2. Determine the largest value of $x$ for which
$$
\left|x^{2}-4 x-39601\right| \geq\left|x^{2}+4 x-39601\right|
$$
2. Answer: 199
By direct computation,
$$
\begin{aligned}
& \left|x^{2}-4 x-39601\right| \geq\left|x^{2}+4 x-39601\right| \\
\Longleftrightarrow & \left(x^{2}-4 x-39601\right)^{2}-\left(x^{2}+4 x-39601\right)^{2} \geq 0 \\
\Longleftrightarrow & 2\left(x^{2}-39601\right)(-8 x) \geq 0 \\
\Longleftrightarrow & x(x+199)(x-199) \leq 0,
\end{aligned}
$$
we conclude that the largest possible value of $x$ is 199 .