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2015 HMIC #2: another Farey-style problem

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April 26, 2015

Farey sequencesStern-Brocot treesHMICvectorHMMT

Problem Statement

Let

m,n

be positive integers with

m \ge n

. Let

S

be the set of pairs

(a,b)

of relatively prime positive integers such that

a,b \le m

and

a+b > m

.
For each pair

(a,b)\in S

, consider the nonnegative integer solution

(u,v)

to the equation

au - bv = n

chosen with

v \ge 0

minimal, and let

I(a,b)

denote the (open) interval

(v/a, u/b)

.
Prove that

I(a,b) \subseteq (0,1)

for every

(a,b)\in S

, and that any fixed irrational number

\alpha\in(0,1)

lies in

I(a,b)

for exactly

n

distinct pairs

(a,b)\in S

.
Victor Wang, inspired by 2013 ISL N7

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