Problem: (2005 Putnam – B1) Find a nonzero polynomial such that for all reals numbers .
Solution: What’s the relation between and ? We can see that either
for all reals (yes, even negative ones). So it suffices to have a polynomial with zeros at and , such as
Comment: This is about the easiest question you’ll see on a Putnam test. Each question is worth 10 points (120 total) and about half of the people who take it get 0 points.
Practice Problem: (2005 Putnam – A1) Show that every positive integer is a sum of one or more numbers of the form , where and are nonnegative integers and no summand divides another. (For example, .)
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