HMMT 十一月 2013 · GEN 赛 · 第 8 题
HMMT November 2013 — GEN Round — Problem 8
题目详情
- [ 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
- [ 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 √ √ ◦