2011 AMC 12B Problem 25

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.