{"status": "success", "data": {"description_md": "Two parabolas have equations $y= x^2 + ax +b$ and $y= x^2 + cx +d$, where $a$, $b$, $c$, and $d$ are integers, each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have at least one point in common?\n\n$\\textbf{(A)}\\ \\frac{1}{2}\\qquad\\textbf{(B)}\\ \\frac{25}{36}\\qquad\\textbf{(C)}\\ \\frac{5}{6}\\qquad\\textbf{(D)}\\ \\frac{31}{36}\\qquad\\textbf{(E)}\\ 1$\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>Two parabolas have equations  <span class=\"katex--inline\">y= x^2 + ax +b</span>  and  <span class=\"katex--inline\">y= x^2 + cx +d</span> , where  <span class=\"katex--inline\">a</span> ,  <span class=\"katex--inline\">b</span> ,  <span class=\"katex--inline\">c</span> , and  <span class=\"katex--inline\">d</span>  are integers, each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have at least one point in common?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{1}{2}\\qquad\\textbf{(B)}\\ \\frac{25}{36}\\qquad\\textbf{(C)}\\ \\frac{5}{6}\\qquad\\textbf{(D)}\\ \\frac{31}{36}\\qquad\\textbf{(E)}\\ 1</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": 3, "problem_name": "2012 AMC 12B Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12B_p14", "prev": "/problem/12_amc12B_p12"}}