1998 AIME Problem 12

Let ABC be equilateral, and D, E, and F be the midpoints of \overline{BC}, \overline{CA}, and \overline{AB}, respectively. There exist points P, Q, and R on \overline{DE}, \overline{EF}, and \overline{FD}, respectively, with the property that P is on \overline{CQ}, Q is on \overline{AR}, and R is on \overline{BP}. The ratio of the area of triangle ABC to the area of triangle PQR is a + b\sqrt {c}, where a, b, and c are integers, and c is not divisible by the square of any prime. What is a^{2} + b^{2} + c^{2}?

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.