HMMT 十一月 2020 · 冲刺赛 · 第 32 题

HMMT November 2020 — Guts Round — Problem 32

[17] The numbers 1 , 2 , . . . , 10 are written in a circle. There are four people, and each person randomly selects five consecutive integers (e.g. 1 , 2 , 3 , 4 , 5, or 8 , 9 , 10 , 1 , 2). If the probability that there exists some number that was not selected by any of the four people is p , compute 10000 p . AE 1

解析

[17] The numbers 1 , 2 , . . . , 10 are written in a circle. There are four people, and each person randomly selects five consecutive integers (e.g. 1 , 2 , 3 , 4 , 5, or 8 , 9 , 10 , 1 , 2). If the probability that there exists some number that was not selected by any of the four people is p , compute 10000 p . Proposed by: Hahn Lheem Answer: 3690 Solution: The unselected numbers must be consecutive. Suppose that { 1 , 2 , . . . , k } are the unselected numbers for some k . In this case, 1 cannot be selected, so there are 5 possible sets of consecutive numbers the people could 4 4 have chosen. This leads to 5 possibilities. Moreover, 10 must be selected, so we must subtract 4 possibilities where neither 1 nor 10 are selected. 4 4 10(5 − 4 ) 3690 Therefore, accounting for the rotation of the unselected numbers, we find p = = . 4 10 10000 AE 1