Question 1164322: The numbers 1,2,3,4,5,6,7,8,9,10 are to be entered into the 10 boxes shown below, so that each number is used exactly once:
P = (blank + blank + blank + blank+ blank)(blank + blank + blank + blank + blank)
What is the maximum value of P? What is the minimum value of P?
Blank stands for the empty boxes that were in the original problem.
Can you solve with the AM GM inequality? I don't really get it.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The sum of the numbers from 1 to 5 is 15, and the sum of the numbers from 6 to 10 is 40. If you average the integers from 1 to 10, you get 27.5, so the closest you can get to having the average on either side is 28 and 27 one way or the other.
The product of the extreme values, 15 and 40 is 600, the product of the mean values 27 and 28 is 756.
The total possibilities for the sums on either side are  and  where  is an integer in the range 0 to 25. That leads you to an integer-valued function \ =\ (15\,+\,x)(40\,-\,x)\ =\ 600\ +\ 25x\ -\ x^2) . The vertex of this concave down parabola is at (12.5,756.25) which is not an integer, but the closest integer points are (12,756) and (13,756), which makes the sums on either side 15 + 12 = 27 or 15 + 13 = 28.
There are two ways to do the extreme values, 1 through 5 in the first set and 6 through 10 in the second set, or vice versa. There are at least 8 different ways to make 28 with 5 of the numbers from 1 to 10, so take your pick.
John
My calculator said it, I believe it, that settles it
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