{"status": "success", "data": {"description_md": "Let $a$, $b$, and $c$ be positive integers with $a\\ge$ $b\\ge$ $c$ such that\n\n$a^{2}-b^{2}-c^{2}+ab=2011$\n$a^{2}+3b^{2}+3c^{2}-3ab-2ac-2bc=-1997$\n\nWhat is $a$?\n\n$\\textbf{(A)}\\ 249\\qquad\\textbf{(B)}\\ 250\\qquad\\textbf{(C)}\\ 251\\qquad\\textbf{(D)}\\ 252\\qquad\\textbf{(E)}\\ 253$\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</span>, <span class=\"katex--inline\">b</span>, and <span class=\"katex--inline\">c</span> be positive integers with <span class=\"katex--inline\">a\\ge</span> <span class=\"katex--inline\">b\\ge</span> <span class=\"katex--inline\">c</span> such that</p>&#10;<p><span class=\"katex--inline\">a^{2}-b^{2}-c^{2}+ab=2011</span><br/>&#10;<span class=\"katex--inline\">a^{2}+3b^{2}+3c^{2}-3ab-2ac-2bc=-1997</span></p>&#10;<p>What is <span class=\"katex--inline\">a</span>?</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}\\ 249\\qquad\\textbf{(B)}\\ 250\\qquad\\textbf{(C)}\\ 251\\qquad\\textbf{(D)}\\ 252\\qquad\\textbf{(E)}\\ 253</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 21", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12A_p22", "prev": "/problem/12_amc12A_p20"}}