{"status": "success", "data": {"description_md": "In rectangle $ABCD,$ $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\\overline{AE}$ and $\\overline{AF}$ intersect $\\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s$ and $t$ is 1. What is $r+s+t$?\n\n$\\textbf{(A) } 7 \\qquad \\textbf{(B) } 9 \\qquad \\textbf{(C) } 12 \\qquad \\textbf{(D) } 15 \\qquad \\textbf{(E) } 20$", "description_html": "<p>In rectangle  <span class=\"katex--inline\">ABCD,</span>   <span class=\"katex--inline\">AB=6</span>  and  <span class=\"katex--inline\">BC=3</span> . Point  <span class=\"katex--inline\">E</span>  between  <span class=\"katex--inline\">B</span>  and  <span class=\"katex--inline\">C</span> , and point  <span class=\"katex--inline\">F</span>  between  <span class=\"katex--inline\">E</span>  and  <span class=\"katex--inline\">C</span>  are such that  <span class=\"katex--inline\">BE=EF=FC</span> . Segments  <span class=\"katex--inline\">\\overline{AE}</span>  and  <span class=\"katex--inline\">\\overline{AF}</span>  intersect  <span class=\"katex--inline\">\\overline{BD}</span>  at  <span class=\"katex--inline\">P</span>  and  <span class=\"katex--inline\">Q</span> , respectively. The ratio  <span class=\"katex--inline\">BP:PQ:QD</span>  can be written as  <span class=\"katex--inline\">r:s:t</span>  where the greatest common factor of  <span class=\"katex--inline\">r,s</span>  and  <span class=\"katex--inline\">t</span>  is 1. What is  <span class=\"katex--inline\">r+s+t</span> ?</p>\n<p> <span class=\"katex--inline\">\\textbf{(A) } 7 \\qquad \\textbf{(B) } 9 \\qquad \\textbf{(C) } 12 \\qquad \\textbf{(D) } 15 \\qquad \\textbf{(E) } 20</span> </p>\n<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": "2016 AMC 10A Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/16_amc10A_p20", "prev": "/problem/16_amc10A_p18"}}