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1989 IMO Longlists
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(1)
An accurate 12-hour analog clock has an hour hand
Source: IMO Longlist 1989, Problem 101
9/18/2008
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
⋅
π
3
.
\frac{2 \cdot \pi}{3}.
3
2
⋅
π
.
Let
f
(
t
)
,
g
(
t
)
,
h
(
t
)
f(t), g(t), h(t)
f
(
t
)
,
g
(
t
)
,
h
(
t
)
be the respective absolute deviations of the separating angles from \frac{2 \cdot \pi}{3} at
t
t
t
hours after 12:00 o’clock. What is the minimum value of
m
a
x
{
f
(
t
)
,
g
(
t
)
,
h
(
t
)
}
?
max\{f(t), g(t), h(t)\}?
ma
x
{
f
(
t
)
,
g
(
t
)
,
h
(
t
)}?
algebra unsolved
algebra