HMMT 十一月 2012 · THM 赛 · 第 3 题

HMMT November 2012 — THM Round — Problem 3

[ 4 ] Find the smallest positive integer n such that 2 > 3 . (The notation 2 is used to denote ︸︷︷︸︸︷︷︸ n n 2 · 2 · 2 2 2 a power tower with n 2’s. For example, 2 with n = 4 would equal 2 .) ︸︷︷︸ n 2 2 ( x +2 x +3) 2 ( x +2 x +3)

解析

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