CMO 1990 Problem 5

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 $f(T)\le 1990$ .