{"status": "success", "data": {"description_md": "The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$?\n\n<center><img class=\"problem-image\" alt='[asy] unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt));  path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle; marker m=marker(scale(5)*d,Fill); path f1=(0,0); path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1); path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1); path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2); path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2); path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)-- (3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3); path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^ (-2,2)--(-3,3);  draw(f1,m); draw(shift(5,0)*f2,m); draw(shift(5,0)*g2); draw(shift(12,0)*f3,m); draw(shift(12,0)*g3); draw(shift(21,0)*f4,m); draw(shift(21,0)*g4); label(\"$F_1$\",(0,-4)); label(\"$F_2$\",(5,-4)); label(\"$F_3$\",(12,-4)); label(\"$F_4$\",(21,-4)); [/asy]' class=\"latexcenter\" height=\"115\" src=\"https://latex.artofproblemsolving.com/d/2/4/d24a6830ee3382e3945990d8af2e9e00ba14c85e.png\" width=\"355\"/></center><br>\n\n$\\textbf{(A)}\\ 401 \\qquad \\textbf{(B)}\\ 485 \\qquad \\textbf{(C)}\\ 585 \\qquad \\textbf{(D)}\\ 626 \\qquad \\textbf{(E)}\\ 761$\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 figures <span class=\"katex--inline\">F_1</span>, <span class=\"katex--inline\">F_2</span>, <span class=\"katex--inline\">F_3</span>, and <span class=\"katex--inline\">F_4</span> shown are the first in a sequence of figures. For <span class=\"katex--inline\">n\\ge3</span>, <span class=\"katex--inline\">F_n</span> is constructed from <span class=\"katex--inline\">F_{n - 1}</span> by surrounding it with a square and placing one more diamond on each side of the new square than <span class=\"katex--inline\">F_{n - 1}</span> had on each side of its outside square. For example, figure <span class=\"katex--inline\">F_3</span> has <span class=\"katex--inline\">13</span> diamonds. How many diamonds are there in figure <span class=\"katex--inline\">F_{20}</span>?</p>&#10;<center><img class=\"latexcenter\" alt=\"[asy] unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(8pt));  path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle; marker m=marker(scale(5)*d,Fill); path f1=(0,0); path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1); path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1); path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2); path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2); path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)-- (3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3); path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^ (-2,2)--(-3,3);  draw(f1,m); draw(shift(5,0)*f2,m); draw(shift(5,0)*g2); draw(shift(12,0)*f3,m); draw(shift(12,0)*g3); draw(shift(21,0)*f4,m); draw(shift(21,0)*g4); label(&#34;$F_1$&#34;,(0,-4)); label(&#34;$F_2$&#34;,(5,-4)); label(&#34;$F_3$&#34;,(12,-4)); label(&#34;$F_4$&#34;,(21,-4)); [/asy]\" height=\"115\" src=\"https://latex.artofproblemsolving.com/d/2/4/d24a6830ee3382e3945990d8af2e9e00ba14c85e.png\" width=\"355\"/></center><br/>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}\\ 401 \\qquad \\textbf{(B)}\\ 485 \\qquad \\textbf{(C)}\\ 585 \\qquad \\textbf{(D)}\\ 626 \\qquad \\textbf{(E)}\\ 761</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": 3, "problem_name": "2009 AMC 12A Problem 11", "can_next": true, "can_prev": true, "nxt": "/problem/09_amc12A_p12", "prev": "/problem/09_amc12A_p10"}}