The fact that you are getting three beeps without memory, means that the POST has bassed teh basic CPU motherboard check, so in theory they are fine. But if you have checked all other components...then it has to be the board or CPU, have you checked the monitor and cables? PSU? also do you have a graphics card, have you checked it?
The monitor is fine (works on all other systems). The video card is fine (checked with multiple, known good cards). Changing to a new, good PSU makes no difference. All components have been verified, except for the memory and CPU (I don't have spares for either).
I am just trying to discern if the motherboard, CPU, or memory is the culprit.