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HMMT 十一月 2012 · THM 赛 · 第 3 题

HMMT November 2012 — THM Round — Problem 3

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

[4] Find the smallest positive integer nsuch that 22··2
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
n>3333
. (The notation 22··2
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
nis used to denote
a power tower with n2’s. For example, 22··2
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
nwithn= 4 would equal 2222
.)

解析

英文解析

  1. [ 4 ] Find the smallest positive integer n such that 2 > 3 . (The notation 2 is used to denote
    ︸ ︷︷ ︸ ︸ ︷︷ ︸
    n n
    ·2
    ·2
    a power tower with n 2’s. For example, 2 with n = 4 would equal 2 .)222
    ︸ ︷︷ ︸
    Answer: 6 Clearly, n ≥ 5. When we take n = 5, we haven
    2 32
    2 16 27 3
    2 2 3 3
    2 = 2 < 3 = 3 .
    On the other hand, when n = 6, we have
    2 322
    2 65536 65535 27 27 3
    2 2 2 4 3 3
    2 = 2 = 4 > 4 > 3 = 3 .
    Our answer is thus n = 6.
    2 ( x +2 x +3)2
    2 ( x +2 x +3)