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As for www.mysite.com and mysite.com, some people link to one, and some link to the other. Google recognizes them as duplicates, and only displays one at a time. But which one it displays is basically random.
What you can do is use a 301 response to redirect all mysite.com hits to the same page at www.mysite.com, or vice versa, then Google will figure out which one you prefer. See here [webmasterworld.com].