{"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 <span class=\"katex--inline\">1</span>. What is <span class=\"katex--inline\">r+s+t</span>?</p>&#10;<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>&#10;", "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"}}