Problem:
Triangle ABC has side lengths AB=9,BC=53, and AC=12. Points A=P0,P1,P2,…,P2450=B are on segment AB with Pk between Pk−1 and Pk+1 for k=1,2,…,2449, and points A=Q0,Q1,Q2,…,Q2450=C are on segment AC with Qk between Qk−1 and Qk+1 for k=1,2,…,2449. Furthermore, each segment PkQk,k=1,2,…,2449, is parallel to 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 PkQk,k=1,2,…,2450, that have rational length.
Solution:
For 1≤k≤2450,[APkQk]=2450k[ABC], where the brackets denote area. Because the ratio of the areas of similar figures is the square of the ratio of the corresponding side lengths,
This last expression is rational if and only if k=6j2 for some positive integer j. Because k≤2450, this is satisfied by j=1,2,3,…,20, giving 20 possible values of k.