2011 iMac not starting, then starting, and then not starting again?

Question

Hi,
I've asked this question at another forum, but I haven't gotten any feedback, so I thought I'd try here.
I have a 2011 iMac which I'm not really happy with at all, to be honest.
The most severe problem is that it refuses to turn on sometimes. It has actually only happened two times so far, with a few weeks in between.
I talked to Apple for over two hours yesterday. After a lot of troubleshooting, PRAM resets, (obviously error-free) diagnostics, etc, they still had absolutely no clue what the problem could be.
The first time it happened, the iMac had been disconnected from the power outlet for a week. Imagine trying to power on an iMac with the power chord unplugged. That's how it was.
After a lot of fiddling around, inserting and reinserting the power cable, and pressing the power button 40-50 times, it finally booted up. I then turned it off again, and it wouldn't power on once again. Then, it suddenly started working as normal, turning on every time.
I happened again yesterday. The exact same issue, despite the fact that it has been plugged into the power all the time. After my long support call with Apple, it was working as usual again, powering on as normal every time. There was this one time when I held in the power button for like 5 seconds, until I heard the fans running. I then released the button, and the fans stopped. Then it powered on when I tried again.
Rebooting always works, going to sleep and waking up always works, and it never powers off by itself once it's on. Trying a different power outlet makes no difference.
I really just want another iMac, or my money back, but I understand that this is the absolute last thing Apple will do.
All I kept hearing from the support reps were "call a service place". But what good is that going to do?
What if it dies again and I really can't get it to work no matter how many times I press the button? It seems to be ALL about pressing the button btw, doing anything else has any impact whatsoever.
If a technician tries to start it, it may start on the first attempt. Or the third. Or the twentieth. I have no idea, since there's no pattern, and nothing to go by.
The only real clue is that it's imposible to NOT get it to start after it's been running for around 15 minutes.
So, what should I do? Should I try to get it fixed/replaced/sold? Should I reinstall Lion (the guys at Apple actually told me to try that)? Should I just accept it and never shut it off (only sleeping/waking and rebooting)?
I would love to hear your thoughts.

Answer 1

Sorry, I forgot to mention - I was instructed to do that. Don't know if it helped or not, only time will tell.
I'm 100% certain that the problem will occur again. It could be in a few weeks, or months, or even years.