Question 59
Let $\zeta = e^{2 \pi i / 99}$ and $\omega = e^{2 \pi i / 101}$. The polynomial
$$
x^{9999} + a_{9998} x^{9998} + \cdots + a_{1} x + a_{0}
$$
has roots $\zeta^{m} + \omega^{n}$ for all pairs of integers $(m, n)$ with $0 \leq m \lt 99$ and $0 \leq n \lt 101$. Compute $a_{9799} + a_{9800} + \cdots + a_{9998}$.
$$
x^{9999} + a_{9998} x^{9998} + \cdots + a_{1} x + a_{0}
$$
has roots $\zeta^{m} + \omega^{n}$ for all pairs of integers $(m, n)$ with $0 \leq m \lt 99$ and $0 \leq n \lt 101$. Compute $a_{9799} + a_{9800} + \cdots + a_{9998}$.