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1. Unit complex numbers with a Re-Im condition ●●●○○ 50 ptsHow many complex numbers $z$ satisfy the following two conditions: $|z|=1$ and $\operatorname{Re}\left(z^{100}\right)=\operatorname{Im}\left(z^{200}\right)$?
2. Count complex solutions of an exponential equation ●●●●○ 100 ptsHow many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation $e^z = \frac{z - 1}{z + 1}$?
3. Smallest n with many solutions ●●●●○ 100 ptsLet $f(x)=|2\{x\}-1|$, where $\{x\}$ denotes the fractional part of $x$. Find the smallest positive integer $n$ such that the equation $n f(x f(x))=x$ has at least $2012$ real solutions.