My LG TV also flashes once, then screen goes black, though sound still works. But I have been able to get it to work by following these steps:
1) Turn off any input into the TV. For me this means turning off the Sky TV box plugged into AV1. (TV input selection is on AV1 when the TV powers up)
2) Turn on the TV. If it flashes and goes dark, then turn it off again, count to 5, then turn it on again. Repeat (for me its usually 2 times) until a flashing box 'No Signal' appears.
3) Leave the TV alone for 4-5 minutes with the flashing box 'No Signal'. This warms up something in the TV so the picture will work.
4) Turn the input back on - for me this is the Sky box. Picture should work now. If it goes dark, you probably didn't wait long enough in step 3. Start again at Step1.
We have now gotten used to this silly routine and have continued using the TV for about 6mos so far since the screen started going black. We have to do it every time, but it works. Hope it helps for you too,