HMMT 二月 2005 · GEN2 赛 · 第 10 题

HMMT February 2005 — GEN2 Round — Problem 10

Let b x c denote the greatest integer less than or equal to x . How many positive integers less than 2005 can be expressed in the form b x b x cc for some positive real x ? 1

解析

Let b x c denote the greatest integer less than or equal to x . How many positive integers less than 2005 can be expressed in the form b x b x cc for some positive real x ? Solution: 990 Let { x } = x − b x c be the fractional part of x . Note that 2 b x b x cc = b ( b x c + { x } ) b x cc = b x c + b{ x } b x cc . Because { x } may take on any value in the half-open interval [0 , 1), the quantity b{ x } b x cc can take on any integer value between 0 and b x c − 1, inclusive. 2 2 2 If b x c = n , then b x b x cc can be any of the numbers n , n + 1 , . . . , n + n − 1. In other words, there are precisely n possible values that b x b x cc can take, and moreover, all of 2 2 2 them are less than ( n + 1) . Because 44 + 43 = 1979 < 2005 and 45 = 2025 > 2005, n can range between 1 and 44, inclusive. Therefore, the answer is 44 ∑ 44 · 45 n = = 990 . 2 n =1 4