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2025 AMC 12B Problem 16

Below is the professionally curated solution for Problem 16 of the 2025 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12B solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:clocktrigonometric identity

Difficulty rating: 1840

16.

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?

00

2−1\sqrt{2} - 1

2−22 - \sqrt{2}

22\dfrac{\sqrt{2}}{2}

3−23 - \sqrt{2}

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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