IMO Team Selection Test #0jik

IMO 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 $\triangle ABC$ be an acute triangle, and let $X$ be a variable interior point on the minor arc $\widehat{BC}$ of its circumcircle. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $CA$ and $CB$ , respectively. Let $R$ be the intersection of line $PQ$ and the perpendicular from $B$ to $AC$ . Let $\ell$ be the line through $P$ parallel to $XR$ . Prove that as $X$ varies along minor arc $\widehat{BC}$ , the line $\ell$ always passes through a fixed point. (Specifically: prove that there is a point $F$ , determined by triangle $ABC$ , such that no matter where $X$ is on arc $\widehat{BC}$ , line $\ell$ passes through $F$ .)