{"status": "success", "data": {"description_md": "Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0$?\n\n$\\textbf{(A) }\\frac{17}{32} \\qquad \\textbf{(B) }\\frac{11}{16} \\qquad \\textbf{(C) }\\frac79 \\qquad \\textbf{(D) }\\frac76\\qquad \\textbf{(E) }\\frac{25}{11}$\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</span>  be a function defined on the set of positive rational numbers with the property that  <span class=\"katex--inline\">f(a\\cdot b)=f(a)+f(b)</span>  for all positive rational numbers  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span> . Suppose that  <span class=\"katex--inline\">f</span>  also has the property that  <span class=\"katex--inline\">f(p)=p</span>  for every prime number  <span class=\"katex--inline\">p</span> . For which of the following numbers  <span class=\"katex--inline\">x</span>  is  <span class=\"katex--inline\">f(x)&lt;0</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }\\frac{17}{32} \\qquad \\textbf{(B) }\\frac{11}{16} \\qquad \\textbf{(C) }\\frac79 \\qquad \\textbf{(D) }\\frac76\\qquad \\textbf{(E) }\\frac{25}{11}</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 12A Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc12A_p19", "prev": "/problem/21_amc12A_p17"}}