Way back in the stone age when I was an undergraduate (the mid 90's), the internet was a germinal thing and that consisted of not much more than e-mail, ftp and the unix "talk" command (as far as I can remember). HTML and web-pages were still germinal. Google wouldn't have had anything to search, had it existed. Nowadays Google is an incredibly convenient way of finding almost anything -- not just solutions to mathematics problems, but even friends you lost track of 20+ years ago.
My question concerns how Google (and to a lesser extent other technological advances) has changed the landscape for you. Specifically, when you're teaching proofs. More details on what I'm getting at:
A "rite of passage" homework problem in the 2nd year multi-variable calc/analysis course at the University Alberta was the Cantor-Schroeder-Bernstein theorem. In the 3rd year there was the Kuratowski closure/14-set theorem. It's not very useful to ask students to prove such theorems on homework assignments nowadays, since the "pull" of Google is too strong. They easily find proofs of these theorems even if they're not deliberately searching for them. The reason I value these "named" traditional problems is primarily that they are fairly significant problems where a student, after they've completed the problem, can look back and know they've proven (on their own) some kind structural theorem - they know they're not just proving meaningless little lemmas, as the theorems have historical significance. As these kinds of accomplishments accumulate, students observe they've learned to some extent how an area develops and what it takes in terms of contributions of new ideas, dogged deduction, and so on.
I'm curious to what extent you've adapted to this new dynamic. I have certainly noticed students being able to look-up not just named theorems but also relatively simple, arbitrary problems. After all, even if you create a problem that you think is novel, it's rather unlikely that this is the case - sometimes students find your problem on a 3-year-old homework assignment on a course webpage half-way around the planet, even if it's new to you.
As Jim Conant mentioned in the comments, this is a relatively new thing. When I was an undergraduate, going to the library meant a 30-minute walk each way, then the decision process of trying to figure out what textbook to look in, frequently a long search that led me to learning something interesting that I hadn't planned on, and frequently not finding what I set out to find. But type in part of your problem into Google and it brings you to the exact line of all the textbooks in which it appears. It brings up all the home-pages where the problem appears and frequently solutions keys, if not Wikipedia pages on the problem -- I've deleted more than one Wikipedia page devoted to solutions to particular homework problems.
Of course there are direct ways to adapt: asking relatively obscure questions. And there's "denying the problem" - the idea that good students won't (deliberately or accidentally) look up solutions. IMO this underestimates how easy it is to find solutions nowadays. And it underestimates how diligent students have to be in order to succeed in mathematics.
Any insights welcome.