Given a finite set X, let f be a rule such that f maps every even-element-subset E of X (i.e. E⊆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∪B)=f(A)+f(B)−1990 holds.
Prove that there exist two subsets P,Q of X satisfying:
(1) P∩Q=∅, P∪Q=X;
(2) for any non-even-element-subset S of P (i.e. S⊆P, ∣S∣ is odd), we have f(S)>1990;
(3) for any even-element-subset T of Q, we have f(T)≤1990. algebra unsolvedalgebracombinatoricsExtremal combinatorics