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I think the original question was fine. It was very open ended, but there were plenty of nice concrete examples. I think the open-endedness was necessary, so there wasn't much wrong with the original. The revisions have ruined things, but it is possible to rollback.
EDIT: A small change in hypothesis like, changing the space from Zygmund class of functions to Log-lipschitz class or looking for an integer solution in a equation where the solution is easy to find in real or complex case. For instance, Fermat's last theorem is easy to solve on real-line or complex-plane but it engaged mathematicians for three centuries in the integer case. (I have just turned around the example in comment by B. Bischof given below and it is suitable here!)
I thought, and still think, that changing from "asking for real solutions" to "asking for integer solutions" is neither a small change nor a loose hypothesis, but a drastic and dramatic change of the problem.
My misgivings are that the question seemed to invite somewhat superficial answers, and would attract more in the future, getting bumped up to the top each time someone had an idea. (Changing from $L^p$ to $L^1$ is not a small change, it is a big change, and not so much a case of a loose hypothesis as a cavalier approach to precision.)
Perhaps a version of the question that Uday might find acceptable would be ask for examples of threshold values or boundaries between certain classes of objects (e.g. Zygmund class, or critical exponents for percolation, etc.)
I don't see how the question could be made interesting. Take pretty much any universally recognized theorem that's considered "hard" in mathematics. Almost always there's a textually (perhaps not conceptually) small modification to the statement that would turn it into an easy true or false statement. My problem with the thread is it pretty much is asking for an endless list of recognized "hard" statements together with various easily-produced modifications of them.
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