It sounds like a great idea to publicize the problem among young people who may have a lot of time on their hands, not a lot of other research ideas, and a desire to demonstrate their abilities through something other than timed contests.
One awkward possibility would be getting stuck in a situation in which someone can't convince you that their solution is correct, but you can't convince them it is incomplete or wrong. Of course this could happen with anyone, but it's particularly likely to occur with someone who is not an experienced researcher. If you offer a bounty, you should be clear about the conditions. (For example, that it all comes down to your judgment, and that you don't commit to spend unlimited amounts of time trying to understand a proposed solution.)
Coauthorship is certainly mandatory if you publish a paper that depends on this solution (unless of course the students publish their solution separately and you cite it). However, you may run into tricky situations in which it's not enough just to offer coauthorship for the first solution. For example, if several people solve the problem independently and around the same time, then they should all be treated evenhandedly. Or what if someone proposes a solution you don't believe, someone else proposes a clearer solution and you offer the second person coauthorship, and then the first person convinces you their solution was correct after all? Of course these problems are not hard to solve (in the last case, you can't retract the authorship of the second person, but you should make the first an author too if you still can). But the important thing to keep in mind is that you may need to make sensible decisions that go a little beyond the "first solver is a coauthor" rule.
A final observation is that if you succeed in attracting students to this problem, you will likely get lots of questions and comments from them along the way. If that appeals to you, then it's an advantage of this approach. On the other hand, if your goal is to get a solution with a minimum amount of time and attention on your part, then asking beginners might not be fruitful.
