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

Sunday, November 20, 2011

Please email your solutions to puzzles@cotpi.com.

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### Credit

This is an original puzzle from cotpi.

### Further reading

The following is a list of resources on related topics:

- Poonen, Bjorn. "The AM-GM inequality." Inequalities. 4 Nov. 2001. University of California, Berkeley. 27 Nov. 2011 <http://mathcircle.berkeley.edu/BMC4/Handouts/inequal/node1.html>.

## 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.