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The numbers 1-10 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? What is the minimum value of P?

 

Please just give me a straight answer

 Oct 15, 2020
 #1
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By AM-GM, there is no minimum value, and the maximum value is 756.25.

 
 Oct 15, 2020
 #2
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I need steps and details.

 
Guest Oct 15, 2020
 #3
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(1+2+3+4+5)(6+7+8+9+10) = 15*40 = 600  smallest

 

Now if the numbers that are multiplied are as close as possible that will give the biggest product.

1+2+3+4+5+6+7+8+9+10=55 = 54+1

half of 54 = 27  

So one will be 27 and the other 28

I also notice that 

1+10=11  first bracket

2+9=11    second bracket

3+8=11    first

4+7=11    second

5+6=11

 

you can finish it.

 
 Oct 19, 2020
edited by Melody  Oct 19, 2020

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