13th Bay Area Mathematical Olympiad #0j81

13th Bay Area Mathematical OlympiadHard

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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:

Three circles $k_{1}$ , $k_{2}$ , and $k_{3}$ intersect in point $O$ . Let $A$ , $B$ , and $C$ be the second intersection points (other than $O$ ) of $k_{2}$ and $k_{3}$ , $k_{1}$ and $k_{3}$ , and $k_{1}$ and $k_{2}$ , respectively. Assume that $O$ lies inside of the triangle $ABC$ . Let lines $AO$ , $BO$ , and $CO$ intersect circles $k_{1}$ , $k_{2}$ , and $k_{3}$ for a second time at points $A'$ , $B'$ , and $C'$ , respectively. If $|XY|$ denotes the length of segment $XY$ , prove that

\frac{|AO|}{|AA'|} + \frac{|BO|}{|BB'|} + \frac{|CO|}{|CC'|} = 1