{"status": "success", "data": {"description_md": "Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that $$\\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \\dfrac{A}{s-p} + \\dfrac{B}{s-q} + \\frac{C}{s-r}$$for all $s\\not\\in\\{p,q,r\\}$. What is $\\tfrac1A+\\tfrac1B+\\tfrac1C$?\n\n$\\textbf{(A) }243\\qquad\\textbf{(B) }244\\qquad\\textbf{(C) }245\\qquad\\textbf{(D) }246\\qquad\\textbf{(E) } 247$", "description_html": "<p>Let  <span class=\"katex--inline\">p</span> ,  <span class=\"katex--inline\">q</span> , and  <span class=\"katex--inline\">r</span>  be the distinct roots of the polynomial  <span class=\"katex--inline\">x^3 - 22x^2 + 80x - 67</span> . It is given that there exist real numbers  <span class=\"katex--inline\">A</span> ,  <span class=\"katex--inline\">B</span> , and  <span class=\"katex--inline\">C</span>  such that  <span class=\"katex--display\">\\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \\dfrac{A}{s-p} + \\dfrac{B}{s-q} + \\frac{C}{s-r}</span> for all  <span class=\"katex--inline\">s\\not\\in\\{p,q,r\\}</span> . What is  <span class=\"katex--inline\">\\tfrac1A+\\tfrac1B+\\tfrac1C</span> ?</p>\n<p> <span class=\"katex--inline\">\\textbf{(A) }243\\qquad\\textbf{(B) }244\\qquad\\textbf{(C) }245\\qquad\\textbf{(D) }246\\qquad\\textbf{(E) } 247</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": 4, "problem_name": "2019 AMC 10A Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc10A_p25", "prev": "/problem/19_amc10A_p23"}}