Problem

Source: China mathematical olympiad 1990 problem5

Tags: algebra unsolved, algebra, combinatorics, Extremal combinatorics



Given a finite set $X$, let $f$ be a rule such that $f$ maps every even-element-subset $E$ of $X$ (i.e. $E \subseteq X$, $|E|$ is even) into a real number $f(E)$. Suppose that $f$ satisfies the following conditions: (I) there exists an even-element-subset $D$ of $X$ such that $f(D)>1990$; (II) for any two disjoint even-element-subsets $A,B$ of $X$, equation $f(A\cup B)=f(A)+f(B)-1990$ holds. Prove that there exist two subsets $P,Q$ of $X$ satisfying: (1) $P\cap Q=\emptyset$, $P\cup Q=X$; (2) for any non-even-element-subset $S$ of $P$ (i.e. $S\subseteq P$, $|S|$ is odd), we have $f(S)>1990$; (3) for any even-element-subset $T$ of $Q$, we have $f(T)\le 1990$.