Is Half - Palindrome subset of a context-free language context-free?
Let $L=\{a^n b^n c^m d d c^k b^k a^m\}$, then $L$ is context-free. I claim 
$L'=\{a^n b^n c^n d\}$ which is not context-free.

Suppose $x\in L'$. Then $x$ must be of the form $a^i b^j c^k d$. Then $x x^R = a^i b^j c^k d d c^k b^j a^i \in L$, which means that $i=j=k$. Conversely, it's clear that $\{a^n b^n c^n d\} \subseteq L'$.