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I recently got a PSP 1001 from a friend. He thought it worked fine, but saddly it doesn't. I tried turning it on when he gave it to me, didn't turn on. I thought this was normal as he lost the AC adapter (i got one laying around house from my psp 3000) so I plug mine in, and the charging light comes on! I let it charge for an hour or 2, then I come back. It was still charging, but I tried to turn it on anyways. No luck. I then remembered that the battery could be shot so I removed it, then tried to turn it on just from AC power. The power light flashes green for about 1/2 second, then goes out.
I know it's not a brick, as I asked him befor buy if he every tried a custom firmware, hack, or if he turned off during an update. He said he doesn't have wireless internet, so he didn't update, and then asked me what the hell custom firmware was.
Based upon some stuff I'v found (people with same problem), it points to the power fuse, but I want to make sure before I open up the PSP and start soldering stuff.