AIME II #0l6m — AIME II · USAMO Guide - Olympiad Mathematics Learning Platform

AIME II #0l6m

AIME IIHard

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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

Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ$ , $\angle ABC = 60^\circ$ , and $\angle ACB = 36^\circ$ . Let $D$ , $E$ , and $F$ be the midpoints of sides $\overline{BC}$ , $\overline{AC}$ , and $\overline{AB}$ , respectively. The circumcircle of $\triangle DEF$ intersects $\overline{BD}$ , $\overline{AE}$ , and $\overline{AF}$ at points $G$ , $H$ , and $J$ , respectively. The points $G$ , $D$ , $E$ , $H$ , $J$ , and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\widehat{DE} + 2 \cdot \widehat{HJ} + 3 \cdot \widehat{FG}$ , where the arcs are measured in degrees.