Berkeley Math Circle Monthly Contest 4 #0hw6

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.

TangentsAngle chasingTriangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circle

Problem

Problem:

Let $ABCDEF$ be a hexagon circumscribing a circle $\omega$ . The sides $AB$ , $BC$ , $CD$ , $DE$ , $EF$ , $FA$ touch $\omega$ at $U$ , $V$ , $W$ , $X$ , $Y$ , and $Z$ respectively; moreover, $U$ , $W$ , and $Y$ are the midpoints of sides $AB$ , $CD$ , and $EF$ , respectively. Prove that $UX$ , $VY$ , and $WZ$ are concurrent.

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