{"status": "success", "data": {"description_md": "Let $f(z)= \\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $\\left| a \\right| = 1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $\\left| b \\right|$?\n\n$\\textbf{(A)}\\ 0 \\qquad<br>\\textbf{(B)}\\ \\sqrt{2}-1 \\qquad<br>\\textbf{(C)}\\ \\sqrt{3}-1 \\qquad<br>\\textbf{(D)}\\ 1 \\qquad<br>\\textbf{(E)}\\ 2$\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(z)= \\frac{z+a}{z+b}</span>  and  <span class=\"katex--inline\">g(z)=f(f(z))</span> , where  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  are complex numbers. Suppose that  <span class=\"katex--inline\">\\left| a \\right| = 1</span>  and  <span class=\"katex--inline\">g(g(z))=z</span>  for all  <span class=\"katex--inline\">z</span>  for which  <span class=\"katex--inline\">g(g(z))</span>  is defined. What is the difference between the largest and smallest possible values of  <span class=\"katex--inline\">\\left| b \\right|</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 0 \\qquad\\textbf{(B)}\\ \\sqrt{2}-1 \\qquad\\textbf{(C)}\\ \\sqrt{3}-1 \\qquad\\textbf{(D)}\\ 1 \\qquad\\textbf{(E)}\\ 2</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": "2011 AMC 12A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/11_amc12A_p24", "prev": "/problem/11_amc12A_p22"}}