{"status": "success", "data": {"description_md": "Let $\\omega=-\\tfrac{1}{2}+\\tfrac{1}{2}i\\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\\omega+c\\omega^2,$ where $0\\leq a \\leq 1,0\\leq b\\leq 1,$ and $0\\leq c\\leq 1.$ What is the area of $S$?\n\n$\\textbf{(A) } \\frac{1}{2}\\sqrt3 \\qquad\\textbf{(B) } \\frac{3}{4}\\sqrt3 \\qquad\\textbf{(C) } \\frac{3}{2}\\sqrt3\\qquad\\textbf{(D) } \\frac{1}{2}\\pi\\sqrt3 \\qquad\\textbf{(E) } \\pi$\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\">\\omega=-\\tfrac{1}{2}+\\tfrac{1}{2}i\\sqrt3.</span>  Let  <span class=\"katex--inline\">S</span>  denote all points in the complex plane of the form  <span class=\"katex--inline\">a+b\\omega+c\\omega^2,</span>  where  <span class=\"katex--inline\">0\\leq a \\leq 1,0\\leq b\\leq 1,</span>  and  <span class=\"katex--inline\">0\\leq c\\leq 1.</span>  What is the area of  <span class=\"katex--inline\">S</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } \\frac{1}{2}\\sqrt3 \\qquad\\textbf{(B) } \\frac{3}{4}\\sqrt3 \\qquad\\textbf{(C) } \\frac{3}{2}\\sqrt3\\qquad\\textbf{(D) } \\frac{1}{2}\\pi\\sqrt3 \\qquad\\textbf{(E) } \\pi</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": "2019 AMC 12B Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12B_p25", "prev": "/problem/19_amc12B_p23"}}