Question 13
Let $$
\begin{aligned} p(x, y) &= a_0 + a_1x + a_2y \\ &\quad {}+ a_3x^2 + a_4xy + a_5y^2 \\ &\quad {}+ a_6x^3 + a_7x^2y \\ &\quad {}+ a_8xy^2 + a_9y^3. \end{aligned}
$$ Suppose that $$
\begin{aligned} p(0, 0) &= p(1, 0) = p(-1, 0) \\ &= p(0, 1) = p(0, -1) \\ &= p(1, 1) = p(1, -1) \\ &= p(2, 2) = 0. \end{aligned}
$$
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 $c \gt 1.$ Find $a + b + c.$
\begin{aligned} p(x, y) &= a_0 + a_1x + a_2y \\ &\quad {}+ a_3x^2 + a_4xy + a_5y^2 \\ &\quad {}+ a_6x^3 + a_7x^2y \\ &\quad {}+ a_8xy^2 + a_9y^3. \end{aligned}
$$ Suppose that $$
\begin{aligned} 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. \end{aligned}
$$
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 $c \gt 1.$ Find $a + b + c.$