Suppose that a council consists of five members and that decisions in this council are made according to a method based on the positive or negative vote of its members. The method used by this council has the following two properties: $\bullet$ Ascension:If the presumptive final decision is favorable and one of the opposing members changes his/her vote, the final decision will still be favorable. $\bullet$ Symmetry: If all of the members change their vote, the final decision will change too. Prove that the council uses a weighted decision-making method ; that is , nonnegative weights $\omega _1 , \omega _2 , \cdots ,\omega _5$ can be assigned to members of the council such that the final decision is favorable if and only if sum of the weights of those in favor is greater than sum of the weights of the rest. Remark. The statement isn't true at all if you replace $5$ with arbitrary $n$ . In fact , finding a counter example for $n=6$ , was appeared in the same year's Iran MO 2nd round P6