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HMMT 二月 2004 · 代数 · 第 3 题

HMMT February 2004 — Algebra — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Compute: ⌊ ⌋ 3 3 2005 2003 − . 2003 · 2004 2004 · 2005
解析
  1. Compute: ⌊ ⌋ 3 3 2005 2003 − . 2003 · 2004 2004 · 2005 Solution: 8 Let x = 2004. Then the expression inside the floor brackets is 3 3 4 4 3 ( x + 1) ( x − 1) ( x + 1) − ( x − 1) 8 x + 8 x 16 x − = = = 8 + . 3 3 ( x − 1) x x ( x + 1) ( x − 1) x ( x + 1) x − x x − x 3 Since x is certainly large enough that 0 < 16 x/ ( x − x ) < 1, the answer is 8. 1