YummY Pizza But tricky distribution

A  inch diameter round pizza has been sliced by  straight cuts, leaving a right triangle in the middle as shown here

The edge of the pizza has been cut into  arcs, one each of the following, in degrees, totaling  degrees

but not necessarily in that order. The maximum area, in square inches, this right triangle piece in the middle can have can be expressed as

where  are integers (which may be negative). Find

Integer  is positive. Do not rely on diagram to guess arcs in degrees.

WHAT!!!! you must be joking

Circle in a chess board

Consider an infinite chessboard, where the squares have side length of 1. The squares are colored black and white alternately. The (finite) radius of the largest circle whose circumference can be drawn completely on the white squares (hence you can see the entire circle) has the form , where  and  are integers,  and  are coprime, and  is not divisible by the square of any prime. What is the value of ?

Details and assumptions

 and  are all allowed to be 1. In particular, if you think the the largest radius is , then your answer to this should be .

Easy...NAAH!! yeah!!

365 can be written as a sum of 2 consecutive perfect squares and also 3 consecutive perfect squares:What is the next number with this property? Give the last 3 digits of the number.

The perfect squares cannot be zero.

Number theory

Find the sum of all positive integral  such that there exists a permutation  of the set  such that

is a rational number.  denotes the -th term of the permutation.