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Part of 1994 Putnam

Problems(2)

Putnam 1994 A5

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7/12/2014
Let (rn)n≥0(r_n)_{n\ge 0} be a sequence of positive real numbers such that lim⁡n→∞rn=0\lim_{n\to \infty} r_n = 0. Let SS be the set of numbers representable as a sum ri1+ri2+⋯+ri1994, r_{i_1} + r_{i_2} +\cdots + r_{i_{1994}} , with i1<i2<⋯<i1994.i_1 < i_2 < \cdots< i_{1994}. Show that every nonempty interval (a,b)(a, b) contains a nonempty subinterval (c,d)(c, d) that does not intersect SS.
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Putnam 1994 B5

Source:

7/13/2014
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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