1983 USAMO Problems/Problem 5
Consider an open interval of length on the real number line, where is a positive integer. Prove that the number of irreducible fractions , with , contained in the given interval is at most .
Let be an open interval of length and the set of fractions with , and .
Assume that . If is such that , and is such that , then Therefore . This means that is the only fraction in with denominator or multiple of .
Therefore, from each of the pairs in at most one element from each can be a denominator of a fraction in .
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