VWO 2003

Two circles $C_1$ and $C_2$ intersect at $S$ . The tangent in $S$ to $C_1$ intersects $C_2$ in $A$ different from $S$ . The tangent in $S$ to $C_2$ intersects $C_1$ in $B$ different from $S$ . Another circle $C_3$ goes through $A, B, S$ . The tangent in $S$ to $C_3$ intersects $C_1$ in $P$ different from $S$ and $C_2$ in $Q$ different from $S$ . Prove that the distance $PS$ is equal to the distance $QS$ .

Attachments:

A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.

Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with $r=2\sqrt2$ goes through 4 points)

Prove that $\forall n\in \mathbb{N}, \exists r$ so that the circle with midpoint 0,0 and radius $r$ goes through at least $n$ points.