Let $O$ be the center of the circle $\omega$, and let $A$ be a point inside this circle, different from $O$. Find all points $P$ on the circle $\omega$ for which the angle $\angle OPA$ acquires the greatest value.
1. **Define the problem geometrically:**
   - Let \( O \) be the center of the circle \(\omega\) with radius \( r \).
   - Let \( A(a, b) \) be a point inside the circle, different from \( O \).
   - We need to find the points \( P \) on the circle \(\omega\) such that the angle \(\angle OPA\) is maximized.

2. **Parameterize the circle:**
   - The circle \(\omega\) can be parameterized as \( P(r \cos \alpha, r \sin \alpha) \) where \(\alpha\) is the angle parameter.

3. **Expression for \(\tan \angle OPA\):**
   - The tangent of the angle \(\angle OPA\) can be given by:
     \[
     \tan \angle OPA = \frac{a \sin \alpha - b \cos \alpha}{r - a \cos \alpha - b \sin \alpha}
     \]

4. **Maximizing the angle \(\angle OPA\):**
   - To find the maximum value of \(\angle OPA\), we need to maximize the expression for \(\tan \angle OPA\).
   - The maximum value of \(\tan \angle OPA\) occurs when the numerator is maximized and the denominator is minimized.

5. **Simplify the expression:**
   - Consider the expression \(\frac{a \sin \alpha - b \cos \alpha}{r - a \cos \alpha - b \sin \alpha}\).
   - To maximize this, we need to find the critical points of the function with respect to \(\alpha\).

6. **Critical points:**
   - The critical points can be found by taking the derivative of the expression with respect to \(\alpha\) and setting it to zero.
   - However, a more intuitive approach is to consider the geometric interpretation.

7. **Geometric interpretation:**
   - The angle \(\angle OPA\) is maximized when \( P \) is such that the line segment \( OP \) is tangent to the circle centered at \( A \) with radius \( OA \).
   - This occurs when \( P \) is at the point where the line through \( O \) and \( A \) intersects the circle \(\omega\).

8. **Finding the points \( P \):**
   - The points \( P \) that maximize \(\angle OPA\) are the points where the line through \( O \) and \( A \) intersects the circle \(\omega\).
   - These points can be found by solving the system of equations for the circle and the line.

9. **Conclusion:**
   - The points \( P \) on the circle \(\omega\) that maximize \(\angle OPA\) are the points where the line through \( O \) and \( A \) intersects the circle.

The final answer is the points where the line through \( \boxed{ O } \) and \( A \) intersects the circle \(\omega\).