BAMO #0hys — BAMO · USAMO Guide - Olympiad Mathematics Learning Platform

BAMO #0hys

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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:

$\triangle ABC$ is inscribed in a circle $k$ with center $O$ so that $\angle ACB = 120^\circ$ .

a. If $H$ is the orthocenter of $\triangle ABC$ , prove that $A, B, O, H$ lie on a circle with center the midpoint of the arc $ACB$ . (The orthocenter of $\triangle ABC$ is the intersection point of its three altitudes.)

b. If $G$ is the centroid of $\triangle ABC$ , and $I$ is the incenter of $\triangle ABH$ , prove that the points $O, G, I, H$ lie on a line. (The centroid of $\triangle ABC$ is the intersection point of its three medians: a median connects a vertex of $\triangle ABC$ with the midpoint of the opposite side; the incenter of $\triangle ABC$ is the intersection of its three angle bisectors.)