{"status": "success", "data": {"description_md": "An ant is placed randomly in a $50\\times20$ rectangle. It then walks $n$ units north, south, east, or west (chosen at random with a $\\frac{1}{4}$ chance for each direction) once, and then stops. Find the minimum positive integer value of $n$, such that the probability that the ant leaves the rectangle is greater than $\\frac12$.\n\n$\\textbf{(A)}~11\\qquad\\textbf{(B)}~12\\qquad\\textbf{(C)}~14\\qquad\\textbf{(D)}~15\\qquad\\textbf{(E)}~18$", "description_html": "<p>An ant is placed randomly in a <span class=\"katex--inline\">50\\times20</span> rectangle. It then walks <span class=\"katex--inline\">n</span> units north, south, east, or west (chosen at random with a <span class=\"katex--inline\">\\frac{1}{4}</span> chance for each direction) once, and then stops. Find the minimum positive integer value of <span class=\"katex--inline\">n</span>, such that the probability that the ant leaves the rectangle is greater than <span class=\"katex--inline\">\\frac12</span>.</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}~11\\qquad\\textbf{(B)}~12\\qquad\\textbf{(C)}~14\\qquad\\textbf{(D)}~15\\qquad\\textbf{(E)}~18</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2024 Mock AMC 12 - Problem 16", "can_next": true, "can_prev": true, "nxt": "/problem/2024_mock_amc12-p17", "prev": "/problem/2024_mock_amc12-p15"}}