USA TST Selection Test for 67th IMO and 15th EGMO #0l8c

USA TST Selection Test for 67th IMO and 15th EGMOHard

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

Let triangle $ABC$ have orthocenter $H$ . Let $B_1, C_1, B_2$ and $C_2$ be collinear points which lie on lines $AB, AC, BH$ , and $CH$ , respectively. Let $\omega_B$ and $\omega_C$ be the circumcircles of triangles $BB_1B_2$ and $CC_1C_2$ , respectively. Prove that the radical axis of $\omega_B$ and $\omega_C$ intersects the line through their centers on the nine point circle of triangle $ABC$ .