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Sequence contains a perfect square

Source: Putnam 1983

June 14, 2017
number theoryPutnamfunctionSequence

Problem Statement

Problem. Let f:R0+→R0+f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+ be a function defined as f(n)=n+⌊n⌋ ∀ n∈R0+.f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+. Prove that for any positive integer m,m, the sequence m,f(m),f(f(m)),f(f(f(m))),…m,f(m),f(f(m)),f(f(f(m))),\ldots contains a perfect square.
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