This topic is linked to Problem 1.
Y by Adventure10, Mango247
Look at my solution to problem 5 in this pdf: http://submissions.usamts.org/Year28/Round1/34279-1476743929-USAMTS%202016%20Matthew%20Roth.pdf
I found some useful results about cyclic quadrilaterals that may generalize to Problem(ii)
EDIT: I'll think about it some more but I think this problem can be solved by breaking n-gons into a bunch of quadrilaterals and maybe 1 triangle (depending on n) and maximizing the areas by making all the quadrilaterals cyclic and the triangle equilateral
I found some useful results about cyclic quadrilaterals that may generalize to Problem(ii)
EDIT: I'll think about it some more but I think this problem can be solved by breaking n-gons into a bunch of quadrilaterals and maybe 1 triangle (depending on n) and maximizing the areas by making all the quadrilaterals cyclic and the triangle equilateral