{"status": "success", "data": {"description_md": "A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br><center><img class=\"problem-image\" alt=\"[asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H);[/asy]\" class=\"latexcenter\" height=\"165\" src=\"https://latex.artofproblemsolving.com/7/2/7/7274a387184a2ce0dc35e8ecaac8b5da8b052e4c.png\" width=\"165\"/></center>\n\n$\\textbf{(A)}\\ \\frac{\\sqrt{2} - 1}{2} \\qquad \\textbf{(B)}\\ \\frac{1}{4} \\qquad \\textbf{(C)}\\ \\frac{2 - \\sqrt{2}}{2} \\qquad \\textbf{(D)}\\ \\frac{\\sqrt{2}}{4} \\qquad \\textbf{(E)}\\ 2 - \\sqrt{2}$\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>A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?<br/><center><img class=\"latexcenter\" alt=\"[asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H);[/asy]\" height=\"165\" src=\"https://latex.artofproblemsolving.com/7/2/7/7274a387184a2ce0dc35e8ecaac8b5da8b052e4c.png\" width=\"165\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{\\sqrt{2} - 1}{2} \\qquad \\textbf{(B)}\\ \\frac{1}{4} \\qquad \\textbf{(C)}\\ \\frac{2 - \\sqrt{2}}{2} \\qquad \\textbf{(D)}\\ \\frac{\\sqrt{2}}{4} \\qquad \\textbf{(E)}\\ 2 - \\sqrt{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": 3, "problem_name": "2011 AMC 12B Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/11_amc12B_p13", "prev": "/problem/11_amc12B_p11"}}