When I've attended math competition discussions, I've often heard people remark "oh, this is a famous problem" or say that it's similar to one. Most of them I've actually never heard of before. Competition books tend to have a vast list of past problems, and I haven't been able to sift through and tell which ones are supposedly well known and which ones aren't. Contest math isn't especially a priority for me, but it would be nice to at least know of certain problems I should be familiar with.
Could someone give a list of some problems and technique that are part of the "folklore", so to speak? I'm interested in any competition topic, but to narrow the discussion, let's try to emphasize competition problems in calculus/analysis (sequences, series, integrals, etc.)