{"status": "success", "data": {"description_md": "For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?\n\n$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 14 \\qquad \\textbf{(C) } 16 \\qquad \\textbf{(D) } 18 \\qquad \\textbf{(E) } 20$\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>For a positive integer  <span class=\"katex--inline\">n</span>  and nonzero digits  <span class=\"katex--inline\">a</span> ,  <span class=\"katex--inline\">b</span> , and  <span class=\"katex--inline\">c</span> , let  <span class=\"katex--inline\">A_n</span>  be the  <span class=\"katex--inline\">n</span> -digit integer each of whose digits is equal to  <span class=\"katex--inline\">a</span> ; let  <span class=\"katex--inline\">B_n</span>  be the  <span class=\"katex--inline\">n</span> -digit integer each of whose digits is equal to  <span class=\"katex--inline\">b</span> , and let  <span class=\"katex--inline\">C_n</span>  be the  <span class=\"katex--inline\">2n</span> -digit (not  <span class=\"katex--inline\">n</span> -digit) integer each of whose digits is equal to  <span class=\"katex--inline\">c</span> . What is the greatest possible value of  <span class=\"katex--inline\">a + b + c</span>  for which there are at least two values of  <span class=\"katex--inline\">n</span>  such that  <span class=\"katex--inline\">C_n - B_n = A_n^2</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 12 \\qquad \\textbf{(B) } 14 \\qquad \\textbf{(C) } 16 \\qquad \\textbf{(D) } 18 \\qquad \\textbf{(E) } 20</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": "2018 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/18_amc12A_p24"}}