The vast majority of textbook exercises are worded essentially in the format:
This assertion is (true/false). Prove this or find a counterexample.
This, of course, is not how mathematics is done. In the "real mathematical world", the questions you pose have unknown answers, and thus there isn't such an obvious structure to solving the problem. While it takes some creativity and frustration to come up with a proof that you know exists (only a Google away!), it seems vastly harder to try and actually solve a problem where the definitive answer is unknown.
Since I know many researchers on this site work on such problems every day, I must ask: how does one go about getting started on a problem like this? Do you try to write a positive proof and see where it fails? search for counterexamples by intuition alone?