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If list S contains nine distinct integers, at least one of w

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If list S contains nine distinct integers, at least one of w

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If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.
(2) The sum of all nine integers in list S is equal to the median of list S.

What's the best way to determine which statement is sufficient? Any experts can help?

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ardz24 wrote:
If list S contains nine distinct integers, at least one of which is negative, is the median of the integers in list S positive?

(1) The product of the nine integers in list S is equal to the median of list S.
(2) The sum of all nine integers in list S is equal to the median of list S.

What's the best way to determine which statement is sufficient? Any experts can help?
Statement 1: if the nine integers are distinct, this can only be true if the median = 0.
(To see this algebraically, imagine the nine elements are the variables a - i, in ascending order. So the median is e. Now we'd know that a * b * c * d * e * f * g * h * i = e. If e weren't 0, then we could divide both sides by e to get a * b * c * d * f * g * h * I = 1. But we can't multiply 8 different integers and get a value of 1. So e MUST be 0.)
This statement alone is sufficient - the median is not positive, giving us a definitive NO as the answer to our question.

Statement 2. The easiest way to construct this set is to have the largest 4 terms and the smallest 4 terms sum to 0. If that's the case, the sum of all the terms will have to be equal to the median.
Case 1 {-6,-5,-4,-3, 0, 3, 4, 5, 6} This gives us a NO, the median is not positive.
Case 2 {-6,-5,-4,-3, 1, 3, 4, 5, 6} This gives us a YES, the median is positive.
Because we can get a NO or a YES, this statement alone is not sufficient to answer the question.

The answer is A

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