Years ago I had the exact same issue on my PC. Turned out my VGA plug was coming loose so the pins on one side of the connector wasn't connecting (this problem took me quite a while to solve, too
). You say you've tried other cables so I'm wondering if it's the port on the Amiga which has some dead pins for some reason? Maybe something needs resoldering? Just guessing here though.
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I am also guessing a dry solder joint on the RGB port, but it might also be other things like the custom chipset. Problem with the A600 is everything is surface mounted so is hard to replace or fix anything. Although replacement motherboards can be found cheap if it turns out to be the problem.
One thing to check is to connect the A600 to a TV using the yellow composite video output on the back. If the picture is still blue then it must be an issue with the hardware itself. If the picture looks in colour and OK then it is probably a problem with the RGB port.
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