{"status": "success", "data": {"description_md": "The number $a = \\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying$$\\lfloor x \\rfloor \\cdot \\{x\\} = a \\cdot x^2$$is $420$, where $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x$ and $\\{x\\} = x - \\lfloor x \\rfloor$ denotes the fractional part of $x$. What is $p + q?$\n\n$\\textbf{(A) } 245 \\qquad \\textbf{(B) } 593 \\qquad \\textbf{(C) } 929 \\qquad \\textbf{(D) } 1331 \\qquad \\textbf{(E) } 1332$\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>The number  <span class=\"katex--inline\">a = \\tfrac{p}{q}</span> , where  <span class=\"katex--inline\">p</span>  and  <span class=\"katex--inline\">q</span>  are relatively prime positive integers, has the property that the sum of all real numbers  <span class=\"katex--inline\">x</span>  satisfying <span class=\"katex--display\">\\lfloor x \\rfloor \\cdot \\{x\\} = a \\cdot x^2</span> is  <span class=\"katex--inline\">420</span> , where  <span class=\"katex--inline\">\\lfloor x \\rfloor</span>  denotes the greatest integer less than or equal to  <span class=\"katex--inline\">x</span>  and  <span class=\"katex--inline\">\\{x\\} = x - \\lfloor x \\rfloor</span>  denotes the fractional part of  <span class=\"katex--inline\">x</span> . What is  <span class=\"katex--inline\">p + q?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 245 \\qquad \\textbf{(B) } 593 \\qquad \\textbf{(C) } 929 \\qquad \\textbf{(D) } 1331 \\qquad \\textbf{(E) } 1332</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": "2020 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/20_amc12A_p24"}}