Irregularity of $\{a^x b^y c^z : x=2y \lor y>z\}$
Suppose towards a contradiction that $L$ is regular, and let $n \ge 1$ be its pumping length.
Consider the word $a^{2n}b^nc \in L$. By the pumping lemma there is some value of $k \in \{1, \dots, n\}$ such that, for every $i \ge 0$, the word $a^{2n+(i-1)k} b^nc$ also belongs to $L$.
Picking $i=0$ yields the sought contradiction since $a^{2n-k} b^n c \not\in L$. We can then conclude that $L$ is not regular.