Let $L\subseteq \Sigma^*$,
$$ \frac12 L=\{w \in \Sigma^* \mid \exists v \in \Sigma^*\ wv\in L \text{ and }|w|=|v|\} $$ $$ L\frac12=\{w\in \Sigma^* \mid \exists v\in \Sigma^*\ vw\in L\text{ and }|w|=|v|\}$$ (1) Is there exists language $L$ such that $L\in P$, but $\frac12L, L\frac12 $ are $NP$-complete ?
(2) Is there exists language $L$ such that $L$ is $NP$-complete, but $\frac12 L, L\frac12$ are in $P$ ?
I don't know what I should think about it. I have no idea how to solve it. Can you help me, please ?