsolvable problem'. The Abel-Ruffini theorem is more similar to the statement "Not all penguins are black-and-white". When you think of a penguin, it is natural to think of a black-and-white bird and it'd be nice if this image you had in mind truly was representative of every possible penguin. The Abel-Ruffini theorem just says "Actually, if you look deeply enough you'll see penguins that are more than just black and white".

Also there are problems that are only solvable numerically with computers but we don't have enough computing power. For example it would be very interesting to be able to simulate the movement of air and the combustion of fuel as it goes through a jet engine, especially if such an analysis included the effect of heat on the stiffness of the materials and the deflection created thereby. And we are pretty close to being able to do that but it takes enormous computing power.