Now that it's been posted, it does look as if it's poorly worded. The setup already says the parts are equal. That precludes the question of "maximal". The product of the lengths is (L/n)^n.

Now, without the word "equal" in the first sentence, I'm sure that the answer remains (L/n)^n, as the product is maximal when the lengths are equal. Suppose you start out with equal parts (or even with unequal parts to begin with), then just take any two pieces. Whatever the product of the rest of the pieces may be, the product of these two pieces, that add up to whatever is not in the remainder of the stick, is maximized when they equal one another. The same is true for any other chosen pair.