The "suddenly won't turn on" thing happened 2 times before throughout the years (I have it since late 2009...), so every time service would't find anything better than replacing the motherboard - which is f***ing expensive.
So I tried this time what @rudenator suggested, and YES !
In detail, all 4 memory slots were taken. Since some time, I had a doubt about it, as system diagnosis showed up only two ram in place. But so far the iMac worked fine, and I omitted to check physically if I was crazy or if the 2 ram had disappeared somehow.
In reality, the two "missing" memory bars seemed to be broken, and yes, they were still in place, they simply didn't show up. These two were in the BOTTOM emplacement. I took them out, as I guessed they were defective since a while.
Next I tried the two remaining individually in the TOP emplacement (relative to the screen which is top). This, because after Apple a single ram can work in the TOP place, but not in the bottom, at least for this model.
THE iMAC REBOOTED with one of them !
Next, I checked them all individually, and in fact even one of the former bottom ram still worked. YES.
I explain myself what happened, by the fact that if you pair a working with a non- working ram diagnosis thinks they are broken both, so if they are in bottom the machine don't shows them, and in top the machine won't boot.
I dont know, but maybe this trying saved me from a new motherboard...(which I wouldn't have paid, as the iMac is 8 years old now), but at least it saved me from buying new ram.