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combinatorics problem

Source: 2023 China TST P10

March 19, 2023
combinatoricsChina TST

Problem Statement

The set of nonempty integers AA is said to be "elegant" if it is for any a∈A,a\in A, 1≤k≤2023,1\leq k\leq 2023, ∣{b∈A:⌊b3k⌋=⌊a3k⌋}∣=2k.\left| \left\{ b\in A:\left\lfloor\frac b{3^k}\right\rfloor =\left\lfloor\frac a{3^k}\right\rfloor\right\}\right| =2^k. Prove that if the intersection of the integer set SS and any "elegant" set is not empty,, then SS contains an "elegant" set.
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