MathDB
2021 Team #8

Source:

June 27, 2021
combinatoricsfloor functionnumber theory

Problem Statement

For each positive real number α\alpha, define ⌊αN⌋:={⌊αm⌋  ∣  m∈N}.\lfloor \alpha \mathbb{N}\rfloor :=\{\lfloor \alpha m \rfloor\; |\; m\in \mathbb{N}\}. Let nn be a positive integer. A set S⊆{1,2,…,n}S\subseteq \{1,2,\ldots,n\} has the property that: for each real β>0\beta >0, if  S⊆⌊βN⌋,then  {1,2,…,n}⊆⌊βN⌋. \text{if}\; S\subseteq \lfloor \beta \mathbb{N} \rfloor, \text{then}\; \{1,2,\ldots,n\} \subseteq \lfloor \beta\mathbb{N}\rfloor. Determine, with proof, the smallest positive size of SS.
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