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Here is the problem I am attempting to prove.

We have $n$ segments whose sum is equal to one, with each segment having length less than one and greater than zero. Prove that making each segment of length $1/n$ maximizes the product $\prod_{i=1}^{n} n_i$.

From empirical observation I have seen that this product is maximized when each segment is $1/n$ but I can thinking of no way to prove it. Does anyone have any ideas?

Thanks

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    The question is not really "number theory" in the ambient sense, although it involves numbers, indeed. Re-titling and re-tagging will help other people understand the situation better, I hope. If not, of course, as always, feel free to revert my edits. – 2017-02-18

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You have some numbers $n_i$ satisfying $\sum_i n_i = 1$, and you want to maximize the product $\prod_i n_i$. Note that the sum and the product are related by the AM-GM inequality:

$\frac{1}{n} \sum_i n_i \geq \Bigl(\prod_i n_i \Bigr)^{1/n}$.

Re-writing this, when $\sum_i n_i = 1$, yields

$\prod_i n_i \leq n^{-n}$.

Since taking $n_i = 1/n$ for all $i$ yields equality, that choice of $n_i$'s achieves the maximum possible value.

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    Hey, thanks man this helps a lot. – 2017-02-18