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First off, optimization problems are sort-of my weakness in contest math. Any ideas on resources I can use to get better?

 

I was having fun playing around with cool problems during spring break, when I ran into the following problem and didn't really know where to start...

 

Given that:

2(10a + 13b + 14c + 15d) − (a^2 + b^2 + c^2 + d^2)/3 = 2020,

what is the maximum possible value of a + b + c + d.

 

I was wondering if someone could please help me figure out this problem? Thanks in advance!

 Mar 20, 2020
 #1
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If the first 2 is meant to be multiplied by the first bracketed term and the division by 3 applies only to the last bracketed term such as I have written it here n=2*((10*a + 13*b + 14*c + 15*d)) - ((a^2 + b^2 +c^2 + d^2)) / 3, then there are infinite number of solutions such as these few ones; You have to clarify your problem by using accurate brackets:

 

a   b   c    d

29 38 40 33
31 38 40 33
29 40 40 33
31 40 40 33
29 37 41 33
 

 Mar 20, 2020
 #2
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Hello, thank you, yes, the division by 3 applies only to the last term. But the problem is asking for the MAXIMUM possible value of a + b + c + d.

Guest Mar 20, 2020
 #3
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There appears to be a maximum value for d which is = 57

 

a     b      c   d

31, 40, 44, 57

Guest Mar 20, 2020
 #4
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Thanks for your help! I think I figured it out!

DannyMath  Mar 21, 2020

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