Well-used iMac suddenly turning off

Question

My not-so-new 20" iMac is turning off on me. I have used this computer, many hours a day, for several years now. For the past 20 months, I've lived in an apartment with old outlets that haven't been replaced, and don't have great contact with the electrical plugs. I've been using a power strip, and the computer only turned off when someone knocked the plug out of the outlet or bumped "reset" on the power strip.
Starting yesterday, my computer started turning itself off (apparent power failure) without anything getting bumped. I'm not seeing any changes in the lights around, me, either, so I don't think that power fluctuations are the proximate cause. I've fiddled with all of the pulgs, unplugged and replugged them, and the problem is getting worse: my computer is shutting off more often. I'm using my spouse's laptop to ask this question, because it's gotten to the point where my comptuer really isn't useable. I've gotten great advice from these forums before, and I'm looking forward to getting good advice this time.
Could a power regulating power strip solve my problems? I'm guessing no, but should I do this in the future, to prevent damage to my computer? More importantly, what is likely wrong with my computer, and about how much would it cost for me to replace the part, or get it fixed at the Apple store? I did not purchase any extended warranties, so this computer is completely out of warranty.

Answer 1

It's been a month since I first posted, and I've got more clues and details now. After I posted my question I realized that my computer is and has been on a surge protector (and the protector light is light), so on later reflection I don't think the problem is the steadines of the power.
den.thed's suggestion seemed to work at first, although I think the longest my computer has gone without shutting down in the last month was about a week, and at this point resetting the SMC doesn't seem to be helping (not that I've turned the computer back on without unplugging it, so I haven't tested this precisely).
I leave my computer on at night (sometimes I put it to sleep, sometimes it puts itself to sleep, but sometimes something happens and the computer doesn't put itself to sleep, or my cats step on the keyboard and wake it up), and it has not shut down at night. It only shuts down when I'm sitting in front of the computer, using it. This made me think I might be having a software problem, so I tried avoiding using specific programs, but I can't isolate a specific program that is always open or always being used when my computer shuts itself down. I think my computer has only shut down while I'm typing, but I'm not certain on that one. I'm using a logitech wireless keyboard (because I spilled water on and destroyed the original keyboard), and (because I only have the one keyboard at the moment) I haven't tried using another keyboard.
Last night I had the realization that the times my computer was going longer without shutting down was when I'd unplugged my backup HD running Time Machine (because of the limited number of USB connections on the computer).  I am not certain if my computer has ever shut itself down with the external HD attached. I have had successful saves to Time Machine on my external HD while dealing with this problem.
But these two clues: maybe shutting down only when external HD is plugged in, and definitely only shutting down when when I'm sitting at the computer, actively using it, don't seem to me to point to the same problem.