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2025 AMC 12B Problem 16

Problem 16 of 25IntermediateGeometryArithmetic

An analog clock starts at midnight and runs for 20252025 minutes before stopping. What is the tangent of the acute angle between the hour hand and the minute hand when the clock stops?

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Solution

20252025 minutes is 3333 hours 4545 minutes, which modulo 1212 hours reads 9:45.9{:}45. The minute hand points at 270∘270^\circ and the hour hand at 9.75×30∘=292.5∘,9.75 \times 30^\circ = 292.5^\circ, so the acute angle between them is 22.5∘.22.5^\circ. Using the half-angle value, tan⁡22.5∘=2−1.\tan 22.5^\circ = \sqrt{2} - 1. Thus, the correct answer is B.
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Tagged: clock · trigonometric identity

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