An extensive list is at MSE, including pointers to dedicated web sites, such as this one.
I might add
The Stanford Mathematics Problem Book by George Polya and Jeremy Kilpatrick.
These 20 sets of intriguing problems test originality and insight rather than routine competence. They involve theorizing and verifying mathematical facts; examining the results of general statements; discovering that highly plausible conjectures can be incorrect; solving sequences of subproblems to reveal theory construction; and recognizing "red herrings," in which obvious relationships among the data prove irrelevant to solutions.