{"status": "success", "data": {"description_md": "$ABCD$ is a square of side length $\\sqrt{3} + 1$. Point $P$ is on $\\overline{AC}$ such that $AP = \\sqrt{2}$. The square region bounded by $ABCD$ is rotated $90^{\\circ}$ counterclockwise with center $P$, sweeping out a region whose area is $\\frac{1}{c} (a \\pi + b)$, where $a$, $b$, and $c$ are positive integers and $\\text{gcd}(a,b,c) = 1$. What is $a + b + c$?\n\n$\\textbf{(A)} \\ 15 \\qquad \\textbf{(B)} \\ 17 \\qquad \\textbf{(C)} \\ 19 \\qquad \\textbf{(D)} \\ 21 \\qquad \\textbf{(E)} \\ 23$\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> <span class=\"katex--inline\">ABCD</span>  is a square of side length  <span class=\"katex--inline\">\\sqrt{3} + 1</span> . Point  <span class=\"katex--inline\">P</span>  is on  <span class=\"katex--inline\">\\overline{AC}</span>  such that  <span class=\"katex--inline\">AP = \\sqrt{2}</span> . The square region bounded by  <span class=\"katex--inline\">ABCD</span>  is rotated  <span class=\"katex--inline\">90^{\\circ}</span>  counterclockwise with center  <span class=\"katex--inline\">P</span> , sweeping out a region whose area is  <span class=\"katex--inline\">\\frac{1}{c} (a \\pi + b)</span> , where  <span class=\"katex--inline\">a</span> ,  <span class=\"katex--inline\">b</span> , and  <span class=\"katex--inline\">c</span>  are positive integers and  <span class=\"katex--inline\">\\text{gcd}(a,b,c) = 1</span> . What is  <span class=\"katex--inline\">a + b + c</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)} \\ 15 \\qquad \\textbf{(B)} \\ 17 \\qquad \\textbf{(C)} \\ 19 \\qquad \\textbf{(D)} \\ 21 \\qquad \\textbf{(E)} \\ 23</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": "2013 AMC 12A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/13_amc12A_p24", "prev": "/problem/13_amc12A_p22"}}