2021 WSMO Accuracy Round Problems/Problem 4
Problem 4
A 12-hour clock has a minute hand that is the same length as the second hand, and an hour hand half the length of the minute hand. In a day, the tip of the minute hand travels a distance of the tip of the second hand travels a distance of
and the tip of the hour hand travels a distance of
The value of
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Solution 1
WLOG, assume that the length of the minute hand is 2. This means that the length of the second hand is 2 and the length of the hour hand is 1. In a day, there are 24 hours, which means that the minute hand travels each day. Also, since there are
hours, the hour hand travels
each day. Finally, there are
minutes in a day, which means that the second hand travels
each day. Thus, the final answer is
~pinkpig
Solution 2
Let the distance traveled by one revolution of the minute hand tip be . Note that
,
, and
. Our desired expression becomes:
This gives us an answer of .
~BigKahuna227