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The numbers $1,$ $2,$ $\dots,$ $10$ are to be entered into the 10 boxes shown below, so that each number is used exactly once: \[P = (\square + \square + \square + \square + \square)(\square + \square + \square + \square + \square).\]What is the maximum value of $P$

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 Jul 1, 2020
 #1
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Make the numerator as large as possible :  10  9 8 7 6

and the denominator as small as possibe      1 2 3 4 5

 

 

Sorry....the lates just converted to soemthing readable.....

   to maximize the product, make the numberes as close together as possible

 

1 - 10 summ to 55    so     27 x 28    

   10   7   5   3   2   *     9   8   6   4   1       I suppose...   anyone else?

 Jul 1, 2020
edited by ElectricPavlov  Jul 1, 2020
edited by ElectricPavlov  Jul 1, 2020
 #2
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I think the maximum value is -726.25.

Lets say that this is the equation. \(P=(\square+ \square+ \square+ \square+ \square) (\square+ \square+ \square+ \square+ \square)\)

Now saying that one of those factors is \(x\)

Now when you add up 1+2+3+4+5+6+7+7+9+10, you will get it equal to 55.

So we can say that 55-\(x\) is equal to the other factor. 

Now when you multiply those out you get: \(x(55-x)=-x^2+55x\)

Then you get it into vertex form, and when you do so, you will get \(-1(x-27.5)^2-27.5^2\)

We know that 27.5 squared is then the maximum value, so computing that, we get that the maximum is -756.25. 

 Jul 1, 2020

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