{"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 $b$. Furthermore, 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$\\textbf{(A)} ~\\frac{17}{32}\\qquad\\textbf{(B)} ~\\frac{11}{16}\\qquad\\textbf{(C)} ~\\frac{7}{9}\\qquad\\textbf{(D)} ~\\frac{7}{6} \\qquad\\textbf{(E)} ~\\frac{25}{11}$", "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> . Furthermore, 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>\n<p> <span class=\"katex--inline\">\\textbf{(A)} ~\\frac{17}{32}\\qquad\\textbf{(B)} ~\\frac{11}{16}\\qquad\\textbf{(C)} ~\\frac{7}{9}\\qquad\\textbf{(D)} ~\\frac{7}{6} \\qquad\\textbf{(E)} ~\\frac{25}{11}</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": 3, "problem_name": "2021 AMC 10A Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc10A_p19", "prev": "/problem/21_amc10A_p17"}}