February 2017 #0jwi — February 2017 · USAMO Guide - Olympiad Mathematics Learning Platform

February 2017 #0jwi

February 2017Hard

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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

Problem: Let $L B C$ be a fixed triangle with $L B = L C$ , and let $A$ be a variable point on arc $L B$ of its circumcircle. Let $I$ be the incenter of $\triangle A B C$ and $\overline{A K}$ the altitude from $A$ . The circumcircle of $\triangle I K L$ intersects lines $K A$ and $B C$ again at $U \neq K$ and $V \neq K$ . Finally, let $T$ be the projection of $I$ onto line $U V$ . Prove that the line through $T$ and the midpoint of $\overline{I K}$ passes through a fixed point as $A$ varies.