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.