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[VA] SRC #008 - 2021 is here !
01-15-2021, 12:50 AM
Post: #39
RE: [VA] SRC #008 - 2021 is here !
.
Hi, Vincent:

(01-14-2021 10:18 PM)Vincent Weber Wrote:  My bad ! I posted this using my phone, not realizing how bad the formatting would be. I have deleted the original post.

That sort of explains it. Thanks for deleting the post, that will make generating a properly formatted PDF much easier.

Quote:Now that the challenge is over, please find below my original text without the code tags, for your kind consideration.

But of course, my pleasure. Let's see:

Quote:First, let's fix the number of numbers the sum is composed of. Let's try the simplest case - 2 numbers. There are x and 2021-x. The product is maximum when the derivative of x*(2021-x) is 0, so when 2021-x-x=0, or x=2021/2. This is not an integer, ok, but then I got the intuition that the product is maximized when the numbers are as close to each other as possible.

Totally correct, and a logical intuition.

Quote:So let's try with 3 numbers. They are x,y,and (2021-x-y).
The product is maximized when both x and y partial derivatives are zero, so when 2021-2x-y=0 and 2021-x-2y=0. This simple system leads to x=y=2021/3. Again, non-integer, but confirms the intution that the numbers need to be equal !

Again, quite correct and indeed reinforces said intuition.

Quote:So I was bald engouh to take this intuition for granted for any number of numbers Smile

Very plausible, the previous cases are motivation enough to hypothesize that if the intuition was correct for the cases N=2 and N=3 it's reasonable to assume that it might also be true for general N and see where that leads us. So far so good.

Quote:So now, what this number of numbers should be ? Well if x is this number, then 2021/x is each equal number contributing to the sum, and the product is (2021/x)^x, which is also equal to e^(x.ln(2021/x)). Composition derivation leads to a derivative of

(ln(2021/x)+x.-2021/x^2*x/2021)*...= (Ln(2021/x)-1)*... ("..." being the original product function which is never going to be zero).

Which leads to an optimal number of numbers of x=2021/e = 743.48... non integer of course.

Correct reasoning, nihil obstat.

Quote:So each number should ideally be equal to e=2.718... the closest integer that comes to mind is obviously 3. You can fit 673 times 3 in 2021. 3×673=2019, remains 2. As for the product, 3^673*2=2.53E321, which is, well, quite a lot Smile maybe not as much as you expected though , Valentin ? Smile

The reasoning is crystal-clear and the conclusion, correct. And I did expect exactly as much from you, Vincent  Smile

Congratulations are in order and I'm glad I finally got to see your original message, it's been an interesting read for me and I'm sure that many other readers of this thread will find it very interesting (and enlightening!) as well.

Thanks again for your fine contribution, it sure helps a lot to achieve my stated main goal for these SRC's. Hope you enjoyed it all and I fully expect to see you participate in the next one !  Smile

Best regards.
V.

  
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Messages In This Thread
RE: [VA] SRC #008 - 2021 is here ! - Gene - 01-02-2021, 01:49 AM
RE: [VA] SRC #008 - 2021 is here ! - robve - 01-03-2021, 06:33 PM
RE: [VA] SRC #008 - 2021 is here ! - robve - 01-05-2021, 03:39 AM
RE: [VA] SRC #008 - 2021 is here ! - Gene - 01-04-2021, 05:56 PM
RE: [VA] SRC #008 - 2021 is here ! - Gene - 01-04-2021, 07:24 PM
RE: [VA] SRC #008 - 2021 is here ! - Gene - 01-06-2021, 02:54 PM
RE: [VA] SRC #008 - 2021 is here ! - EdS2 - 01-08-2021, 01:32 PM
RE: [VA] SRC #008 - 2021 is here ! - Valentin Albillo - 01-15-2021 12:50 AM



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