{"status": "success", "data": {"description_md": "Let $k$ be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with $k+1$ digits. Every time Bernardo writes a number, Silvia erases the last $k$ digits of it. Bernardo then writes the next perfect square, Silvia erases the last $k$ digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let $f(k)$ be the smallest positive integer not written on the board. For example, if $k = 1$, then the numbers that Bernardo writes are $16, 25, 36, 49, 64$, and the numbers showing on the board after Silvia erases are $1, 2, 3, 4,$ and $6$, and thus $f(1) = 5$. What is the sum of the digits of $f(2) + f(4)+ f(6) + ... + f(2016)$?\n\n$\\textbf{(A)}\\ 7986\\qquad\\textbf{(B)}\\ 8002\\qquad\\textbf{(C)}\\ 8030\\qquad\\textbf{(D)}\\ 8048\\qquad\\textbf{(E)}\\ 8064$\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\">k</span>  be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with  <span class=\"katex--inline\">k+1</span>  digits. Every time Bernardo writes a number, Silvia erases the last  <span class=\"katex--inline\">k</span>  digits of it. Bernardo then writes the next perfect square, Silvia erases the last  <span class=\"katex--inline\">k</span>  digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let  <span class=\"katex--inline\">f(k)</span>  be the smallest positive integer not written on the board. For example, if  <span class=\"katex--inline\">k = 1</span> , then the numbers that Bernardo writes are  <span class=\"katex--inline\">16, 25, 36, 49, 64</span> , and the numbers showing on the board after Silvia erases are  <span class=\"katex--inline\">1, 2, 3, 4,</span>  and  <span class=\"katex--inline\">6</span> , and thus  <span class=\"katex--inline\">f(1) = 5</span> . What is the sum of the digits of  <span class=\"katex--inline\">f(2) + f(4)+ f(6) + ... + f(2016)</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 7986\\qquad\\textbf{(B)}\\ 8002\\qquad\\textbf{(C)}\\ 8030\\qquad\\textbf{(D)}\\ 8048\\qquad\\textbf{(E)}\\ 8064</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": "2016 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/16_amc12A_p24"}}