2008 AIME I Problem 13

Let

p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.

Suppose that

p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1) = p(1,1) = p(1, - 1) = p(2,2) = 0.

There is a point \left(\frac {a}{c},\frac {b}{c}\right) for which p\left(\frac {a}{c},\frac {b}{c}\right) = 0 for all such polynomials, where a, b, and c are positive integers, a and c are relatively prime, and c > 1. Find a + b + c.

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.