Rug Backing on Bath Floor


I currently need to get a bath painted and freshened that had the bath rugs stuck to the floor. The backing has come off crumbly and remained on the linoleum (sp?). I might believe the folks who have used it for the last year when they say that the toilet tank kept sweating and getting the rugs wet as there is a grayed area under the tank and along the sides up to the rug... but there is absolutely *no* smell that would indicate it is anything but water.
It puzzles me, actually, that the rest of the room seems very well kept but they do not pick up the runner, though... maybe they used something to clean that made it stick and don't want to take the blame and it's easier to say it's the toidy and not their fault? I do know that this tank had some sweating a few summers back but my understanding is that is was not critical. Maybe they leave the regular mat while they are still dripping wet and leave the room a soggy mess?
I guess I'm saying I feel sure it's not whiz and likely water but leaving the door open that it may be a cleaner or something. At any rate, here are my questions:
1. How can I remove the backing without destroying the linoleum under it, not knowing what stuck it there? That flooring is only about 10 years old and needs to go (and should have gone) another 10.
2. If I can save that strip... how can I get the gray out of the flooring under the tank?
3. Outside of plumbing in a hot line to the cold supply, are there any ways to stop the sweat? (This is well water from a deep well if that affects the temp any).
Thanks
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rdoc wrote:

Probably don't have god news for you but you could try mineral spirits to remove the stuck backing. Should not harm the flooring but might break the bond between it and the pad. Make sure to have good ventilation while doing this. The gray under the tank is probably mold/mildew transferring through the back of the vinyl from being constantly damp. Only way to fix is to replace flooring after stopping the sweating. No good ideas on how to stop sweating.
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i've had rubber backed mats get sticky as they age just through normal use. they started sticking to things so they got tossed, so i have no idea how to remove the residue. try mild things first and work up, as anything that may also dissolved the rubber will also probably attack the linoleum.
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1 - Lighter Fluid (not the BBQ type, the type for "flint & wick" lighters) and WD40 are both pretty good for dissolving adhesives without harming most surfaces. Try it in a inconspicuous spot first.
2 - The grey floor? I'm thinking you might be out of luck if it's mold/ mildew. You might try a little bleach in a small area and see what happens.
3 - A tempering valve would work the best. Here's a neat one: http://searchwarp.com/swa413729.htm
Other than that, you could try insulating the tank:
http://www.ehow.com/how_117384_stop-toilet-tank.html
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Your floor is probably vinyl. Never use rubber backed rugs on vinyl. A chemical reaction results that usually yellows the vinyl.
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