Yeah, once the computer starts doing that there's obviously problems afoot. :p Most likely its the power supply, as thats exactly what prior computers I've serviced have done when said supply was blown. Next it could be the motherboard. Just make sure the big cable plugged into the motherboard is secure there. If its even a little bit loose, it won't start the computer.

You might wanna see about snatching up a spare power supply from somewhere and checking that out too.