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Putnam 1994 B5

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July 13, 2014
Putnamfloor functioncollege contests

Problem Statement

For each α∈R\alpha\in \mathbb{R} define fα(x)=⌊αx⌋f_{\alpha}(x)=\lfloor{\alpha x}\rfloor. Let n∈Nn\in \mathbb{N}. Show there exists a real α\alpha such that for 1≤ℓ≤n1\le \ell \le n : fαℓ(n2)=n2−ℓ=fαℓ(n2). f_{\alpha}^{\ell}(n^2)=n^2-\ell=f_{\alpha^{\ell}}(n^2). Here fαℓ(x)=(fα∘fα∘⋯∘fα)(x)f^{\ell}_{\alpha}(x)=(f_{\alpha}\circ f_{\alpha}\circ \cdots \circ f_{\alpha})(x) where the composition is carried out ℓ\ell times.
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