Bay Area Mathematical Olympiad #0jbt

Bay Area Mathematical OlympiadHard

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.

Problem

Problem:

Given a segment $A B$ in the plane, choose on it a point $M$ different from $A$ and $B$ . Two equilateral triangles $\triangle A M C$ and $\triangle B M D$ in the plane are constructed on the same side of segment $A B$ . The circumcircles of the two triangles intersect in point $M$ and another point $N$ . (The circumcircle of a triangle is the circle that passes through all three of its vertices.)

a. Prove that lines $A D$ and $B C$ pass through point $N$ .

b. Prove that no matter where one chooses the point $M$ along segment $A B$ , all lines $M N$ will pass through some fixed point $K$ in the plane.