{"status": "success", "data": {"description_md": "Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<f(100)<5000(k+1)$ for some integer $k$. What is $k$?\n\n$\\textbf{(A)}\\ 1 \\qquad<br>\\textbf{(B)}\\ 2 \\qquad<br>\\textbf{(C)}\\ 3 \\qquad<br>\\textbf{(D)}\\ 4 \\qquad<br>\\textbf{(E)}\\ 5$\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\">f(x)=ax^2+bx+c</span> , where  <span class=\"katex--inline\">a</span> ,  <span class=\"katex--inline\">b</span> , and  <span class=\"katex--inline\">c</span>  are integers. Suppose that  <span class=\"katex--inline\">f(1)=0</span> ,  <span class=\"katex--inline\">50&lt;f(7)&lt;60</span> ,  <span class=\"katex--inline\">70&lt;f(8)&lt;80</span> ,  <span class=\"katex--inline\">5000k&lt;f(100)&lt;5000(k+1)</span>  for some integer  <span class=\"katex--inline\">k</span> . What is  <span class=\"katex--inline\">k</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2 \\qquad\\textbf{(C)}\\ 3 \\qquad\\textbf{(D)}\\ 4 \\qquad\\textbf{(E)}\\ 5</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": "2011 AMC 12A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/11_amc12A_p21", "prev": "/problem/11_amc12A_p19"}}