When Does This Point Not Exist?

Geometry Level 5

Let ABC\triangle ABC be an acute angles triangle with BAC=80.\angle BAC = 80^{\circ}. Let MM be the midpoint of BC,BC, and let OO be the circumcenter of ABC.\triangle ABC. Suppose there exists no point XX in the plane of ABC\triangle ABC which satisfies the following conditions.

  • XO,XAX \neq O, X \neq A
  • BAX=CAM\angle BAX = \angle CAM
  • AXO=90\angle AXO = 90^{\circ}

Let ABC=N.\angle ABC = N^{\circ}. Find N2.\dfrac{N}{2}.

Details and assumptions
- The diagram shows the point X,X, which shouldn't exist.
- The first condition means XX must be different from OO and A.A.

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