1. Let $x, y$ satisfy $\left\{\begin{array}{l}2 x+y \geq 4 \\ x-y \geq 1 \\ x-2 y \leq 2\end{array}\right.$, if $z=a x+y$ attains its minimum value only at point $A(2,0)$, then the range of the real number $a$
Answer $\left(-\frac{1}{2}, 2\right)$.
By analyzing and plotting the feasible region, it is known from the graph that when the line $z=a x+y$ is between the lines $x-2 y=2$ and the $x$-axis, or between the lines $2 x+y=4$ and the $x$-axis, it meets the conditions. At this time, $0-2$, i.e., $-\frac{1}{2}<a<0$ or $0 \leq a<2$, which means $-\frac{1}{2}<a<2$. Therefore, the answer is $\left(-\frac{1}{2}, 2\right)$.