2013/14 AHSMC Part 1 Problem 9

Problem 9 of 16

Two bus routes stop at a certain bus stop. The A bus comes in one-hour intervals while the B bus also comes in regular intervals of a different length. When grandma rests on the bench by the bus stop, one A bus and two B buses come by. Later, grandpa rests on the same bench and eight A buses come by. The minimum number of B buses that must have come by during that time is

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To minimize the number of B buses coming by, we stretch the length of their intervals as far as possible. Suppose grandma sees the A bus that comes at $10$ : $00$ . Then she does not see the ones that come at $9$ : $00$ or $11$ : $00$ . Thus the length of the interval between two B buses is strictly less than $2$ hours. The two B buses grandma sees may have come at $9$ : $01$ and $10$ : $59$ , in which case the interval is of length $118$ minutes. Grandpa is on the bench for at least $7$ hours. This is longer than three intervals for the B buses, so he must have seen at least $3$ of them. Suppose he sees the A buses at $11$ : $00$ , $12$ : $00$ , $13$ : $00$ , $14$ : $00$ , $15$ : $00$ , $16$ : $00$ , $17$ : $00$ and $18$ : $00$ . Then he will see only the B buses at $12$ : $57$ , $14$ : $55$ and $16$ : $53$ . The answer is (A).

University of Alberta

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