{"status": "success", "data": {"description_md": "A sequence of complex numbers $z_{0}, z_{1}, z_{2}, ...$ is defined by the rule\n\n$$z_{n+1} = \\frac {iz_{n}}{\\overline {z_{n}}},$$<br>where $\\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. How many possible values are there for $z_{0}$?\n\n$\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 2005 \\qquad \\textbf{(E)}\\ 2^{2005}$\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>A sequence of complex numbers <span class=\"katex--inline\">z_{0}, z_{1}, z_{2}, ...</span> is defined by the rule</p>&#10;<p><span class=\"katex--display\">z_{n+1} = \\frac {iz_{n}}{\\overline {z_{n}}},</span><br/>where <span class=\"katex--inline\">\\overline {z_{n}}</span> is the complex conjugate of <span class=\"katex--inline\">z_{n}</span> and <span class=\"katex--inline\">i^{2}=-1</span>. Suppose that <span class=\"katex--inline\">|z_{0}|=1</span> and <span class=\"katex--inline\">z_{2005}=1</span>. How many possible values are there for <span class=\"katex--inline\">z_{0}</span>?</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}\\ 1 \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 2005 \\qquad \\textbf{(E)}\\ 2^{2005}</span></p>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2005 AMC 12B Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/05_amc12B_p23", "prev": "/problem/05_amc12B_p21"}}