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Part of 2015 HMIC

Problems(1)

2015 HMIC #2: another Farey-style problem

Source:

4/26/2015
Let m,nm,n be positive integers with m≥nm \ge n. Let SS be the set of pairs (a,b)(a,b) of relatively prime positive integers such that a,b≤ma,b \le m and a+b>ma+b > m.
For each pair (a,b)∈S(a,b)\in S, consider the nonnegative integer solution (u,v)(u,v) to the equation au−bv=nau - bv = n chosen with v≥0v \ge 0 minimal, and let I(a,b)I(a,b) denote the (open) interval (v/a,u/b)(v/a, u/b).
Prove that I(a,b)⊆(0,1)I(a,b) \subseteq (0,1) for every (a,b)∈S(a,b)\in S, and that any fixed irrational number α∈(0,1)\alpha\in(0,1) lies in I(a,b)I(a,b) for exactly nn distinct pairs (a,b)∈S(a,b)\in S.
Victor Wang, inspired by 2013 ISL N7
Farey sequencesStern-Brocot treesHMICvectorHMMT