MathDB
n-good sets

Source: 2024 CTST P14

March 24, 2024
combinatorics

Problem Statement

For a positive integer nn and a subset SS of {1,2,…,n}\{1, 2, \dots, n\}, let SS be "nn-good" if and only if for any xx, y∈Sy\in S (allowed to be same), if x+y≤nx+y\leq n, then x+y∈Sx+y\in S. Let rnr_n be the smallest real number such that for any positive integer m≤nm\leq n, there is always a mm-element "nn-good" set, so that the sum of its elements is not more than m⋅rnm\cdot r_n. Prove that there exists a real number α\alpha such that for any positive integer nn, ∣rn−αn∣≤2024.|r_n-\alpha n|\leq 2024.
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