{"status": "success", "data": {"description_md": "Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\\tfrac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\\tfrac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\\overline{PQ}$. What is $AR+BR+CR+DR$ ?\n\n$\\textbf{(A)}\\; 180 \\qquad\\textbf{(B)}\\; 184 \\qquad\\textbf{(C)}\\; 188 \\qquad\\textbf{(D)}\\; 192\\qquad\\textbf{(E)}\\; 196$\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>Four circles, no two of which are congruent, have centers at  <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> , and points  <span class=\"katex--inline\">P</span>  and  <span class=\"katex--inline\">Q</span>  lie on all four circles. The radius of circle  <span class=\"katex--inline\">A</span>  is  <span class=\"katex--inline\">\\tfrac{5}{8}</span>  times the radius of circle  <span class=\"katex--inline\">B</span> , and the radius of circle  <span class=\"katex--inline\">C</span>  is  <span class=\"katex--inline\">\\tfrac{5}{8}</span>  times the radius of circle  <span class=\"katex--inline\">D</span> . Furthermore,  <span class=\"katex--inline\">AB = CD = 39</span>  and  <span class=\"katex--inline\">PQ = 48</span> . Let  <span class=\"katex--inline\">R</span>  be the midpoint of  <span class=\"katex--inline\">\\overline{PQ}</span> . What is  <span class=\"katex--inline\">AR+BR+CR+DR</span>  ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\; 180 \\qquad\\textbf{(B)}\\; 184 \\qquad\\textbf{(C)}\\; 188 \\qquad\\textbf{(D)}\\; 192\\qquad\\textbf{(E)}\\; 196</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": "2015 AMC 12B Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/15_amc12B_p25", "prev": "/problem/15_amc12B_p23"}}