Computer either turns on, then off, or on with no display and fan sped up

[SentriCast]

When I last shut down my computer, and I turned it back on, the display would not work. After ~20 seconds, the fan started spinning faster than normal and louder than normal. I restarted the computer, and it failed to boot again but turned off on its own and back on again. This has been happening repeatedly.
System specs:
HP Compaq Pro 6300
20 GB RAM, 3 memory sticks
Seagate Barracuda HDD (500 GB)
Intel HD 4000 Graphics
While the computer is off, there is a light on the motherboard that is active, and stays active for a while after the cable to the PSU is unplugged (nested between two of the memory stick slots).

 

[hang-the-9]

Try reseating all connections, including RAM, try another known good power supply, reset the BIOS to defaults (unplug system, pull the CMOS battery out for about 10 seconds).