{"status": "success", "data": {"description_md": "How many integers $n \\geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that\n\n$$|z_1| = |z_2| = ... = |z_n| = 1 \\text{    and    } z_1 + z_2 + ... + z_n = 0,$$<br>then the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane?\n\n$\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2 \\qquad\\textbf{(C)}\\ 3 \\qquad\\textbf{(D)}\\ 4 \\qquad\\textbf{(E)}\\ 5$\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>How many integers  <span class=\"katex--inline\">n \\geq 2</span>  are there such that whenever  <span class=\"katex--inline\">z_1, z_2, ..., z_n</span>  are complex numbers such that</p>&#10;<p> <span class=\"katex--display\">|z_1| = |z_2| = ... = |z_n| = 1 \\text{    and    } z_1 + z_2 + ... + z_n = 0,</span> <br/>then the numbers  <span class=\"katex--inline\">z_1, z_2, ..., z_n</span>  are equally spaced on the unit circle in the complex plane?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 1 \\qquad\\textbf{(B)}\\ 2 \\qquad\\textbf{(C)}\\ 3 \\qquad\\textbf{(D)}\\ 4 \\qquad\\textbf{(E)}\\ 5</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": "2020 AMC 12B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/20_amc12B_p24", "prev": "/problem/20_amc12B_p22"}}