{"status": "success", "data": {"description_md": "Let $g(x)$ be a polynomial with leading coefficient $1,$ whose three roots are the reciprocals of the three roots of $f(x)=x^3+ax^2+bx+c,$ where $1<a<b<c.$ What is $g(1)$ in terms of $a,b,$ and $c?$\n\n$\\textbf{(A) }\\frac{1+a+b+c}c \\qquad \\textbf{(B) }1+a+b+c \\qquad \\textbf{(C) }\\frac{1+a+b+c}{c^2}\\qquad \\textbf{(D) }\\frac{a+b+c}{c^2} \\qquad \\textbf{(E) }\\frac{1+a+b+c}{a+b+c}$\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>Let  <span class=\"katex--inline\">g(x)</span>  be a polynomial with leading coefficient  <span class=\"katex--inline\">1,</span>  whose three roots are the reciprocals of the three roots of  <span class=\"katex--inline\">f(x)=x^3+ax^2+bx+c,</span>  where  <span class=\"katex--inline\">1&lt;a&lt;b&lt;c.</span>  What is  <span class=\"katex--inline\">g(1)</span>  in terms of  <span class=\"katex--inline\">a,b,</span>  and  <span class=\"katex--inline\">c?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }\\frac{1+a+b+c}c \\qquad \\textbf{(B) }1+a+b+c \\qquad \\textbf{(C) }\\frac{1+a+b+c}{c^2}\\qquad \\textbf{(D) }\\frac{a+b+c}{c^2} \\qquad \\textbf{(E) }\\frac{1+a+b+c}{a+b+c}</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": "2021 AMC 12B Problem 16", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc12B_p17", "prev": "/problem/21_amc12B_p15"}}