7. In trapezoid $ABCD$, points $G$ and $H$ are taken on the base $AD$, and points $E$ and $F$ are taken on the base $BC$. Segments $BG$ and $AE$ intersect at point $K$, segments $EH$ and $GF$ intersect at point $L$, and segments $FD$ and $HC$ intersect at point $M$. The area of quadrilateral $ELGK$ is 4, and the area of quadrilateral $FMHL$ is 8. Find the greatest and least possible values of the area of triangle $CDM$ given that the area of this triangle is expressed as an integer.
Solution. Let the areas of triangles $ABK, KEG$ be denoted by $x$, the areas of triangles $ELG, FHL$ by $y$, and the areas of triangles $FMH, CDM$ by $z$. Then $x+y=4, y+z=8$. From these equations and the fact that $z$ is an integer, it follows that $x, y$ are also integers. Since $y \geq 1$, the possible values for $x$ are $x=3, x=2, x=1$, corresponding to $z=7, z=6, z=5$, respectively. Therefore, the smallest and largest values sought are the numbers 5 and 7.

Answer: $z=7, z=5$.

| Criteria for evaluating the performance of the task | Points |
| :--- | :---: |
| A correct and justified sequence of all steps of the solution is provided. All transformations and calculations are performed correctly. The correct answer is obtained. | $\mathbf{1 2}$ |
| A correct sequence of all steps of the solution is provided. There are gaps in the justification of the solution (for example, the fact of the equality of the areas of pairs of triangles $ABK$ and $KEG$, $ELG$ and $FHL$, $FMH$ and $CDM$ is used, but this fact is not proven) or a computational error or typo that does not affect the further course of the solution. As a result of this error or typo, an incorrect answer may be obtained. | $\mathbf{8}$ |
| The problem is not solved, but its solution is significantly advanced, i.e.: - a significant part of the solution is performed correctly, possibly inaccurately (for example, the equality of the areas of pairs of triangles is correctly noted); - another part is either not performed or performed incorrectly, possibly even with logical errors (for example, the possible values of these areas are not found). The solution may be incomplete. | $\mathbf{4}$ |
| All cases of the solution that do not correspond to the above criteria for evaluation of 2 and 4 points. | $\mathbf{0}$ |