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HMMT 十一月 2013 · GEN 赛 · 第 8 题

HMMT November 2013 — GEN Round — Problem 8

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 6 ] How many of the first 1000 positive integers can be written as the sum of finitely many distinct
    0 1 2
    numbers from the sequence 3 , 3 , 3 , . . . ?
    √ √
    °
解析

英文解析

  1. [ 6 ] How many of the first 1000 positive integers can be written as the sum of finitely many distinct
    0 1 2
    numbers from the sequence 3 , 3 , 3 , . . . ?
    Answer: 105 We want to find which integers have only 0’s and 1’s in their base 3 representation.
    Note that 1000 = 1101001 . We can construct a bijection from all such numbers to the binary strings,
    10 3
    by mapping x ↔ x . Since 1101001 = 105 , we conclude that the answer is 105.
    3 2 2 10
    √ √
    °