AMC12 2009 A

AMC12 2009 A · Q24

AMC12 2009 A · Q24. It mainly tests Exponents & radicals, Patterns & sequences (misc).

The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ for which \[\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}\] is defined?

二的塔函数递归定义如下：$T(1) = 2$ 且对 $n\ge1$ 有 $T(n + 1) = 2^{T(n)}$。令 $A = (T(2009))^{T(2009)}$，$B = (T(2009))^A$。求最大的整数 $k$，使得 \[\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ 次}}\] 有定义。

(A) 2009 2009

(B) 2010 2010

(D) 2012 2012

(E) 2013 2013

Answer

Correct choice: (E)

正确答案：（E）

Solution

Testing the first two (or three) positive integers instead of 2009, $k$ seems to always be 4 more. Put E and go on to tackle #25 :)

把 2009 换成前两个（或三个）正整数试一下，$k$ 似乎总是多 4。选 E 然后继续做第 25 题吧 :)

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