Thanks finncapt.
As for Tosh McCaber ... if you're not comfortable fiddling around in a methodical manner with hardware, I suggest you find yourself a better repair shop as I'm not too sure of capabilities of your present one based on your description.
If you are happy to fiddle around, follow the suggestions from finncapt.... take removable stuff out, put stuff in.... one by one... see where it breaks. If swapping ram, graphics cards etc doesn't help... then you're probably looking at motherboard and/or CPU as finncapt said.... then if you want to persue the self-help route its merely a case of whether or not you are competent with an oscilloscope and a soldering iron.
I would normally suggest booting off a hardware test CD, but if your computer is really turning itself off at the earliest opportunity then obviously that's a no-go.
Good luck !