Do graduate schools care about published solutions of contest problems in Mathematical magazines? The math club at my university spends its time solving problems from mathematical magazines (e.g. Mathematics Magazine). Let's assume I, as an undergraduate student, submit a solution to a challenge problem from such a magazine, and my solution is published. Is there any advantage to having one or more published solutions like this when applying to graduate school?

For reference, assume this is graduate school in math.
Sure, that would certainly be viewed favorably.  How favorably would depend on the rest of the application and the institution (though I can't think of any situation in which it would play a decisive role), but if the question is having done it whether to list it, then the answer is a clear yes.

If on the other hand you haven't done the problem-solving yet but are wondering whether you should engage in this activity for the value it has in your future grad school application: I don't recommend that, but I can think of plenty worse ways to spend your time.