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HMMT 十一月 2015 · 冲刺赛 · 第 23 题

HMMT November 2015 — Guts Round — Problem 23

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 12 ] Compute the smallest positive integer n for which
    √ √
    4 41
    0 < n − b n c < .
    2015
解析

英文解析

  1. [ 12 ] Compute the smallest positive integer n for which
    √ √
    4 41
    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
    √ √
    4 41
    n − b n c <
    2015
    √
    a + b − a <441
    2015
    √
    a + b < a +441
    2015
    ( )
    a + b < a +414
    2015
    3 2
    4 a 6 a 4 a 1
    4 4
    a + b < a + + + +
    2 3 4
    2015 2015 2015 2015
    To minimize n = a + b , we clearly should minimize b , which occurs at b = 1. Then 4
    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
    4 a 20483
    a = 8, we have = > 1, so a = 8 is the minimum.
    2015 2015
    Hence, the minimum n is 8 + 1 = 4097 .4