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2017 China TSTST 1 Day 1 Problem 3

Source: 2017 China TSTST 1 Day 1 Problem 3

March 7, 2017
set theorySubsetscombinatoricsinequalitiesalgebra

Problem Statement

Suppose S={1,2,3,...,2017}S=\{1,2,3,...,2017\},for every subset AA of SS,define a real number f(A)0f(A)\geq 0 such that: (1)(1) For any A,BSA,B\subset S,f(AB)+f(AB)f(A)+f(B)f(A\cup B)+f(A\cap B)\leq f(A)+f(B); (2)(2) For any ABSA\subset B\subset S, f(A)f(B)f(A)\leq f(B); (3)(3) For any k,jSk,j\in S,f({1,2,,k+1})f({1,2,,k}{j});f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\}); (4)(4) For the empty set \varnothing, f()=0f(\varnothing)=0. Confirm that for any three-element subset TT of SS,the inequality f(T)2719f({1,2,3})f(T)\leq \frac{27}{19}f(\{1,2,3\}) holds.