Berkeley Math Circle: Monthly Contest 4 #0hwh

Berkeley Math Circle: Monthly Contest 4Hard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Problem

Problem:

Let $A X Y Z B$ be a convex pentagon inscribed in a semicircle with diameter $\overline{A B}$ , and let $K$ be the foot of the altitude from $Y$ to $\overline{A B}$ . Let $O$ denote the midpoint of $\overline{A B}$ and $L$ the intersection of $\overline{X Z}$ with $\overline{Y O}$ . Select a point $M$ on line $K L$ with $M A = M B$ , and finally, let $I$ be the reflection of $O$ across $\overline{X Z}$ . Prove that if quadrilateral $X K O Z$ is cyclic then so is quadrilateral $Y O M I$ .