Berkeley Math Circle Monthly Contest 2 #0htt

Berkeley Math Circle Monthly Contest 2Hard

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.

Cyclic quadrilateralsAngle chasing

Problem

Problem:

Let $ABCD$ be a square. Consider four circles $k_{1}, k_{2}, k_{3}, k_{4}$ which pass respectively through $A$ and $B$ , $B$ and $C$ , $C$ and $D$ , $D$ and $A$ , and whose centers are outside the square. Circles $k_{4}$ and $k_{1}$ , $k_{1}$ and $k_{2}$ , $k_{2}$ and $k_{3}$ , $k_{3}$ and $k_{4}$ intersect respectively at $L$ , $M$ , $N$ , $P$ inside the square. Prove that quadrilateral $LMNP$ can be inscribed in a circle.

← Back to all problems