{"status": "success", "data": {"description_md": "Let $f(x) = \\log_{10} \\left(\\sin(\\pi x) \\cdot \\sin(2 \\pi x) \\cdot \\sin (3 \\pi x) \\cdots \\sin(8 \\pi x)\\right)$. The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is $n$?\n\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 18 \\qquad \\textbf{(D)}\\ 22 \\qquad \\textbf{(E)}\\ 36$\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) = \\log_{10} \\left(\\sin(\\pi x) \\cdot \\sin(2 \\pi x) \\cdot \\sin (3 \\pi x) \\cdots \\sin(8 \\pi x)\\right)</span> . The intersection of the domain of  <span class=\"katex--inline\">f(x)</span>  with the interval  <span class=\"katex--inline\">[0,1]</span>  is a union of  <span class=\"katex--inline\">n</span>  disjoint open intervals. What is  <span class=\"katex--inline\">n</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 18 \\qquad \\textbf{(D)}\\ 22 \\qquad \\textbf{(E)}\\ 36</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": 5, "problem_name": "2010 AMC 12A Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/10_amc12A_p25", "prev": "/problem/10_amc12A_p23"}}