I was wondering why you couldn't assume that point O was the center of the circle. Does the question have to directly mention that fact …
No, you cannot. Anything that is NOT stated by the question statement or specifically marked on graphics is considered unknown. (However basic facts, such as: lines, which look straight, are straight; how the points are notated, etc. can be assumed based on the graphics.)
But maybe you could prove (based on the given facts) that O is the center of the circle? Unfortunately no. There are no grounds for that.
… , because I assumed that it was the center and therefore took OD and OA as being radii. Which ultimately gives you an answer of 3:π.
Even if O was the center of the circle, then 3:π would still NOT be the correct answer. OA and OD would be the radii indeed, but the triangle AOD would be just an isosceles one. And as the explanation shows, the radius differs from the length of AD, so AOD would NOT be an equilateral triangle.