The linear programming problem is given by:

Maximize $z = x_1 + 5x_2$

Subject to:

1. $x_1 + x_2 \geq 3$
2. $x_1 - 2x_2 \leq 0$
3. $2x_1 - x_2 \geq 0$
4. $x_1 \geq 0$
5. $x_2 \geq 0$

We need to determine the feasible region and find the solution.

1. Analyze the constraints:

   • Constraint 2 rearranges to $x_1 \leq 2x_2$.
   • Constraint 3 rearranges to $2x_1 \geq x_2$.

2. Feasibility check:

   • All constraints intersect in a manner that provides a bounded feasible region.
   • There is a solution where the constraints overlap, forming a feasible region.

3. Evaluate the function at possible vertices:

   • Solve the system of equations representative of the vertices or extrema.

4. Comparing choices:

   • Check which option accurately describes the situation.
   • Configurations evaluated at key points, $(x_1, x_2)$, which satisfy all inequalities will likely yield the correct answer.

By solving and comparing values, the answer matches:

• There exists an optimal solution at $(x_1, x_2) = (1, 2)$ with $Z_{\text{max}} = 11$.

So, the correct answer is:

• имеет единственное решение $Z_{\text{max}} = (1;2) = 11$