What does it take to get a perfect score on the Putnam? There are many guides on how to prepare for the Putnam, and so many people participate in the Putnam. However scoring perfect scores or anything reasonably close to a perfect score on the Putnam is so rare. It seems to me that the present guides make the basic assumption that answering everything on the Putnam is impossible, thus people who follow those guides would end up scoring 'normal' scores, just like everyone else. So, what specific steps should one take in order to get a perfect score on the Putnam? Is it important to master all of undergraduate mathematics and some graduate mathematics? Or is it really just a matter of practicing endless practice problems, just like all the other participants who don't get anything close to a perfect score?

There is no useful advice, because the grading rubric on the Putnam makes it difficult to gain complete marks for an essentially correct solution.   It is not all that unrealistic for some of the top competitors to have solved the important parts of every question, but to gain maximum points they need also to have checked every boundary case or small $n$, and not made some minor errors.  You would need a combination of extreme preparation, tremendous speed, checking and re-checking of details, and still depend on luck to deliver up an exam "easy" enough (in the year to year variation) to be able to really solve everything.  The last is probably the limiting factor, as there are enough multiple IMO gold winners taking the test now, every year, and an effectively unlimited amount of preparation material.
