Triangle $ABC$ has side lengths $AB = 9$, $BC =$ $5\sqrt{3}$, and $AC = 12$. Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$, and points $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C$ are on segment $\overline{AC}$ with $Q_{k}$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1, 2, ..., 2449$. Furthermore, each segment $\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2449$, 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 $\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2450$, that have rational length.
For each $k$ between $2$ and $2450$, the area of the trapezoid with $\overline{P_kQ_k}$ as its bottom base is the difference between the areas of two triangles, both similar to $\triangle{ABC}$. Let $d_k$ be the length of segment $\overline{P_kQ_k}$. The area of the trapezoid with bases $\overline{P_{k-1}Q_{k-1}}$ and $P_kQ_k$ is $\left(\frac{d_k}{5\sqrt{3}}\right)^2 - \left(\frac{d_{k-1}}{5\sqrt{3}}\right)^2 = \frac{d_k^2-d_{k-1}^2}{75}$ times the area of $\triangle{ABC}$. (This logic also applies to the topmost triangle if we notice that $d_0 = 0$.) However, we also know that the area of each shape is $\frac{1}{2450}$ times the area of $\triangle{ABC}$. We then have $\frac{d_k^2-d_{k-1}^2}{75} = \frac{1}{2450}$. Simplifying, $d_k^2-d_{k-1}^2 = \frac{3}{98}$. However, we know that $d_0^2 = 0$, so $d_1^2 = \frac{3}{98}$, and in general, $d_k^2 = \frac{3k}{98}$ and $d_k = \frac{\sqrt{\frac{3k}{2}}}{7}$. The smallest $k$ that gives a rational $d_k$ is $6$, so $d_k$ is rational if and only if $k = 6n^2$ for some integer $n$.The largest $n$ such that $6n^2$ is less than $2450$ is $20$, so $k$ has $\boxed{020}$ possible values.
Solution by zeroman