{"status": "success", "data": {"description_md": "Four regular hexagons surround a square with a side length $1$, each one sharing an edge with the square, as shown in the figure below. The area of the resulting $12$-sided outer nonconvex polygon can be written as $m\\sqrt{n} + p$, where $m$, $n$, and $p$ are integers and $n$ is not divisible by the square of any prime. What is $m + n + p$?<br><center><img class=\"problem-image\" alt=\"[asy]         import geometry;         unitsize(3cm);         draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle);         draw(shift((1/2,1-sqrt(3)/2))*polygon(6));         draw(shift((1/2,sqrt(3)/2))*polygon(6));         draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6));         draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*polygon(6)); \t\tdraw((0,1-sqrt(3))--(1,1-sqrt(3))--(3-sqrt(3),sqrt(3)-2)--(sqrt(3),0)--(sqrt(3),1)--(3-sqrt(3),3-sqrt(3))--(1,sqrt(3))--(0,sqrt(3))--(sqrt(3)-2,3-sqrt(3))--(1-sqrt(3),1)--(1-sqrt(3),0)--(sqrt(3)-2,sqrt(3)-2)--cycle,linewidth(2)); [/asy]\" class=\"latexcenter\" height=\"355\" src=\"https://latex.artofproblemsolving.com/c/c/f/ccfd1265c72b286aa453b9ee0080021d484e6d25.png\" width=\"355\"/></center>\n\n$\\textbf{(A) } -12 \\qquad\\textbf{(B) }-4 \\qquad \\textbf{(C) } 4 \\qquad\\textbf{(D) }24 \\qquad\\textbf{(E) }32$\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>Four regular hexagons surround a square with a side length <span class=\"katex--inline\">1</span>, each one sharing an edge with the square, as shown in the figure below. The area of the resulting <span class=\"katex--inline\">12</span>-sided outer nonconvex polygon can be written as <span class=\"katex--inline\">m\\sqrt{n} + p</span>, where <span class=\"katex--inline\">m</span>, <span class=\"katex--inline\">n</span>, and <span class=\"katex--inline\">p</span> are integers and <span class=\"katex--inline\">n</span> is not divisible by the square of any prime. What is <span class=\"katex--inline\">m + n + p</span>?<br/><center><img class=\"latexcenter\" alt=\"[asy]         import geometry;         unitsize(3cm);         draw((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle);         draw(shift((1/2,1-sqrt(3)/2))*polygon(6));         draw(shift((1/2,sqrt(3)/2))*polygon(6));         draw(shift((sqrt(3)/2,1/2))*rotate(90)*polygon(6));         draw(shift((1-sqrt(3)/2,1/2))*rotate(90)*polygon(6)); &#9;&#9;draw((0,1-sqrt(3))--(1,1-sqrt(3))--(3-sqrt(3),sqrt(3)-2)--(sqrt(3),0)--(sqrt(3),1)--(3-sqrt(3),3-sqrt(3))--(1,sqrt(3))--(0,sqrt(3))--(sqrt(3)-2,3-sqrt(3))--(1-sqrt(3),1)--(1-sqrt(3),0)--(sqrt(3)-2,sqrt(3)-2)--cycle,linewidth(2)); [/asy]\" height=\"355\" src=\"https://latex.artofproblemsolving.com/c/c/f/ccfd1265c72b286aa453b9ee0080021d484e6d25.png\" width=\"355\"/></center></p>&#10;<p><span class=\"katex--inline\">\\textbf{(A) } -12 \\qquad\\textbf{(B) }-4 \\qquad \\textbf{(C) } 4 \\qquad\\textbf{(D) }24 \\qquad\\textbf{(E) }32</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": "2022 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/22_amc12B_p24"}}