HMMT 二月 2002 · 团队赛 · 第 5 题

英文解析

5. [20] Prove for integers n that
   ⌊ ⌋ ⌊ ⌋ ⌊ ⌋
   n n + 1 n 2 = .
   2 2 4
   Solution. Suppose n = 2 m is even; then b n/ 2 c = b m c = m and b ( n + 1) / 2 c = b m + 1 / 2 c = m ,
   ⌊ ⌋ ⌊ ⌋
   2 2 2
   whose product is m = m = (2 m ) / 4 . Otherwise n = 2 m + 1 is odd, so that b n/ 2 c =
   b m + 1 / 2 c = m and b ( n + 1) / 2 c = b m + 1 c = m + 1, whose product is m + m . On the other side,2
   we find that
   ⌊ ⌋ ⌊ ⌋ ⌊ ⌋
   2 2
   n 4 m + 4 m + 1 1
   2 2 = = m + m + = m + m,
   4 4 4
   as desired.
   In problems 6–7 you may use without proof the known summations
   L L
   ∑ ∑
   3 2 2
   n = n ( n + 1) / 2 and n = n ( n + 1) / 4 for positive integers L .
   n =1 n =1
   ∑ L