[carstimon]

It depends on what you mean by a math problem. It seems a common idea is: A math problem is a question asking for the set of solutions to some mathematical equation, and only the actual solution is useful.

But I would include all of the following as math problems: -Prove that some equation actually has solutions -Find some bounds on solutions to that equation -How can I compute an approximate solution to that equation? -How good is that approximation going to be? -What's a way to try to separate large data sets into clusters?

With regards to the first four: many equations people are interested in simply have no hope of getting a solution you can write down. Simple example: sqrt(2). A more complicated example: solutions to the Navier-Stokes equations.