{"status": "success", "data": {"description_md": "The parallelogram bounded by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$, and $y=bx+d$ has area $18$. The parallelogram bounded by the lines $y=ax+c$, $y=ax-d$, $y=bx+c$, and $y=bx-d$ has area $72$. Given that $a$, $b$, $c$, and $d$ are positive integers, what is the smallest possible value of $a+b+c+d$?\n\n$\\mathrm {(A)} 13\\qquad \\mathrm {(B)} 14\\qquad \\mathrm {(C)} 15\\qquad \\mathrm {(D)} 16\\qquad \\mathrm {(E)} 17$\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 parallelogram bounded by the lines  <span class=\"katex--inline\">y=ax+c</span> ,  <span class=\"katex--inline\">y=ax+d</span> ,  <span class=\"katex--inline\">y=bx+c</span> , and  <span class=\"katex--inline\">y=bx+d</span>  has area  <span class=\"katex--inline\">18</span> . The parallelogram bounded by the lines  <span class=\"katex--inline\">y=ax+c</span> ,  <span class=\"katex--inline\">y=ax-d</span> ,  <span class=\"katex--inline\">y=bx+c</span> , and  <span class=\"katex--inline\">y=bx-d</span>  has area  <span class=\"katex--inline\">72</span> . Given that  <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 positive integers, what is the smallest possible value of  <span class=\"katex--inline\">a+b+c+d</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm {(A)} 13\\qquad \\mathrm {(B)} 14\\qquad \\mathrm {(C)} 15\\qquad \\mathrm {(D)} 16\\qquad \\mathrm {(E)} 17</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": "2007 AMC 12B Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/07_amc12B_p21", "prev": "/problem/07_amc12B_p19"}}