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IMOR 2017 - Problem 1

Source: 1st International Mathematical Olympic Revenge

July 22, 2017
IMORnumber theory

Problem Statement

Let f(x)f(x) be the distance from xx to the nearest perfect square. For example, f(π)=4−πf(\pi) = 4 - \pi. Let α=3+52\alpha = \frac{3 + \sqrt{5}}{2} and let mm be an integer such that the sequence an=f(m  αn)a_n = f(m \; \alpha^n) is bounded. Prove that either m=k2m=k^2 or m=5k2m = 5k^2 for some integer kk.
Proposed by Rodrigo Sanches Angelo (rsa365), Brazil.
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