Please help in solving problem:

We consider the n-point space of elementary outcomes , it is given a convex set of nonnegative functions . The probability of each point is . It is known that , where . Prove that there exist a point such that .

this implies that no more than one point function may be greater than 1, so the conclusion seemed obvious. But it comes to throwing a universal reference point for all functions at once, so this evidence does not pass. What then?