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Thread: What is the maximum possible product?

  1. #1
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    What is the maximum possible product?

    .


    Suppose there is a set of positive real numbers, where the numbers are not necessarily all the same, that add to 270.

    What is their maximum product?






    You should express the answer in scientific notation and round the rightmost digit (to the right of the decimal point):

    #.## X 10^(exponent)
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  2. #2
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    Re: What is the maximum possible product?

    It appears to be

    $e^{270/e} \approx 1.37203\times 10^{43}$

    This is based on the idea that symmetric maximization occurs when the elements are identical.

    So for a set of $n$ reals the maxima of the product will be $\left(\dfrac{270}{n}\right)^n$

    I know you said the numbers can't be equal but you can tweak them slightly with random noise of infinitesimal magnitude and it won't affect the product.

    Using the usual method of finding the maximum of this expression yields the expression on line 2.
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    Re: What is the maximum possible product?

    Quote Originally Posted by romsek View Post

    I know you said the numbers can't be equal but . . .
    No, I stated "...where the numbers are not necessarily all the same..."

    Thank you so far.


    I will look for additional and different ideas (read: including different answers) from different users now.
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    Re: What is the maximum possible product?

    Quote Originally Posted by greg1313 View Post
    No, I stated "...where the numbers are not necessarily all the same..."

    Thank you so far.


    I will look for additional and different ideas (read: including different answers) from different users now.
    I confess the difference eludes me since not being the same will provide a smaller product than all of them being equal.

    At any rate a bit of sim shows that the max product is definitely in the ballpark of the figure given.

    Good luck!
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    Re: What is the maximum possible product?

    Quote Originally Posted by romsek View Post
    I confess the difference eludes me since not being the same will provide a smaller product than all of them being equal.
    Let me see if later today, or tomorrow, if I can come up with a counter(example) to that (but not the optimum ... yet).
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    Re: What is the maximum possible product?

    Quote Originally Posted by romsek View Post
    It appears to be

    $e^{270/e} \approx 1.37203\times 10^{43}$
    You don't have 270/e e's to multiply together, because that isn't
    an integer.

    270 = 99e + 0.89009898... -->

    $e^{99}*(0.89009898...) \ \approx \ 8.80 \times 10^{42}$

    -----------

    270 = 98e + 3.60838... -->

    $e^{98}*(3.60838...) \ \approx \ 1.31 \times 10^{43}$

    ------------

    270 = 97e + 6.326662... -->

    270 = 97e + 3.16333... + 3.16333... -->

    $e^{97}*(3.16333...)^2 \ \approx \ 1.34 \times 10^{43}$


    It seems your answer is an upper bound approximation.
    Last edited by greg1313; Mar 25th 2018 at 09:24 AM.
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    Re: What is the maximum possible product?

    $\left( \dfrac{270}{99} \right)^{99} \approx 1.37\times 10^{43} $

    As far as your problem is concerned, it seems like this is the solution.
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