{"status": "success", "data": {"description_md": "For every $m$ and $k$ integers with $k$ odd, denote by $\\left[\\frac{m}{k}\\right]$ the integer closest to $\\frac{m}{k}$. For every odd integer $k$, let $P(k)$ be the probability that\n\n$$\\left[\\frac{n}{k}\\right] + \\left[\\frac{100 - n}{k}\\right] = \\left[\\frac{100}{k}\\right]$$
for an integer $n$ randomly chosen from the interval $1 \\leq n \\leq 99!$. What is the minimum possible value of $P(k)$ over the odd integers $k$ in the interval $1 \\leq k \\leq 99$?\n\n$\\textbf{(A)}\\ \\frac{1}{2} \\qquad \\textbf{(B)}\\ \\frac{50}{99} \\qquad \\textbf{(C)}\\ \\frac{44}{87} \\qquad \\textbf{(D)}\\ \\frac{34}{67} \\qquad \\textbf{(E)}\\ \\frac{7}{13}$\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": "

For every m and k integers with k odd, denote by \\left[\\frac{m}{k}\\right] the integer closest to \\frac{m}{k} . For every odd integer k , let P(k) be the probability that

\\left[\\frac{n}{k}\\right] + \\left[\\frac{100 - n}{k}\\right] = \\left[\\frac{100}{k}\\right]
for an integer n randomly chosen from the interval 1 \\leq n \\leq 99! . What is the minimum possible value of P(k) over the odd integers k in the interval 1 \\leq k \\leq 99 ?

\\textbf{(A)}\\ \\frac{1}{2} \\qquad \\textbf{(B)}\\ \\frac{50}{99} \\qquad \\textbf{(C)}\\ \\frac{44}{87} \\qquad \\textbf{(D)}\\ \\frac{34}{67} \\qquad \\textbf{(E)}\\ \\frac{7}{13}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2011 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/11_amc12B_p24"}}