Obviously, I'm not sure. I have never seen nor touched your PC, that is the only way I'd be sure.
But usually, when the PC starts up like that, then shuts down within 4-5 seconds, that means it overheating.
Are you sure the Heatsink is properly mounted and making full contact across the CPU?
And if your using the stock cooler, then it would have come with thermal paste. So that may not be it.
But like I said, normally that's a sign of overheating.
Have you tried jumping the on/off pins with a screwdriver (any metal thing really)?
Is that when it started up? When you do it again for the second time, does it take even less time to shut down?
If so, then I'd say it's a overheating problem of some kind..