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Pingsheng Number

Source: 2023 China Southeast MO Grade 10 Day 1 Problem 4

July 30, 2023
combinatorics

Problem Statement

Given an integer n≥2n\geq 2. Call a positive integer T{T} Pingsheng Number, if there exists pairwise different non empty subsets A1,A2,⋯ ,AmA_1,A_2,\cdots ,A_m (m≥3)(m\geq 3) of set S={1,2,⋯ ,n},S=\{1,2,\cdots ,n\}, satisfying T=∑i=1m∣Ai∣,T=\sum\limits_{i=1}^m|A_i|, and for ∀p,q,r∈{1,2,⋯ ,m},p≠q,q≠r,r≠p,\forall p,q,r\in\{1,2,\cdots ,m\},p\neq q,q\neq r,r\neq p, we have Ap∩(Aq△Ar)=∅A_p\cap(A_q\triangle A_r)=\varnothing or Ap⊆(Aq△Ar).A_p\subseteq (A_q\triangle A_r). Find the max Pingsheng Number.
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