1. Find all integer values of $x$ that are solutions to the equation $\sin (\pi(2 x-1))=\cos (\pi x / 2)$.
1. Answer: $x=4 t \pm 1, t \in Z$. Solution.

$\sin (\pi(2 x-1))=\sin (\pi(1-x) / 2) \rightarrow\left\{\begin{array}{l}2 x-1=(1-x) / 2+2 m \rightarrow 5 x / 2-2 m=3 / 2 \\ 2 x-1=(x+1) / 2+2 n \rightarrow 3 x / 2-2 n=3 / 2\end{array}\right.$

The equation $5 x-4 m=3$ in integers has the following solutions $\left\{\begin{array}{c}x=4 t+3 \\ m=5 t+3, t \in Z\end{array}\right.$.

The second equation $3 x-4 n=3$ in integers has solutions $\left\{\begin{array}{c}x=4 t+1 \\ m=3 t, t \in Z\end{array}\right.$.

Combining the solutions, we get the answer $x=4 t \pm 1, t \in Z$.