MathDB

4

Part of 2023 South East Mathematical Olympiad

Problems(2)

Inequality of decimal part

Source: 2023 China Southeast MO Grade11 Day1 Problem 4

7/30/2023
Find the largest real number cc, such that for any integer s>1s>1, and positive integers m,nm, n coprime to ss, we have∑j=1s−1{jms}(1−{jms}){jns}(1−{jns})≥cs \sum_{j=1}^{s-1} \{ \frac{jm}{s} \}(1 - \{ \frac{jm}{s} \})\{ \frac{jn}{s} \}(1 - \{ \frac{jn}{s} \}) \ge cs where {x}=x−⌊x⌋\{ x \} = x - \lfloor x \rfloor .
inequalitiesnumber theory
Pingsheng Number

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

7/30/2023
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.
combinatorics