Team Selection Test #0j8c

Team Selection TestHard

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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 $ABC$ be an acute scalene triangle inscribed in circle $\Omega$ . Circle $\omega$ , centered at $O$ , passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$ , respectively. Point $P$ lies on major arc $\widehat{BAC}$ of $\Omega$ . Prove that lines $BD$ , $CE$ , $OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.

(This problem was suggested by Qiang Song.)