HMMT February 2023 #0kw5

HMMT February 2023Hard

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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.

Spiral similarityIsogonal/isotomic conjugates, barycentric coordinatesAngle chasingDistance chasing

Problem

Problem:

Let $ABC$ be a triangle with $\angle BAC > 90^{\circ}$ . Let $D$ be the foot of the perpendicular from $A$ to side $BC$ . Let $M$ and $N$ be the midpoints of segments $BC$ and $BD$ , respectively. Suppose that $AC = 2$ , $\angle BAN = \angle MAC$ , and $AB \cdot BC = AM$ . Compute the distance from $B$ to line $AM$ .

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