I have already posted this question time before on stackexchange, but didn't receive a definitive solution.

So this is the game: consider a positive integer number $n$ and divide it in a finite number of parts $m$, then take the product of all the $m$ numbers you obtained and multiply them. For example for $n=10$ and $m=5$ the product will be $32$. The game consists in finding a number $m$ (positive integer) which maximize this product.

Now it's trivial to prove that the function $f(x)=(n/x)^x$, with $(0,+∞)$ as domain, increases on $(0,n/e]$ and decreases on $[n/e,+∞)$, so it has a max in $x=n/e$, thus you'll find the $m$ you're looking for by evaluating the function in the two integers nearest to $n/e$ and comparing these two values.

What is not easy to prove is that if you take the nearest positive integer to $n/e$ it will always be the best $m$ for the game, or at least for all the cases I've taken into consideration (with computational experiments too).

So my question is: how to prove (or disprove) that this fact holds for any positive integer $n$?