Consider all pairs of distinct points on the Cartesian plane $(A, B)$ with integer coordinates. Among these pairs of points, find all for which there exist two distinct points $(X, Y)$ with integer coordinates, such that the quadrilateral $AXBY$ is convex and inscribed.

[i]Proposed by Anton Trygub[/i]
1. **Define the points and distance condition:**
   Let the points be \( A = (a_1, a_2) \) and \( B = (b_1, b_2) \). We need to find pairs of points \( (A, B) \) such that there exist two distinct points \( (X, Y) \) with integer coordinates, making the quadrilateral \( AXBY \) convex and inscribed.

2. **Case when \( d(A, B) = 1 \):**
   If the distance between \( A \) and \( B \) is 1, then:
   \[
   d(A, B) = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2} = 1
   \]
   This implies that \( A \) and \( B \) are adjacent points on the Cartesian grid. The smallest circle that can pass through both points would have a radius \( r \leq \frac{1}{2} \), which cannot contain any other integer points. Therefore, no such points \( X \) and \( Y \) exist in this case.

3. **Case when \( a_1 \neq b_1 \) and \( a_2 \neq b_2 \):**
   If \( a_1 \neq b_1 \) and \( a_2 \neq b_2 \), we can construct points \( X \) and \( Y \) as follows:
   \[
   X = (b_1, a_2) \quad \text{and} \quad Y = (a_1, b_2)
   \]
   The quadrilateral \( AXBY \) forms a rectangle, which is both convex and inscribed in a circle. This is because the diagonals of a rectangle are equal and bisect each other, ensuring that all four vertices lie on a circle.

4. **Case when \( a_2 = b_2 \) and \( a_1 < b_1 + 1 \):**
   If \( a_2 = b_2 \) and \( a_1 < b_1 + 1 \), we can construct points \( X \) and \( Y \) as follows:
   \[
   X = (a_1 + 1, a_2 + 1) \quad \text{and} \quad Y = (a_1 + 1, a_2 + 1 - b_1 + a_1)
   \]
   Here, \( BX = AY \) and \( AB = XY \), ensuring that the points \( A, B, X, Y \) lie on the same circle. This construction guarantees that \( AXBY \) is convex and inscribed.

5. **Conclusion:**
   For all pairs of points \( (A, B) \) except those with \( d(A, B) = 1 \), such points \( X \) and \( Y \) exist, making the quadrilateral \( AXBY \) convex and inscribed.

The final answer is all pairs \( \boxed{ (A, B) } \) except those with \( d(A, B) = 1 \).