AMC 12 2011b 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} \textbf{(B)}\ \frac{50}{99} \textbf{(C)}\ \frac{44}{87} \textbf{(D)}\ \frac{34}{67} \textbf{(E)}\ \frac{7}{13}