{"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<img src=\"https://latex.artofproblemsolving.com/0/0/6/006ecf77f807aa89b66a8e2e9f281e579e7cd062.png\" width=\"500\">\n\nRearranging 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;<img src=\"https://latex.artofproblemsolving.com/0/0/6/006ecf77f807aa89b66a8e2e9f281e579e7cd062.png\" width=\"500\"/>&#10;<p>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/>&#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": "2012 AMC 12A Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12A_p25", "prev": "/problem/12_amc12A_p23"}}