HMMT 二月 2000 · GEN 赛 · 第 8 题

HMMT February 2000 — GEN Round — Problem 8

Bob o the lo wn w as juggling his spheri al o ws again when he realized that when he drops a o w is related to ho w man y o ws he started o juggling. If he juggles 1, he drops it after 64 se onds. When juggling 2, he drops one after 55 se onds, and the other 55 se onds later. In fa t, he w as able to reate the follo wing table: o ws started juggling 1 2 3 4 5 6 7 8 9 10 11 se onds he drops after 64 55 47 40 33 27 22 18 14 13 12 o ws started juggling 12 13 14 15 16 17 18 19 20 21 22 se onds he drops after 11 10 9 8 7 6 5 4 3 2 1 He an only juggle up to 22 o ws. T o juggle the o ws the longest, what n um b er of o ws should he start o juggling? Ho w long (in min utes) an he juggle for?

解析

The time that Bob o an juggle is the n um b er of o ws times se onds. So, w e get the follo wing table: o ws started juggling 1 2 3 4 5 6 7 8 9 10 11 total time 64 110 141 160 165 162 154 144 126 130 132 o ws started juggling 12 13 14 15 16 17 18 19 20 21 22 total time 132 130 126 120 112 102 90 76 60 42 22 Th us, w e see that the maxim um o urs with 5 o ws , and the total time is 165 se onds 3 = 2 min utes . 4 x