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Back in high school, I came across the following contest problem:

Question: Let $S$ be a set of positive integers totaling 20.  What is the maximum value of

$\displaystyle\prod_{x \in S} x$

It’s a fun problem, so don’t rush past the spoiler tags too fast.  When you’re ready, I’ll spoil the solution to the question above, and discuss a “continuous” version of the question above.  Namely, what happens when $S$ is allowed to include positive real numbers?