2005 AMC 8 Problems/Problem 20
Problem
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
Solution
Alice moves steps and Bob moves steps, where is the turn they are on. Alice and Bob coincide when the number of steps they move collectively, , is a multiple of . Since this number must be a multiple of , as stated in the previous sentence, has a factor , must have a factor of . The smallest number of turns that is a multiple of is .
This is such a beautiful solution!!!! - sleepypuppy
See Also
2005 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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