1. Is it possible to place six different integers at the vertices and midpoints of the sides of a triangle so that each number at a vertex is equal to the sum of the numbers at the other two vertices and the number at the midpoint of the opposite side?
Answer: Yes, it is possible.

Solution: For example: A(0), B(3), C(1), $\mathrm{A}_{1}(-4), \mathrm{B}_{1}(2), \mathrm{C}_{1}(-2)$. Here $\mathrm{A}_{1}$ is the midpoint of side $\mathrm{BC}, \mathrm{B}_{1}$ is the midpoint of side $\mathrm{AC}, \mathrm{C}_{1}$ is the midpoint of side AB.

Grading Criteria: Correct example with verification - 7 points, without verification - 6 points, in other cases - 0 points.