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I just a bought a LG 55" tv real cheap as it had this same problem switching off after 10ish minutes. Took back cover off and noticed the heatsink for the graphics card part was quite hot. Put a fan against it and bingo problem gone. Terrible thermal design in the tv. I cut a hole then glued a little USB silent computer fan to the back cover (I butchered a laptop cooler stand for the fan). When the tv is turned on it powers up the fan, when tv is off, its off. So stoked!
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