USA TSTST #0kb1 — USA TSTST · USAMO Guide - Olympiad Mathematics Learning Platform

USA TSTST #0kb1

USA TSTSTHard

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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 triangle with circumcircle $\Omega$ and orthocenter $H$ . Points $D$ and $E$ lie on segments $AB$ and $AC$ respectively, such that $AD = AE$ . The lines through $B$ and $C$ parallel to $DE$ intersect $\Omega$ again at $P$ and $Q$ , respectively. Denote by $\omega$ the circumcircle of $\triangle ADE$ .

a. Show that lines $PE$ and $QD$ meet on $\omega$ .

b. Prove that if $\omega$ passes through $H$ , then lines $PD$ and $QE$ meet on $\omega$ as well.