HMMT 二月 2002 · 团队赛 · 第 6 题

HMMT February 2002 — Team Round — Problem 6

[20] For positive integers L , let S = b n/ 2 c . Determine all L for which S is a square L L n =1 number. ⌊ ⌋ ∑ L 3

解析

[20] For positive integers L , let S = b n/ 2 c . Determine all L for which S is a square L L n =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) . L 2 2 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 not 2 square. Now suppose L = 2 k is even, where k ≥ 1. Then S = S + k = k is always square. L L − 1 Hence S is square exactly when L = 1 or L is even . L ⌊ ⌋ ∑ L 3