{"status": "success", "data": {"description_md": "Consider the set of $30$ parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\\in \\{-2,-1,0,1,2\\}$ and $b\\in \\{-3,-2,-1,1,2,3\\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?\n\n$\\textbf{(A)}\\ 720\\qquad\\textbf{(B)}\\ 760\\qquad\\textbf{(C)}\\ 810\\qquad\\textbf{(D)}\\ 840\\qquad\\textbf{(E)}\\ 870$\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>Consider the set of  <span class=\"katex--inline\">30</span>  parabolas defined as follows: all parabolas have as focus the point  <span class=\"katex--inline\">(0,0)</span>  and the directrix lines have the form  <span class=\"katex--inline\">y=ax+b</span>  with  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  integers such that  <span class=\"katex--inline\">a\\in \\{-2,-1,0,1,2\\}</span>  and  <span class=\"katex--inline\">b\\in \\{-3,-2,-1,1,2,3\\}</span> . No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 720\\qquad\\textbf{(B)}\\ 760\\qquad\\textbf{(C)}\\ 810\\qquad\\textbf{(D)}\\ 840\\qquad\\textbf{(E)}\\ 870</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": "2013 AMC 12B Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/13_amc12B_p22", "prev": "/problem/13_amc12B_p20"}}