HMMT-Feb 2023 geo, Problem 2

Points $X, Y$, and $Z$ lie on a circle with center $O$ such that $X Y=12$. Points $A$ and $B$ lie on segment $X Y$ such that $O A=A Z=Z B=B O=5$. Compute $A B$.

Solution: Solution: Let the midpoint of $X Y$ be $M$. Because $O A Z B$ is a rhombus, $O Z \perp A B$, so $M$ is the midpoint of $A B$ as well. Since $O M=\frac{1}{2} O X, \triangle O M X$ is a $30-60-90$ triangle, and since $X M=6$, $O M=2 \sqrt{3}$. Since $O A=5$, the Pythagorean theorem gives $A M=\sqrt{13}$, so $A B=2 \sqrt{13}$. $\fbox{2 \sqrt{13}}$.

Answer: 2 \sqrt{13}