I have been working on the following two problems: 1) Given any context free language L, form a new language by taking symbols at the odd positions, i.e. $w=a_1a_2\dots a_n \mapsto w'=a_1 a_3 a_5 \dots a_k$ if n is even then k=n-1 otherwise n. Is the resulting language context free for any given context free language $L$?
2) How to show $\{a^n b^n : n\geq 0\} \cup \{a^n b^{2n}: n\geq 0\}$ is not deterministic context free?
For the second problem, my initial thought was trying to prove its complement (maybe intersecting with some regular languages) is not context free. But it did not work out so well. So I think it might be reasonable to prove that it or its complement is inherently ambiguous then it is not deterministic context free. However, the ideas are quite hard to implement. Any thoughts?