{"status": "success", "data": {"description_md": "The vertices $V$ of a centrally symmetric hexagon in the complex plane are given by $$V=\\left\\{   \\sqrt{2}i,-\\sqrt{2}i, \\frac{1}{\\sqrt{8}}(1+i),\\frac{1}{\\sqrt{8}}(-1+i),\\frac{1}{\\sqrt{8}}(1-i),\\frac{1}{\\sqrt{8}}(-1-i) \\right\\}.$$ For each $j$, $1\\leq j\\leq 12$, an element $z_j$ is chosen from $V$ at random, independently of the other choices. Let $P={\\prod}_{j=1}^{12}z_j$ be the product of the $12$ numbers selected. What is the probability that $P=-1$?\n\n$\\textbf{(A) } \\dfrac{5\\cdot11}{3^{10}} \\qquad \\textbf{(B) } \\dfrac{5^2\\cdot11}{2\\cdot3^{10}} \\qquad \\textbf{(C) } \\dfrac{5\\cdot11}{3^{9}} \\qquad \\textbf{(D) } \\dfrac{5\\cdot7\\cdot11}{2\\cdot3^{10}} \\qquad \\textbf{(E) } \\dfrac{2^2\\cdot5\\cdot11}{3^{10}}$\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>The vertices  <span class=\"katex--inline\">V</span>  of a centrally symmetric hexagon in the complex plane are given by  <span class=\"katex--display\">V=\\left\\{   \\sqrt{2}i,-\\sqrt{2}i, \\frac{1}{\\sqrt{8}}(1+i),\\frac{1}{\\sqrt{8}}(-1+i),\\frac{1}{\\sqrt{8}}(1-i),\\frac{1}{\\sqrt{8}}(-1-i) \\right\\}.</span>  For each  <span class=\"katex--inline\">j</span> ,  <span class=\"katex--inline\">1\\leq j\\leq 12</span> , an element  <span class=\"katex--inline\">z_j</span>  is chosen from  <span class=\"katex--inline\">V</span>  at random, independently of the other choices. Let  <span class=\"katex--inline\">P={\\prod}_{j=1}^{12}z_j</span>  be the product of the  <span class=\"katex--inline\">12</span>  numbers selected. What is the probability that  <span class=\"katex--inline\">P=-1</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } \\dfrac{5\\cdot11}{3^{10}} \\qquad \\textbf{(B) } \\dfrac{5^2\\cdot11}{2\\cdot3^{10}} \\qquad \\textbf{(C) } \\dfrac{5\\cdot11}{3^{9}} \\qquad \\textbf{(D) } \\dfrac{5\\cdot7\\cdot11}{2\\cdot3^{10}} \\qquad \\textbf{(E) } \\dfrac{2^2\\cdot5\\cdot11}{3^{10}}</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": "2017 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/17_amc12A_p24"}}