1972 USAMO Problems/Problem 4
Let denote a non-negative rational number. Determine a fixed set of integers , such that for every choice of ,
Note that when approaches , must also approach for the given inequality to hold. Therefore
which happens if and only if
We cross multiply to get . It's not hard to show that, since , , , , , and are integers, then , , and .
Note, however, that this is a necessary but insufficient condition. For example, we must also have to ensure the function does not have any vertical asymptotes (which would violate the desired property). A simple search shows that , , and works.
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