1. Points $A$ and $B$ lie on a circle with center $O$ and radius 6, and point $C$ is equidistant from points $A, B$, and $O$. Another circle with center $Q$ and radius 8 is circumscribed around triangle $A C O$. Find $B Q$. Answer: 10
Solution. Since $A Q=O Q=8$ and $A C=O C$, triangles $\triangle A C Q$ and $\triangle O C Q$ are equal. Therefore, points $A$ and $O$ are symmetric with respect to the line $C Q$, so $A O \perp C Q$.

Next, $\angle B O Q=\angle B O C+\angle C O Q=\angle A O C+\angle O C Q=\pi / 2$. Therefore, $B Q=\sqrt{B O^{2}+O Q^{2}}=\sqrt{6^{2}+8^{2}}=10$.