HMMT 二月 2003 · 代数 · 第 9 题

英文解析

9. For how many integers n , for 1 ≤ n ≤ 1000, is the number even?
   2 n
   Solution: 990
   ( )
   2 n
   In fact, the expression is always even, and it is not a multiple of four if and onlynif n is a power of 2, and there are 10 powers of 2 between 1 and 1000.
   Let f ( N ) denote the number of factors of 2 in N . Thus,
   ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ ⌊ ⌋
   ∞
   ∑
   n n n nf ( n !) = + + + · · · = .
   2 4 8 2 kk =1
   Also, it is clear that f ( ab ) = f ( a ) + f ( b ) and f ( ) = f ( a ) − f ( b ) for integers a, b . Nowabm m +1
   for any positive integer n , let m be the integer such that 2 ≤ n < 2 . Then
   (( )) ( ) ( )
   ⌊ ⌋ ⌊ ⌋
   ∞ ∞
   ∑ ∑
   2 n (2 n )! 2 n nf = f = − 2
   k kn n ! n ! 2 2
   k =1 k =1
   ( )
   ⌊ ⌋ ⌊ ⌋
   ∞ ∞
   ∑ ∑
   n n = − 2
   k − 1 k
   2 2
   k =1 k =1
   ( )
   ⌊ ⌋
   ∞
   ∑
   = b n c −nkk =12
   ( )
   ⌊ ⌋
   ∑mn = n −
   2 kk =1
   ( )
   ∑mn
   ≥ n −
   2 kk =1
   ( )
   2 − 1 nm = n − n = ≥ 1 .
   m m
   2 2
   ( )
   2 nm
   Both equalities hold when n = 2 , and otherwise, f ( ) > 1. n