{"status": "success", "data": {"description_md": "The solutions to the equations $z^2=4+4\\sqrt{15}i$ and $z^2=2+2\\sqrt 3i,$ where $i=\\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\\sqrt q-r\\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p+q+r+s?$\n\n$\\textbf{(A) } 20 \\qquad <br>\\textbf{(B) } 21 \\qquad <br>\\textbf{(C) } 22 \\qquad <br>\\textbf{(D) } 23 \\qquad <br>\\textbf{(E) } 24$\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 solutions to the equations  <span class=\"katex--inline\">z^2=4+4\\sqrt{15}i</span>  and  <span class=\"katex--inline\">z^2=2+2\\sqrt 3i,</span>  where  <span class=\"katex--inline\">i=\\sqrt{-1},</span>  form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form  <span class=\"katex--inline\">p\\sqrt q-r\\sqrt s,</span>  where  <span class=\"katex--inline\">p,</span>   <span class=\"katex--inline\">q,</span>   <span class=\"katex--inline\">r,</span>  and  <span class=\"katex--inline\">s</span>  are positive integers and neither  <span class=\"katex--inline\">q</span>  nor  <span class=\"katex--inline\">s</span>  is divisible by the square of any prime number. What is  <span class=\"katex--inline\">p+q+r+s?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 20 \\qquad \\textbf{(B) } 21 \\qquad \\textbf{(C) } 22 \\qquad \\textbf{(D) } 23 \\qquad \\textbf{(E) } 24</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 22", "can_next": true, "can_prev": true, "nxt": "/problem/18_amc12A_p23", "prev": "/problem/18_amc12A_p21"}}