{"status": "success", "data": {"description_md": "Let $S$ be a square of side length 1. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\\frac12$ is $\\frac{a-b\\pi}{c}$, where $a,b,$ and $c$ are positive integers and $\\text{gcd}(a,b,c) = 1$. What is $a+b+c$?\n\n$\\textbf{(A)}\\ 59 \\qquad\\textbf{(B)}\\ 60 \\qquad\\textbf{(C)}\\ 61 \\qquad\\textbf{(D)}\\ 62 \\qquad\\textbf{(E)}\\ 63$\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\">S</span>  be a square of side length 1. Two points are chosen independently at random on the sides of  <span class=\"katex--inline\">S</span> . The probability that the straight-line distance between the points is at least  <span class=\"katex--inline\">\\frac12</span>  is  <span class=\"katex--inline\">\\frac{a-b\\pi}{c}</span> , where  <span class=\"katex--inline\">a,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)}\\ 59 \\qquad\\textbf{(B)}\\ 60 \\qquad\\textbf{(C)}\\ 61 \\qquad\\textbf{(D)}\\ 62 \\qquad\\textbf{(E)}\\ 63</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": "2015 AMC 12A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/15_amc12A_p24", "prev": "/problem/15_amc12A_p22"}}