{"status": "success", "data": {"description_md": "Real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x$, $y$, and $z$ are within 1 unit of each other is greater than $\\frac {1}{2}$. What is the smallest possible value of $n$?\n\n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 9\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 11$", "description_html": "<p>Real numbers  <span class=\"katex--inline\">x</span> ,  <span class=\"katex--inline\">y</span> , and  <span class=\"katex--inline\">z</span>  are chosen independently and at random from the interval  <span class=\"katex--inline\">[0,n]</span>  for some positive integer  <span class=\"katex--inline\">n</span> . The probability that no two of  <span class=\"katex--inline\">x</span> ,  <span class=\"katex--inline\">y</span> , and  <span class=\"katex--inline\">z</span>  are within 1 unit of each other is greater than  <span class=\"katex--inline\">\\frac {1}{2}</span> . What is the smallest possible value of  <span class=\"katex--inline\">n</span> ?</p>\n<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ 9\\qquad\\textbf{(D)}\\ 10\\qquad\\textbf{(E)}\\ 11</span> </p>\n<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": 4, "problem_name": "2012 AMC 10A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/12_amc10A_p24"}}