HMMT 二月 2005 · GEN2 赛 · 第 10 题

英文解析

10. Let b x c denote the greatest integer less than or equal to x . How many positive integersless 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 thatb x b x cc = b ( b x c + { x } ) b x cc = b x c + b{ x } b x cc .2
    Because { x } may take on any value in the half-open interval [0 , 1), the quantityb{ 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 otherwords, 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 .
    n =12 4