{"status": "success", "data": {"description_md": "Let $\\{a_k\\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general, \n\n$$a_k=\\begin{cases}<br>(0.\\underbrace{20101\\cdots 0101}_{k+2\\text{ digits}})^{a_{k-1}} & \\text{if }k\\text{ is odd,}\\\\<br>(0.\\underbrace{20101\\cdots 01011}_{k+2\\text{ digits}})^{a_{k-1}}& \\text{if }k\\text{ is even.}<br>\\end{cases}$$<br>Rearranging the numbers in the sequence  $\\{a_k\\}_{k=1}^{2011}$ in decreasing order produces a new sequence  $\\{b_k\\}_{k=1}^{2011}$.  What is the sum of all integers $k$, $1\\le k \\le 2011$, such that $a_k=b_k?$\n\n$\\textbf{(A)}\\ 671\\qquad\\textbf{(B)}\\ 1006\\qquad\\textbf{(C)}\\ 1341\\qquad\\textbf{(D)}\\ 2011\\qquad\\textbf{(E)}\\ 2012$\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>Let  <span class=\"katex--inline\">\\{a_k\\}_{k=1}^{2011}</span>  be the sequence of real numbers defined by  <span class=\"katex--inline\">a_1=0.201,</span>   <span class=\"katex--inline\">a_2=(0.2011)^{a_1},</span>   <span class=\"katex--inline\">a_3=(0.20101)^{a_2},</span>   <span class=\"katex--inline\">a_4=(0.201011)^{a_3}</span> , and in general,</p>&#10;<p> <span class=\"katex--display\">a_k=\\begin{cases}(0.\\underbrace{20101\\cdots 0101}_{k+2\\text{ digits}})^{a_{k-1}} &amp; \\text{if }k\\text{ is odd,}\\\\(0.\\underbrace{20101\\cdots 01011}_{k+2\\text{ digits}})^{a_{k-1}}&amp; \\text{if }k\\text{ is even.}\\end{cases}</span> <br/>Rearranging the numbers in the sequence   <span class=\"katex--inline\">\\{a_k\\}_{k=1}^{2011}</span>  in decreasing order produces a new sequence   <span class=\"katex--inline\">\\{b_k\\}_{k=1}^{2011}</span> .  What is the sum of all integers  <span class=\"katex--inline\">k</span> ,  <span class=\"katex--inline\">1\\le k \\le 2011</span> , such that  <span class=\"katex--inline\">a_k=b_k?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 671\\qquad\\textbf{(B)}\\ 1006\\qquad\\textbf{(C)}\\ 1341\\qquad\\textbf{(D)}\\ 2011\\qquad\\textbf{(E)}\\ 2012</span> </p>&#10;<hr><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>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2012 AMC 12A Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12A_p25", "prev": "/problem/12_amc12A_p23"}}