What is the largest real number that can be factorized into a product of positive real numbers such that their sum is 27?

[SOLVED]

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quasi solved this puzzle:

From the AM-GM inequality for n numbers, for each positive integer n, the maximum product is (27 / n)n.

For x ≥ 1, the function f(x) = (27 / x)x is increasing for 1 ≤ x ≤ 27 / e and decreasing for x ≥ 27 / e.

Since 27 / e is between 9 and 10, it suffices to compare (27 / 9)9 and (27 / 10)10.

The bigger one is (27 / 10)10. So, the maximum value is (27 / 10)10 which is 20589.1132094649.

Credit

This is an original puzzle from cotpi.

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