Travel problems on circular tracks: relative speed, repeated meetings, reversals, scheduled rests, polygonal routes, and a special diameter shortcut.
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Questions 1–10 · 6 points each · 60 points
As shown, Runners A and B practice on a circular track in the clockwise direction. Runner A completes one lap in 12 minutes, and Runner B completes one lap in 15 minutes. They start at the same time from opposite endpoints of a diameter. After how many minutes will Runner A catch Runner B?
Opposite starting points; one shared direction.
Diagram not necessarily to scale.
A city has a ring road. A public bus traveling counterclockwise leaves the station every 10 minutes. Driver Wang’s truck travels clockwise on the same road at the same speed as the buses. Assume the regular service is already operating, with buses evenly spaced along the route. During a 30-minute observation period, what is the greatest number of bus encounters Driver Wang can have? Count encounters exactly at the beginning and the end of the period.
A circular track is 400 metres long. Runners A and B start together from the same point. When they run in opposite directions, they meet for the first time after 1 minute. When they start together and run in the same direction, the first catch occurs after 10 minutes. Runner A is faster than Runner B. Find both speeds.
Units: metres per minute.
A circular track has circumference 240 metres. Runners A and B move in the same direction, while Runner C moves in the opposite direction. All three start from the same point. Their speeds are 8 m/s, 5 m/s, and 7 m/s respectively. When all three meet for the first time after starting, how far has Runner C run, expressed in laps? Your answer may include a fraction.
Points A and B are the endpoints of a diameter. Runner A starts at A and Runner B starts at B. They start together and move in opposite directions. They meet for the first time at C and for the second time at D. Runner A travels 80 metres to the first meeting at C, before reaching B. By the second meeting at D, Runner A has reached B for the first time and traveled a further 60 metres in the same direction. Find the circumference of the circular track.
Use directed distances along the circular route.
Diagram not necessarily to scale.
On a circular track, Runner A starts from point A and Runner B starts from point B at the same time, moving in opposite directions. They first meet after 6 minutes. Four minutes later, Runner A first reaches Runner B’s starting point. Eight minutes after that, the runners meet for the second time. How many minutes does each runner take to complete one lap?
Units: minutes.
Track the time from each event to the next.
Runners A and B practice on a 400-metre circular track. They start together from point A and run in opposite directions. After their first meeting, both continue in their original directions. Runner A increases the original speed by 2 m/s, while Runner B decreases the original speed by 2 m/s. They then each take 24 seconds to reach point A for the next time, arriving together. What was Runner A’s original speed? You may give a fraction or round to two decimal places.
Both leave the same point in opposite directions.
The diagram shows a square garden path with side length 100 metres. Runners A and B start together from point A. Runner A travels counterclockwise at 75 m/min, and Runner B travels clockwise at 45 m/min. Their first meeting on side CD, not at C or D, is which numbered meeting after they start?
Side CD is the bottom side.
Cars A and B begin at points A and B on a 360-metre circular track. The shorter arc AB shown is 90 metres. Car A travels at 20 m/min. Follow these events in order:
After this last reversal, how many more minutes pass before the cars meet again?
The marked lower arc between A and B is 90 metres.
Diagram not necessarily to scale.
Two electric toy cars, A and B, have an initial gap of 100 metres along their direction of travel on a circular track. They start at the same time and travel in the same direction, with A in front and B behind. After 5 minutes, B catches A for the first time. Twenty minutes later, B catches A for the second time. At that moment, A has traveled exactly one full lap from its own starting position. What is Car B’s speed?
Questions 11–13 · 20 points each · 60 points
A circular track has circumference 500 metres. Runners A and B start together from the same point and run clockwise. Runner A runs at 60 m/min, and Runner B runs at 50 m/min. Both begin running immediately, and each must rest for 1 minute after every 200 metres run. How many minutes does Runner A need to gain one full lap on Runner B for the first time? Do not count the starting instant or any earlier shared rest stop as this lapping catch.
The path shown is a regular pentagon with five equal sides. Runners A and B walk along the path. The distance Runner A covers in 3 minutes takes Runner B 7 minutes. They start together from vertex A and both travel clockwise. When Runner A catches Runner B for the third time, on which side of the pentagon does the catch occur?
Five equal sides; vertices are named clockwise A–B–C–D–E.
Use the route relationships supplied by the problem; do not measure the drawing.
In a myth, a rabbit lives beside a circular sacred lake whose circumference is initially 1 kilometre. Points A and B are the endpoints of a diameter, dividing the lake into two semicircular arcs. The rabbit starts at B and moves counterclockwise around the lake in jumps. Each jump covers 3/8 kilometre measured along the circumference, followed by exactly one counted rest. The starting position is not a rest. Whenever the rabbit lands exactly at A at a rest point, it slides through the special passage AB to B and continues jumping from B. Passing through AB adds no jump or rest. Each time the rabbit uses the passage, the lake’s radius doubles immediately; it then resumes from B on the enlarged lake. After the rabbit has rested 1000 times, what is the lake’s circumference?
AB is a diameter and also the special passage.