I left class today with a riddle. Find a set of numbers (does not specify what type) that add to 25, but must multiply to a number as high as possible. For example, $10+10+5 = 25, 10\times10\times5 = 500$.
I started with $5+5+5+5+5 \ (5^5 = 3125)$ but I realized I can break up these numbers more. It ended up with $2.7182$ adding to itself $9.1969$ times, which resulted in a product of $9866.4344$
I also realized that it should be a number added and multiplied by itself. What is this number? Or is it a set of different numbers?
From the question I suppose you're allowed to use non-integer numbers which also may appear arbitrary number of times. Then the question really makes no sense, because such list of numbers doesn't exist. It doesn't exist because (if we assume that numbers in the list are finite) we can always find another list whose sum is $25$, but whose product is bigger than the given list.
For example, let $n\to\infty$ be a real number, then the list $$(2n,-n,-n,25)$$ sums up to $25$, but it's product tends to infinity.