2025 AMC 12B Problem 16

Question

An analog clock starts at midnight and runs for

2025

minutes before stopping. What is the tangent of the acute angle between the hour hand and the minute hand when the clock stops?

0

\sqrt{2} - 1

2 - \sqrt{2}

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

3 - \sqrt{2}

Reduce

2025

minutes modulo

12

hours to find where the hands point

Find each hand’s angle, take the acute difference, then evaluate

\tan 22.5^\circ

2025

minutes is

33

hours

45

minutes, which modulo

12

hours reads

9{:}45.

The minute hand points at

270^\circ

and the hour hand at

9.75 \times 30^\circ = 292.5^\circ,

so the acute angle between them is

22.5^\circ.

Using the half-angle value,

\tan 22.5^\circ = \sqrt{2} - 1.

Thus, the correct answer is B.

Problem 16 in other years