2013 AIME I Problem 13

Triangle AB_0C_0 has side lengths AB_0 = 12, B_0C_0 = 17, and C_0A = 25. For each positive integer n, points B_n and C_n are located on \overline{AB_{n-1}} and \overline{AC_{n-1}}, respectively, creating three similar triangles \triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}. The area of the union of all triangles B_{n-1}C_nB_n for n\geq1 can be expressed as \tfrac pq, where p and q are relatively prime positive integers. Find q.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.