The first small problem: connecting BO, because it is a circumscribed circle, so OA=OB=OC, according to the isosceles triangle, the base angles are equal, ∠OAB =∠ OBA, ∠OCB=∠OBC, ∠ABC= 120, so ∠.
The second small problem: the shadow area can be regarded as: sector AOC area minus triangle AOC area plus triangle APC area, ∠AOC= 120, so the sector area is one third of the circle area: 4π/3; If the height of the AC side is ∠AOC= 120, then ∠OCA=∠OAC=30, and the radius is 2, that is, OC=2. You can use Pythagorean theorem to calculate ON= 1, AC=2* root number 3, and triangle AOC area. APC area of triangle =0.5*AC*X=X* root number 3. So y=4π/3- radical number 3+X* radical number 3, and the value range of x is from 0 to ON+ radius, that is, (0,3).