The fix is relatively easy (i.e. black screen, but fan and led lights on computer working). You don't need to be a technician to fix this. This is basically a plain english repeat of what I learned from a technician's post elsewhere (he deserves all the credit). It absolutely worked. I believe the problem stems from the little cheap piece of foam on the gpu (graphics processor chip) wearing out over time. The heatsink system then ceases to function properly. As a result you get black screen on startup. To fix:
Step 1: go to Ebay (or elswhere) and order some copper shims (they are just little squares of copper that conduct heat efficiently). I put 2.5 mm thickness of shims on the gpu (i.e. one 1.5 mm thick shim+ one 1mm thick shim). I put one 1 mm thick shim on the cpu chip. (that's 3 shims total). Also order some thermal grease.
Step 2: open up the bottom access cover of the computer, then remove the "fan and heatsink" from the computer (ez, 4 screws; ie. the fan and heatsink are permanently attached to each other, unplug the fan's miniplug)
Step 3: you will see 2 chips that were covered by the heatsink. The smaller one is the gpu, the bigger one is the cpu. Discard the little piece of foam that sits on the gpu chip (it's worn out).
Step 4: simply stack the two shims as noted in Step 1 on the gpu, and the one shim as noted in Step 1 on the cpu. That's basically the trick. Easy. BUT: before doing so, make sure you have applied thermal grease to ALL the contact surfaces (between the chips and shims, between the two shims over the gpu, between the shim and heatsink).
Step 5: screw back "fan and heatsink" securely. Excess thermal grease will ooze out between contact surfaces. A little bit oozing out shouldn't cause any problem.
Step 6: keep the fan unplugged for now, keep the computer upside down, turn on power and let the heatsink get hot for 10 minutes. This consolidates your new heat diversion system (with the grease and shims). Turn off computer. Let everything cool for 20 minutes. Plug fan in. Startup computer. It should work.
I found on day 1 startup success was now 50% of the time. By day 2 startup success was now 100% of the time.