Here is a problem:

You have a paper disk. You can make multiple cuts from the edge of the circle to the center to leave you with multiple sectors. You then wrap each sector into a cone. How many cone should you make, and how big should each one be, to maximize the total volume of all the cones?

It's a very interesting problem, but ...

I have the formula for the volume from a given sector; I found the sector who gives de biggest volume. But I don't know how maximize a sum of unknown number of terms.

I keep trying.