Can I make an exam question for graduate & undergraduate students to find an idea in making a breakthrough in my research?

I find the whole idea as proposed in the first point quite underhanded. Just announce that you are looking for ideas on how to solve a specific problem and that you'd be happy to provide support for developing it to a thesis and/or publication to any students who got a promising idea. That way you create a win-win situation and avoid all potential ethical issues.

Even ignoring the elephant in the room (that I am unsure why you expect your students to have a reasonable shot to very quickly solve an issue that has apparently been stumping you for some time), this sounds like a pretty bad idea:

• It does not sound ethical. Fundamentally, in an exam, you are expected to know the answer to the questions you ask. How are you going to evaluate different proposals? Does a student who writes a simplistic answer that won't work get more points than a student who recognises how difficult the issue is and consequently is unable to come up with a comprehensive solution (and writes nothing at all)?
• It probably won't work. An exam is not a brainstorming exercise. Your students are under time pressure, and they will assume that there is a reasonably simple solution to the problem. They are not gonna throw crazy ideas that might just work at you, but instead they are going to waste a lot of precious exam time trying to find the "obvious" solution that you yourself have not yet found. An exam is not the right frame for creative problem solving.

I think you can do it in a homework exercise, but only as a bonus question that is not part of the ordinarily graded questions. Adding it to an exam is not fair, for reasons mentioned in other answers. But rather than using it to find a breakthrough in your own research, use it to find hidden geniuses among the students. Make it very clear that those questions are completely optional and harder than the main homework questions; you may or may not state that they are actually open problems.

There are historical examples of students who solved open problems in homework exercises. For example, George Dantzig:

An event in Dantzig's life became the origin of a famous story in 1939 while he was a graduate student at UC Berkeley. Near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard. When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue.

Six weeks later, Dantzig received a visit from an excited professor Neyman, who was eager to tell him that the homework problems he had solved were two of the most famous unsolved problems in statistics.

Of course, when you do against all odds find a hidden genius this way, you can offer to supervise him or her in writing a publication, which should land you co-authorship on the paper.