PDA for the language of words $uv$ such that $|u| \geq |v|$ and $v$ contains 1
A pushdown automaton accepting this language proceeds as follows:

Push a marker $S$ to the top of the stack.
Push $A$ to the stack for each input symbol read.
At some point, nondeterministically transition to the second phase.
Pop $A$ from the stack for each input symbol read, marking whether you ever read $1$.
Fail if you encounter $S$.
Succeed if you successfully read the entire input, and encountered $1$ during the second phase.

It is also easy to give a context-free grammar for this language, using $\Sigma$ as a shortcut for the alphabet:
\begin{align}
&amp;S \to \Sigma S \mid \Sigma S \Sigma \mid \Sigma T 1 \\
&amp;T \to \Sigma T \Sigma \mid \epsilon
\end{align}