{"status": "success", "data": {"description_md": "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 $k$ such that\n\n$$\\log_2\\log_2\\log_2\\ldots\\log_2B_{k}$$where there are $k$ $\\log_2$'s, is defined?\n\n$\\textbf{(A)}\\ 2009\\qquad \\textbf{(B)}\\ 2010\\qquad \\textbf{(C)}\\ 2011\\qquad \\textbf{(D)}\\ 2012\\qquad \\textbf{(E)}\\ 2013$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>The &#8216;&#8216;tower function of twos&#8217;&#8217; is defined recursively as follows: <span class=\"katex--inline\">T(1) = 2</span> and <span class=\"katex--inline\">T(n + 1) = 2^{T(n)}</span> for <span class=\"katex--inline\">n\\ge1</span>. Let <span class=\"katex--inline\">A = (T(2009))^{T(2009)}</span> and <span class=\"katex--inline\">B = (T(2009))^A</span>. What is the largest integer <span class=\"katex--inline\">k</span> such that</p>&#10;<p><span class=\"katex--display\">\\log_2\\log_2\\log_2\\ldots\\log_2B_{k}</span>where there are <span class=\"katex--inline\">k</span> <span class=\"katex--inline\">\\log_2</span>'s, is defined?</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}\\ 2009\\qquad \\textbf{(B)}\\ 2010\\qquad \\textbf{(C)}\\ 2011\\qquad \\textbf{(D)}\\ 2012\\qquad \\textbf{(E)}\\ 2013</span></p>&#10;<hr/>&#10;<p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2009 AMC 12A Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/09_amc12A_p25", "prev": "/problem/09_amc12A_p23"}}