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Re: If list S contains nine distinct integers, at least one of w [#permalink]

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24 Jun 2013, 09:43

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Rock750 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.

Let the set of 9 distinct integers in increasing order be\(a_1\) , \(a_2\) ... \(a_9\).

From F.S 1, we know that \(a_1*a_2*....a_4*a_5....a_9\) = \(a_5\) \(\to\) \(a_5(a_1*a_2*...a_4....a_9-1)\) = 0.

Thus, either \(a_5\) = 0 OR \((a_1*a_2*...*a_5....a_9-1)\) = 0, the latter is not possible as no product of 8 distinct integers can ever equal 1.

Thus, the median,\(a_5\) = 0 and not positive. Sufficient.

From F. S 2, for -4,-3,-2,-1,0,1,2,3,4,the median is 0 and a NO for the question stem. Again, for the series -10,-3,-1,0,1,2,3,4,5, the median is 1, which is positive, and hence a YES for the question stem. Thus, Insufficient.

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Re: If list S contains nine distinct integers, at least one of w [#permalink]

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24 Jun 2013, 14:05

mau5 wrote:

Rock750 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.

Let the set of 9 distinct integers be\(a_1\) , \(a_2\) ... \(a_9\).

From F.S 1, we know that \(a_1*a_2*....a_4*a_5....a_9\) = \(a_5\) \(\to\) \(a_5(a_1*a_2*...a_5....a_9-1)\) = 0.

Thus, either \(a_5\) = 0 OR \((a_1*a_2*...*a_4....a_9-1)\) = 0, the latter is not possible as no product of 8 distinct integers can ever equal 1.

Thus, the median,\(a_5\) = 0 and not positive. Sufficient.

From F. S 2, for -4,-3,-2,-1,0,1,2,3,4,the median is 0 and a NO for the question stem. Again, for the series -10,-3,-1,0,1,2,3,4,5, the median is 1, which is positive, and hence a YES for the question stem. Thus, Insufficient.

A.

How did u come up with this : \(a_1*a_2*....a_4*a_5....a_9\) = \(a_5\) \(\to\) \(a_5(a_1*a_2*...a_4....a_9-1)\) = 0. ?
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Re: If list S contains nine distinct integers, at least one of w [#permalink]

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14 Aug 2015, 20:54

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Re: If list S contains nine distinct integers, at least one of w [#permalink]

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02 Apr 2017, 00:26

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Re: If list S contains nine distinct integers, at least one of w [#permalink]

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19 Apr 2017, 18:39

Rock750 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.

Statement 1:

Two possible numbers to start with are 1 and 0. We cannot have a set of 9 numbers where product of all numbers equals the median if we use 1 (e.x --3 x -2 x -1 x 1 x 2 x 3 x4 ); however, any number times 0 is 0 so if we use 0 as a median we can simply take any product of four negative numbers and multiply them by the product of four positive numbers and then by zero to yield a number equal to the mean or vice versa the product of any four negative numbers times zero times the product of any four positive numbers equals 0 a.k.a the median

Sufficient.

Statement 2

We could have a set of numbers where 0 is the median (1,2,3,4,0,-1,-2,-3,-4) the result of which would be the media- however, we could also have --6,-4,-3,-2,-1,2,3,5,7 which would equal -1 which is also the median of that set and a negative number

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.

Statement 1: Product if integers = Median Which is true only if either all terms are 1 or ,-1 or Median is zero

Since integers are distinct so median has to be zero Sufficient

Statement 2: set may be

(-5,-4,-3,-2,1,2,3,4,5) Or (-5,-4,-3,-2,0,2,3,4,5)

Hence median may be 1 or zero or likewise Not sufficient

Answer Option A
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