HMMT 2013 #0jfs

HMMT 2013Hard

From module Geometry with Complex Numbers

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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.

Complex numbers in geometryTriangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circle

Problem

Problem:

Let $ABCD$ be a quadrilateral inscribed in a unit circle with center $O$ . Suppose that $\angle AOB = \angle COD = 135^{\circ}$ , $BC = 1$ . Let $B'$ and $C'$ be the reflections of $A$ across $BO$ and $CO$ respectively. Let $H_1$ and $H_2$ be the orthocenters of $AB'C'$ and $BCD$ , respectively. If $M$ is the midpoint of $OH_1$ , and $O'$ is the reflection of $O$ about the midpoint of $MH_2$ , compute $OO'$ .

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