2011 AMC 12B Question 25

For every $m$ and $k$ integers with $k$ odd, denote by $\left[\dfrac{m}{k}\right]$ the integer closest to $\dfrac{m}{k}.$ For every odd integer $k,$ let $P(k)$ be the probability that $$
\left[\dfrac{n}{k}\right]+\left[\dfrac{100-n}{k}\right]=\left[\dfrac{100}{k}\right]
$$ for an integer $n$ randomly chosen from the interval $1\le n\le99!.$ What is the minimum possible value of $P(k)$ over the odd integers $k$ in the interval $1\le k\le99?$