1. **State the problem:** We need to find the size of angle $x$ at point $A$ in a cyclic quadrilateral $ABCD$. The quadrilateral is inscribed in a circle, $BD$ passes through the center $O$, and we are given angles $\angle B = 33^\circ$ and $\angle D = 57^\circ$. 2. **Recall properties of cyclic quadrilaterals:** - The opposite angles of a cyclic quadrilateral sum to $180^\circ$. 3. **Use the property to find $\angle A$:** Since $ABCD$ is cyclic, $$\angle A + \angle C = 180^\circ$$ 4. **Find $\angle C$:** Since $BD$ passes through the center $O$, $BD$ is a diameter of the circle. By the Thales' theorem, angle subtended by a diameter is a right angle. Thus, $$\angle BAD = \angle BCD = 90^\circ$$ Here, $\angle C$ is $90^\circ$ because it is subtended by the diameter $BD$. 5. **Calculate $\angle A$ using the cyclic quadrilateral property:** $$\angle A + 90^\circ = 180^\circ$$ $$\angle A = 180^\circ - 90^\circ = 90^\circ$$ 6. **Conclusion:** The angle $x$ at point $A$ is $90^\circ$. Therefore, the size of angle $x$ is $90$ degrees.