This is the problem (use Latex to see clearly the encoded word)...

1. ) For n>=0, let $P(z)\in\math frak{L}_n=\dis playstyle\{P(z )=\sum_{i=0}^n a_iz^i:a_i\in {\pm\}~\mbo x{and}~z=e^{i\ theta}\}$. Then $$\displaystyl e\frac{1}{2^{n +1}}\sum_{P\i n\mathfrak{L}_ n}\frac{1}{2\ pi}\int_0^{2\ pi}\left|P(z) right|^2P(\ov erline{z})^2~d theta=?.$$

2.) Let $P(z)\in\math frak{L}_n=\dis playstyle\{P(z )=\sum_{i=0}^n a_iz^i:a_i\in {\pm\}~\mbo x{and}~z=e^{i\ theta}\}$. Then for what values of $m$ does $$\displaystyl e\frac{1}{2^{n +1}}\sum_{P\i n\mathfrak{L}_ n}\frac{1}{2\ pi}\int_0^{2\ pi}z^{2m}\left|P(z) right|^2P(\ov erline{z})^2~d theta=0$$ for all $n\geq 0$.

Note: The answer must be a function of $n$. Also, one can use any kind of programming. If one can solve this problem, he or she has a reward from me......hehehehe