返回题库

HMMT 二月 2002 · 冲刺赛 · 第 29 题

HMMT February 2002 — Guts Round — Problem 29

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [8] A sequence s , s , s , s , . . . is defined by s = s = 1 and, for every positive integer
    0 1 2 3 0 1
    n , s = s , s = s , s = s + s /s . What is the value of s ?2
    2 n n 4 n +1 2 n +1 4 n − 1 2 n − 1 n − 1 1000
    2 n − 1
解析

英文解析

  1. A sequence s , s , s , s , . . . is defined by s = s = 1 and, for every positive integer
    0 1 2 3 0 1
    n , s = s , s = s , s = s + s /s . What is the value of s ?2
    2 n n 4 n +1 2 n +1 4 n − 1 2 n − 1 n − 1 1000
    2 n − 1
    Solution: 720 Some experimentation with small values may suggest that s = k !, wherenk is the number of ones in the binary representation of n , and this formula is in fact provableby a straightforward induction. Since 1000 = 1111101000 , with six ones, s = 6! = 720.
    10 2 1000