Find The Largest Product
This problem is from Zachary M. Franco, Butler University.
Write 53 as a sum of positive real numbers so that their product is as large as possible.
Submit your answers to mathpotw@acu.edu. Details for submissions can be found here.
Solution to Find The Largest Product
Correct solutions were submitted by: Wyatt Witemeyer.
If S = x + y, then you can always do better by replacing x and y with their average. Thus, the answer will consist of n pieces all equal to 53/n.
Since the function (53/x)x has a maximum at approximately x = 19.5, we test the nearest discrete cases to determine that the largest product is (53/19)19 > 291,691,050.
It is interesting to observe that for all integers k from 1 to 52, the largest product is (k/n)n, where n is such that k/n is closest to e. But this fails for 53. For 53, we use n where n is the closest integer to k/e.
The counterexample list is: 53, 246, 439, 632, 12973, … (which seems to arise from the continued fraction for e). See Z. Franco, Mathematics Magazine, Feb 2000, for a proof of the theorem, “use n where n is the closest integer to k/e,” and for the connection to continued fractions.
