How to judge a Ph.D. applicant that arrives "out of thin air" I just received an email from a student from a distant country, asking me to accept him as a Ph.D. student. I do not know this person, his advisors, his references or even the university he studied in. 
His master's thesis looks good at first glance, but I cannot be sure since it is not in my field. He still wants to do Ph.D. research in my field. 
I asked him to send me some refereed publications, but he replied that he has none yet, and then asked me to give him some work by which his application can be assessed. 

I am very unsure about this - I am afraid to get stuck with a bad student that takes too much time to supervise. On the other hand, I do not want to block him entirely. So my plan is to give him an open problem that I and some colleagues tried to solve in the past but failed. If he fails and gives up, then he will have to look for another advisor; 
if he does manage to solve the open problem, then I will be more than happy to be his Ph.D. advisor, and his solution will be a part of his thesis.

My reservation about this plan is that it might be unethical/unprofessional to give him such a difficult problem to start with. But, he asked for it, and of course he is not obliged to take it if he does not want to.

Is this plan reasonable? Is there a better plan?

UPDATE: based on the answers, I changed the plan as follows: (a) referred the student to register through my university's system; (b) asked the student what papers he has read on my research field, and what ideas he has for extending them; (c) Based on the student's answer, I decided on a research problem that was not solved yet in a publication, but for which I already thought of a solution, so that I can guide the student in case he gets stuck; (d) scheduled a video call. My plan is to start working together remotely in parallel to the registration process, and see how it goes.

Thanks a lot for all the insights and ideas!
I would tell them what you expect from a PhD student: prove theorems/run empirical tests/conduct human studies or all the above. If their profile is a clear mismatch, I wouldn’t bother. 

I would next ask them to review two papers of their choice out of your recent publications. One page review each. The review should include at least one idea for future directions.  It tells you whether they’re serious, what they want to work on, and whether they can reason about ideas in your field. It also guarantees that the problem is at least somewhat tractable (you’ve solved it after all).

If you’re happy with the write ups, schedule a video interview. You both are committing to 4+ years together, best to make sure you can discuss things together.

Good luck!