Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 12F / iii / Solution

The language $\{wabw:w\in \{a,b{\}}^{\ast }\}$ is not context free. Intersecting with a suitable regular language that forces both copies into long homogeneous blocks reduces to a marked copy language; the context-free pumping lemma then places the two pumped pieces within only finitely many local blocks and cannot change the two copies identically. Equivalently, the standard copy-language proof applies with the fixed marker $ab$.