Can all problems really be solved? I believe so. I want to believe so.
But solving a problem might include research, innovating, constructing. And it can take time. So there can be long delay between where the problem solving starts to when the solution is ready.
(And after that we need to consider whether to implement solution)
Is cancer a problem? Lets say it is. Can we solve it? I think so. And the solving problem has started. There are many researching programs working to solve the "cancer problem". As is with any problem we really want to solve.
When solving we need skills, methods and knowledge. But when solving and researching we will also generate new methods, skills and knowledge.
Any problem can be solved. Might take some time though. And might need researching and innovating. But isn't that what problem solving is all about?