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HMMT 十一月 2015 · 冲刺赛 · 第 23 题

HMMT November 2015 — Guts Round — Problem 23

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 12 ] Compute the smallest positive integer n for which √ √ 1 4 4 0 < n − b n c < . 2015
解析
  1. [ 12 ] Compute the smallest positive integer n for which √ √ 1 4 4 0 < n − b n c < . 2015 Proposed by: Alexander Katz Answer: 4097 4 3 2 Let n = a + b where a, b are integers and 0 < b < 4 a + 6 a + 4 a + 1. Then √ √ 1 4 4 n − b n c < 2015 √ 1 4 4 a + b − a < 2015 √ 1 4 4 a + b < a + 2015 ( ) 4 1 4 a + b < a + 2015 3 2 4 a 6 a 4 a 1 4 4 a + b < a + + + + 2 3 4 2015 2015 2015 2015 4 To minimize n = a + b , we clearly should minimize b , which occurs at b = 1. Then 3 2 4 a 6 a 4 a 1 1 < + + + . 2 3 4 2015 2015 2015 2015 2 3 2 3 6 a 4 a 1 1 4 a 6 a 4 a 1 4 · 7 +3 If a = 7, then , , < , so + + + < < 1, so a ≥ 8. When 2 3 4 2 3 4 2015 2015 2015 2015 2015 2015 2015 2015 2015 3 4 a 2048 a = 8, we have = > 1, so a = 8 is the minimum. 2015 2015 4 Hence, the minimum n is 8 + 1 = 4097 .