幾何與角

In the figure, $AOB$ is a diameter of the circle, centre $O$. $CD$ is the perpendicular bisector of $OA$. Which of the angles $a$, $b$, $c$, $d$ is/are equal to $30^\circ$?

In the figure, circle $O$ is inscribed in $\triangle ABC$, touching $BC$ at $X$. Which of the following must be true?

I. $OX\perp BC$ II. $OA$ bisects $\angle A$ III. $AO$ produced bisects $BC$

In the figure, $AC$ and $BC$ are diameters of two semi-circles touching each other internally at $C$. $PQC$ is a straight line. If $AB=1$, then $PQ=$

In the figure, $O$ is the centre of the circle. $PAB$ is a straight line. $x+y=$