I've already bought all the parts on eBay, and I'm eager to begin working on my old iMac!
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=== Update (02/06/2017) ===
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Ok, I did it too! I replaced the CPU (from X7900 to T9300), the Wireless card (wi-fi + bluetooth) and even the PRAM battery. Everything went fine (apart from a little problem I had installing Sierra, as described at the end of this post).
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My iMac (6Gb of ram, 256GB SSD) is now snappier than before, EVEN now that processes like photoanalysisd are still using a lot of the CPU power (but I don't have so many photos to analyse, so it should stop in a couple of days).
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Thank you very much, nandor690. Only a couple of months ago I was thinking that my iMac was too old and that I had to buy a new one, but thanks to you, now I have a faster machine, up to date, and that can also do all the pretty stuff like Airdrop, Continuity, Handoff. I'm happy! :-D
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P.S.
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I had a problem when I installed Sierra (with the macOS Sierra Patch Tool) on the internal disk, because at the point when I had to reboot once again from the installer drive to use the "macOS Post install" (to add the specific patch for this machine), the system wasn't able to reboot from the USB stick and just showed the prohibitory symbol (the circle with a slash that means that it's trying to start from the wrong version of macOS). Pressing C at startup didn't do anything, but pressing X worked: I was able to chose the USB stick as the startup volume and after that I applied the patch and now everything is fine.