HMMT 十一月 2017 · 团队赛 · 第 3 题

HMMT November 2017 — Team Round — Problem 3

[ 25 ] Michael writes down all the integers between 1 and N inclusive on a piece of paper and discovers that exactly 40% of them have leftmost digit 1. Given that N > 2017, find the smallest possible value of N .

解析

[ 25 ] Michael writes down all the integers between 1 and N inclusive on a piece of paper and discovers that exactly 40% of them have leftmost digit 1. Given that N > 2017, find the smallest possible value of N . Proposed by: Michael Tang Answer: 1 , 481 , 480 Let d be the number of digits of N . Suppose that N does not itself have leftmost digit 1. Then the number of integers 1 , 2 , . . . , N which have leftmost digit 1 is d 10 − 1 2 d − 1 1 + 10 + 10 + . . . + 10 = , 9 d 10 − 1 2 N d so we must have = , or 5(10 − 1) = 18 N . But the left-hand side is odd, so this is impossible. 9 5 Thus N must have leftmost digit 1. In this case, the number of integers 1 , 2 , . . . , N which have leftmost digit 1 is 2 d − 2 d − 1 1 + 10 + 10 + . . . + 10 + ( N − 10 + 1) d − 1 10 − 1 d − 1 = + N − 10 + 1 9 ( ) d − 1 10 − 1 = N − 8 . 9 ( ) ( ) d − 1 d − 1 d − 1 10 − 1 2 40 10 − 1 10 − 1 Therefore we need N − 8 = N , or N = . Then, must be divisible by 9 5 3 9 9 d − 1 10 − 1