Harvard-MIT Mathematics Tournament #0j65

Harvard-MIT Mathematics TournamentHard

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:

Let $ABCD$ be a quadrilateral circumscribed about a circle with center $O$ . Let $O_{1}, O_{2}, O_{3}$ , and $O_{4}$ denote the circumcenters of $\triangle AOB$ , $\triangle BOC$ , $\triangle COD$ , and $\triangle DOA$ . If $\angle A = 120^{\circ}$ , $\angle B = 80^{\circ}$ , and $\angle C = 45^{\circ}$ , what is the acute angle formed by the two lines passing through $O_{1}O_{3}$ and $O_{2}O_{4}$ ?