2018 AIME II Problem 7

Triangle $ABC$ has side lengths $AB = 9,$ $BC = 5\sqrt{3},$ and $AC = 12.$ Points $A = P_0,$ $P_1,$ $P_2,$ $\ldots,$ $P_{2450} = B$ are on segment $\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1,$ $2,$ $\ldots,$ $2449,$ and points $A = Q_0,$ $Q_1,$ $Q_2,$ $\ldots,$ $Q_{2450} = C$ are on segment $\overline{AC}$ with $Q_k$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1,$ $2,$ $\ldots,$ $2449.$ Furthermore, each segment $\overline{P_kQ_k},$ $k = 1,$ $2,$ $\ldots,$ $2449,$ is parallel to $\overline{BC}.$ The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions has the same area. Find the number of segments $\overline{P_kQ_k},$ $k = 1,$ $2,$ $\ldots,$ $2450,$ that have rational length.