2018 AIME II Problems/Problem 7

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.

2018 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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2018 AIME II Problems/Problem 7