My iMac20 powers off by itself and won't switch back on

Question

I'm sorry this is a bit of a saga but I'm at the end of my tether with so will be really grateful if anyone has any experience of this issue and can give me some help?
The problem started in May when, after an hour or so of use, my iMac 20 suddenly started to turn itself off and wouldn't turn back on again until the following day. This happened around half a dozen times so I took the machine to the local Apple store and was told that it was probably the thermal trip and if I removed the power cord and left it disconnected for a while, it would power up again more quickly. That evening when it cut out, I followed the advice I was given and removed the power cord from the back of the machine but it made no difference and the machine would not restart until the following morning. Back to the Apple store I went and this time they took the machine in to the workshop; a day or two later I was told that they had cleaned out the fans and tweaked some software and had run it successfully on the bench for several hours with no problem. After paying £49, I collected the machine, took it home, ran it for an hour or so and off it went again and as before, refused to turn on. Back again to the Apple store; "Had I reloaded the operating system?" Well no, because I hadn't been told that I needed to. And anyway why would I... surely the thermal cut-out was a hardware issue? The response was that the problem was definitely software-based and nothing to do with overheating or a faulty power supply unit (which was/is my favourite candidate for villain of the piece). So off home I went, backed up my stuff and ran Disc One successfully and started running Disc Two. After about 20 minutes into Disc Two the machine did its usual trick and powered off, and I went looking for a bottle of wine.
Back to the Apple store. The guy controlling the 'guest list' for the Genius Bar said that if I wanted to return the machine to the workshop that day it would be considered a new incident and there would be a further charge for any repair work. However, if I made a fresh appointment (for a couple of days later) then the work would be done as part of the previous 'repair'. Somebody at Apple really needs to look at the way the stores deal with dissatisfied customers.
So, Saturday the 11th of July arrives and I head back to the Genius Bar for my free appointment. I explained to a very helpful guy that this is my fifth visit and I would like to get to the bottom of the problem. He referred to the tech person who dealt with the original workshop repair and she came out to explain that the issue is caused by 'something' lurking in my Library folder and that if I wipe my hard drive and re-install the OS from scratch but without the Library folder, the problem should be solved.
I spent Sunday wiping the hard drive, running the OS discs and then reinstating my iTunes, photos etc and waiting for the machine to turn itself off - but it didn't! I left my external hard drive returning all my files and documents overnight and when I woke up on Monday, everything was still up and running... bliss!
Then, this afternoon, after downing loading a music file in flac format, I found that it wouldn't go into iTunes without the help of Fluke software. I downloaded the Fluke stuff and then as I dragged the flac file into iTunes - boof! The machine went off and won't switch back on again.
So finally... has anyone seen any similar behaviour with their iMac and was it caused by software or hardware? And if it is software that's causing the problem, why can't I switch the machine on again immediately?
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Answer 1

Amsys plc, the company Apple uses for off-site repairs returned my iMac20 today and the accompanying advice note states that the fault was resolved by replacing the power supply unit. I am really grateful to Apple for finally getting to the bottom of the problem, particularly as the repair was done free of charge, although I had already paid £49 for a useless workshop investigation at the Kingston store. The service given by Amsys was exemplary... the collection and return of the machine was carried out at the times promised and whenever I contacted the company by phone, each person I spoke to seemed to have knowledge of the repair without having to refer to the case notes (and none of the 'press 1 for service, 2 for sales...' rubbish).
So once again BDAqua, many thanks for suggesting I call Apple... it's just a shame that the staff of the Kingston store refused to accept what was obvious to me.