HMMT 二月 2002 · 团队赛 · 第 6 题

英文解析

6. [20] For positive integers L , let S = b n/ 2 c . Determine all L for which S is a square
   L Ln =1
   number.
   Solution. We distinguish two cases depending on the parity of L . Suppose first that L = 2 k − 1
   is odd, where k ≥ 1. Then
   ⌊ ⌋
   ∑ ∑
   n k ( k − 1)
   S = = 2 m = 2 · = k ( k − 1) .
   2 2L
   1 ≤ n ≤ 2 k − 1 0 ≤ m<k
   2 2
   If k = 1, this is the square number 0. If k > 1 then ( k − 1) < k ( k − 1) < k , so k ( k − 1) is notsquare. Now suppose L = 2 k is even, where k ≥ 1. Then S = S + k = k is always square.2
   L L − 1
   Hence S is square exactly when L = 1 or L is even .
   ⌊ ⌋L
   ∑
   3L