I have started solving problems from the American Mathematical Monthly. While I will definitely continue this, I am not sure how much this will help for math graduate school admissions (assuming I submit my solutions and they are recognized by the Monthly). How significant would multiple solutions to Monthly problems be in admissions? What about having the published solution to a problem? One issue is that I don't know how difficult the problems are presumed to be: all I know is that I personally have solved at least one of them!
It probably depends on the committee members, but I wouldn't expect it to help much.
If you get a really clever solution published, or they publish a problem you submit, that could make a difference. It certainly wouldn't mean as much as a research paper, but it could be viewed as the same sort of thing on a smaller scale.
Otherwise, the potential impact is not large. Solving Monthly problems would be viewed favorably, as evidence of talent and effort, but it's not likely to mean the difference between admission and rejection. (Still, it's worth listing on your CV.)
I agree with the other answer, but just to add another data point:
You might as well list it on your CV, but you shouldn't expect it to help very much.
I did (along with a few others) graduate admissions for the math department at UGA for four years -- and in fact am about to get involved in it again later this week, despite not being on the committee anymore -- and I remember exactly one student who listed it on his CV. It was sort of interesting but not particularly impressive, and if I remember it right we did not admit him in the end. He had a rather distinctive name and, while flipping rapidly through the Monthly problems in the months and years since then, I've noticed that he has submitted several more problems as well. I am starting to wonder what happened to this student...but I wouldn't go so far as to say that I regret the decision we made.
In general I have to say -- and this is a very personal opinion, not a professional one -- that it seems to me that "problems" sections in journals like this are a bit old-fashioned. They do not seem to play a nontrivial role in contemporary mathematical life. I remember having exactly two conversations about Monthly problems:
1) As a first-year graduate student, I did solve a Monthly problem. (This was the one and only Monthly problem that I can remember having thought about for more than five minutes, and I think it is telling that, while I usually have quite a good memory for mathematical minutiae, I remember precisely nothing about the problem.) Rather I remember standing in a mezzanine outside of the mathematics department and telling a fellow student that I had solved a Monthly problem. She politely congratulated me. I asked her whether I should actually submit the solution. She said that she couldn't see why not. I ended up not submitting the solution (thus I can't be completely sure that I correctly solved the problem...and it is telling that I don't care very much!). Okay, that was not my most riveting anecdote.
2) This is slightly more amusing. As a postdoc I remember having lunch with one of my close friends. He told me that he was flipping through the problem section of the Monthly and his eye was caught on a problem that had been proposed...by him. He racked his brains about this and did not succeed in recalling anything about the problem or his submitting it. He did have a friend who was involved with editing the problems section at the time, so he guessed that must have had something to do with it (his friend was very conscientious; it is not plausible that he would have done it as a joke or prank).
I'm not saying that most Monthly problems are easy: on the contrary, I am a relatively experienced, relatively successful research mathematician, and I still sometimes at least pass my eyes over these problems while flipping towards the reviews, and I rarely if ever see one that I think "Oh, surely I could solve that very easily." I'm just not really sure what the point of solving them is: I have plenty of other math problems that I'm trying to solve! It's a little like math contests, only adults can participate too.
To be honest, I think that MathOverflow has significantly overtaken solving Monthly problems as being a minor way for young people to show their talent. There are a small number of undergraduates that I would admit in a heartbeat because I have come to see their brilliance on MO (and to a lesser but still probably sufficient extent, on math.SE). Most of these students are so brilliant that they do not condescend to apply to my graduate program, and I assume that if they did the rest of their application would be so superior that I would not have to spend much time explaining to my colleagues doing the admissions why their performance on math Q&A websites makes me confident that they will be excellent graduate students...but still.
I also think that participation on MO and math.SE is really better than solving problems or doing well in math contests...not better in an absolute sense, but closer to what mathematicians actually do and thus more indicative of academic mathematical career potential (as opposed to raw talent; certain kinds of raw mathematical talent are less useful to a career mathematician than one might think!). Still not that close, of course: the fact that I have a higher reputation on MO than any of several Fields Medalists and other true luminaries that regularly contribute there is ample evidence of that.