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Piraldo's soccer matches

Source: Brazilian Math Olympiad 2006, Problem 6

October 30, 2006
algebra unsolvedgeometric sequencefloor function

Problem Statement

Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of mm goals to nn goals, m≥nm\geq n, is tough when m≤f(n)m\leq f(n), where f(n)f(n) is defined by f(0)=0f(0) = 0 and, for n≥1n \geq 1, f(n)=2n−f(r)+rf(n) = 2n-f(r)+r, where rr is the largest integer such that r<nr < n and f(r)≤nf(r) \leq n. Let ϕ=1+52\phi ={1+\sqrt 5\over 2}. Prove that a match with score of mm goals to nn, m≥nm\geq n, is tough if m≤ϕnm\leq \phi n and is not tough if m≥ϕn+1m \geq \phi n+1.
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