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China mathematical olympiad 1990 problem5

Source: China mathematical olympiad 1990 problem5

October 20, 2013
algebra unsolvedalgebracombinatoricsExtremal combinatorics

Problem Statement

Given a finite set XX, let ff be a rule such that ff maps every even-element-subset EE of XX (i.e. E⊆XE \subseteq X, ∣E∣|E| is even) into a real number f(E)f(E). Suppose that ff satisfies the following conditions: (I) there exists an even-element-subset DD of XX such that f(D)>1990f(D)>1990; (II) for any two disjoint even-element-subsets A,BA,B of XX, equation f(A∪B)=f(A)+f(B)−1990f(A\cup B)=f(A)+f(B)-1990 holds. Prove that there exist two subsets P,QP,Q of XX satisfying: (1) P∩Q=∅P\cap Q=\emptyset, P∪Q=XP\cup Q=X; (2) for any non-even-element-subset SS of PP (i.e. S⊆PS\subseteq P, ∣S∣|S| is odd), we have f(S)>1990f(S)>1990; (3) for any even-element-subset TT of QQ, we have f(T)≤1990f(T)\le 1990.
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