1983 USAMO Problems/Problem 5
Problem
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
.
Solution
This problem references the Farey Sequence of order . We wish to show that no open interval of length
contains more than
consecutive terms of the
th Farey sequence.
Lemma: If and
are consecutive terms of the
th Farey sequence and
, then
.
Proof: It suffices to show that, for any reduced fraction with
, we can find an ordered pair of integers
with
such that
. Then
.
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See Also
1983 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
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