Berkeley Math Circle #0i0w

Berkeley Math CircleHard

Community difficultyNo ratings yetBe the first to rate this problem.

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.

Radical axis theoremCyclic quadrilaterals

Problem

Problem:

Circles $k_{1}$ , $k_{2}$ , $k_{3}$ intersect as follows: $k_{1} \cap k_{2} = \{A, D\}$ , $k_{1} \cap k_{3} = \{B, E\}$ , $k_{2} \cap k_{3} = \{C, F\}$ . Also, $A B C D E F$ is a non-self-intersecting hexagon. Prove that

A B \cdot C D \cdot E F = B C \cdot D E \cdot F A.

(Hint: first prove that $A D$ , $B E$ , $C F$ meet at some point $O$ .)

← Back to all problems