2000 AIME I Problem 14

In triangle ABC, it is given that angles B and C are congruent. Points P and Q lie on \overline{AC} and \overline{AB}, respectively, so that AP = PQ = QB = BC. Angle ACB is r times as large as angle APQ, where r is a positive real number. Find \lfloor 1000r \rfloor.

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.