Can $\{a^mb^nc^n\mid m,n \ge 1\}$ be proved non-regular using the pumping lemma? $\{a^mb^nc^n\mid m,n \ge 1\}$ intuitively seems like a non-regular language. It looks like the machine needs to remember the number of $b$s (which isn't limited). The pumping lemma can be used to prove a language is non-regular. In reference to the Wikipedia article linked (see "Use of Lemma"), every string I have tried doesn't give a contradiction. For example, a string like $a^1b^pc^p$ (where $p$ is the pumping length) seems to nicely split into $x = \varepsilon$, $y = a^1$, $z = b^pc^p$. What am I missing here?
You've forgotten that $y$ can be pumped any number of times, including zero. The string $xy^0z = b^pc^p$ is not in the language.