HMMT 二月 2002 · CALC 赛 · 第 7 题
HMMT February 2002 — CALC Round — Problem 7
题目详情
英文原题
- Denote by 〈 x 〉 the fractional part of the real number x (for instance, 〈 3 . 2 〉 = 0 . 2). Apositive integer N is selected randomly from the set { 1 , 2 , 3 , . . . , M } , with each integer having
〈 〉
the same probability of being picked, and N is calculated. This procedure is repeated 87
M times and the average value A ( M ) is obtained. What is lim A ( M )?303
M →∞
√
( 2 − 1) / 2
∫
d x
√
解析
英文解析
- Denote by 〈 x 〉 the fractional part of the real number x (for instance, 〈 3 . 2 〉 = 0 . 2).
A positive integer N is selected randomly from the set { 1 , 2 , 3 , . . . , M } , with each integer
〈 〉
having the same probability of being picked, and N is calculated. This procedure is 87
repeated M times and the average value A ( M ) is obtained. What is lim A ( M )?303
M →∞
Solution: This method of picking N is equivalent to uniformly randomly selecting a
〈 〉
positive integer. Call this the average value of N for N a positive integer. In lowest 87
87 29 0 1 100303
terms, = , so the answer is the same as the average value of , , . . . , , which is
303 101 101 101 101
100 · 101 / 2
1+2+ ··· +100 50 = = .
101 · 101 101 · 101 101
√
( 2 − 1) / 2
∫
d x
√