This is another race puzzle from the Maths Masters team, Burkard Polster (aka Mathologer) and Marty Ross as part of their “Summer Quizzes” offerings for 2013.
“In a 100 meter race, Jacob can beat Johann by 5 meters, and Johann can beat Nicolaus by 10 meters. By how much can Jacob beat Nicolaus?”
See Yet Another Race
This is a somewhat challenging problem from the 1997 American Invitational Mathematics Exam (AIME).
“A car travels due east at 2/3 miles per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at √2/2 miles per minute. At time t = 0, the center of the storm is 110 miles due north of the car. At time t = t1 minutes, the car enters the storm circle, and at time t = t2 minutes, the car leaves the storm circle. Find (t1 + t2)/2.”
See the Storm Chaser Problem
This is a problem from the 1987 American Invitational Mathematics Exam (AIME).
“Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al’s speed of walking (in steps per unit time) is three times Bob’s walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)”
See the Escalator Puzzle
Puzzles and Problems: AIME, travel puzzles
This is a nice variation on a racing problem by Geoffrey Mott-Smith from 1954.
“On one side of the playground some of the children were holding foot-races, under a supervisor who handicapped each child according to age and size. In one race, she placed the big boy at the starting line, the little boy a few paces in front of the line, and she gave the little girl twice as much headstart over the little boy as he had over the big boy. The big boy won the race nevertheless. He overtook the little boy in 6 seconds, and the little girl 4 seconds later.
Assuming that all three runners maintained a uniform speed, how long did it take the little boy to overtake the little girl?”
See the Handicap Racing
This puzzle from the Scottish Mathematical Council (SMC) Senior Mathematics Challenge seems at first to have insufficient information to solve.
“Ant and Dec had a race up a hill and back down by the same route. It was 3 miles from the start to the top of the hill. Ant got there first but was so exhausted that he had to rest for 15 minutes. While he was resting, Dec arrived and went straight back down again. Ant eventually passed Dec on the way down just half a mile before the finish.
Both ran at a steady speed uphill and downhill and, for both of them, their downhill speed was one and a half times faster than their uphill speed. Ant had bet Dec that he would beat him by at least a minute.
Did Ant win his bet?”
See the Close Race Puzzle
(Update 1/2/2023) Alternative Solution from Oscar Rojas Continue reading
This is a fun problem from the 1989 American Invitational Mathematics Exam (AIME).
“Two skaters, Allie and Billie, are at points A and B, respectively, on a flat, frozen lake. The distance between A and B is 100 meters. Allie leaves A and skates at a speed of 8 meters per second on a straight line that makes a 60° angle with AB. At the same time Allie leaves A, Billie leaves B at a speed of 7 meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?”
See the Skating Rendezvous Problem
This puzzle from the Scottish Mathematical Council (SMC) Middle Mathematics Challenge has an interesting twist to it.
“Two young mountaineers were descending a mountain quickly at 6 miles per hour. They had left the hostel late in the day, had climbed to the top of the mountain and were returning by the same route. One said to the other “It was three o’clock when we left the hostel. I am not sure if we will be back before nine o’clock.” His companion replied “Our pace on the level was 4 miles per hour and we climbed at 3 miles per hour. We will just make it.” What is the total distance they would cover from leaving the hostel to getting back there?”
See the Mountain Climbing Puzzle.
This is a slightly challenging problem from the 1993 American Invitational Mathematics Exam (AIME).
“Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Find t, the amount of time in seconds, before Jenny and Kenny can see each other again.”
See the Parallel Stroll Problem
Here is another Brainteaser from the Quantum math magazine.
“Nick left Nicktown at 10:18 A.M. and arrived at Georgetown at 1:30 P.M., walking at a constant speed. On the same day, George left Georgetown at 9:00 A.M. and arrived at Nicktown at 11:40 A.M., walking at a constant speed along the same road. The road crosses a wide river. Nick and George arrived at the bridge simultaneously, each from his side of the river. Nick left the bridge 1 minute later than George. When did they arrive at the bridge?”
See Meeting on the Bridge
Here is another elegant Quantum math magazine Brainteaser problem.
“A raft and a motorboat set out downstream from a point A on the riverbank. At the same moment a second motorboat of the same type sets out from point B to meet them. When the first motorboat arrives at B, will the raft (floating with the current) be closer to point A or to the second motorboat? (G. Galperin)”
See the River Traffic Problem