Problem Analysis #4

To solve the problem, we analyze the constraints on point $P$ on the $xy$-plane:

1. Distance Constraint: $OP = 2$, so $P$ lies on the circle $x^2 + y^2 = 4$.

2. Angle Constraints:

   • $\angle AOP \ge \frac{2\pi}{3}$: This means $P$ must be in the region where the angle between $OA$ (along the positive $y$-axis) and $OP$ is at least $120^\circ$. This corresponds to the lower half of the circle.
   • $\angle OAP \le \frac{\pi}{3}$: This means $P$ must be in the region where the angle between $OA$ and $AP$ is at most $60^\circ$. This corresponds to the upper half of the circle.

3. Intersection of Regions: The intersection of these two regions is the lower semicircle of the circle $x^2 + y^2 = 4$.

4. Arc Length Calculation: The total arc length $L$ of the lower semicircle is half the circumference of the circle:

   $$L = \frac{1}{2} \times 2\pi \times 2 = 2\pi.$$

5. Expressing $L$ in the form $p\pi/q$: Here, $p = 2$ and $q = 1$.

6. Final Answer: $p + q = 2 + 1 = 3$.