IMO 1989 LL USA103

Origin: USA

Problem

An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. It is well known, and not difficult to prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle 2\pi/3. Let f(t), g(t), h(t) be the respective absolute deviations of the separating angles from 2\pi/3 at t hours after 12:00 o’clock. What is the minimum value of max{f(t), g(t), h(t)}?