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?

This problem is very difficult.

You must study the function f(x)=x².sqrt(1-x²) and then the function g(x)=f(x)+f(1-x)

Calculating g'(x) to find the extrema, you get a 7th degree equation that you can reduce to a 3rd degree equation. The expression of the roots is in a complex form but the interesting one is 0.67598...

The maximum volume with only two sectors is with sectors of 2𝜋(0.67598...) and 2𝜋(0.32402...).

You can then easily show that no division get a better result that the biggest sector.

So, the maximum volume is reached with 2 sectors : ~ 4.2473... & ~ 2.0359... radians.

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.