HMMT 二月 2002 · CALC 赛 · 第 7 题
HMMT February 2002 — CALC Round — Problem 7
题目详情
- 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 〈 〉 87 the same probability of being picked, and N is calculated. This procedure is repeated 303 M times and the average value A ( M ) is obtained. What is lim A ( M )? M →∞ √ ( 2 − 1) / 2 ∫ d x √
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 〈 〉 87 having the same probability of being picked, and N is calculated. This procedure is 303 repeated M times and the average value A ( M ) is obtained. What is lim A ( M )? M →∞ Solution: This method of picking N is equivalent to uniformly randomly selecting a 〈 〉 87 positive integer. Call this the average value of N for N a positive integer. In lowest 303 87 29 0 1 100 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 √