The problem is in Bulgarian and involves a trapezoid $ABCD$ with bases on $AB$ and $CD$, with $AB = 3 \cdot CD$. The point $M$ is the midpoint of $AB$. We are asked to express specific vectors using other vectors in the trapezoid.

Let's go through each part of the question as outlined:

1. $\overrightarrow{AM}$ through $\overrightarrow{DC}$: Since $M$ is the midpoint of $AB$, the vector $\overrightarrow{AM}$ can be expressed as: $\overrightarrow{AM} = \frac{1}{2} \overrightarrow{AB}$ However, since $AB = 3 \cdot CD$, we could also express $\overrightarrow{AB}$ in terms of $\overrightarrow{CD}$: $\overrightarrow{AM} = \frac{3}{2} \overrightarrow{CD}$

2. $\overrightarrow{DM}$ through $\overrightarrow{AD}$ and $\overrightarrow{AM}$: To express $\overrightarrow{DM}$, we can use the relation: $\overrightarrow{DM} = \overrightarrow{DA} + \overrightarrow{AM}$ Substituting $\overrightarrow{AM} = \frac{3}{2} \overrightarrow{CD}$, we get: $\overrightarrow{DM} = \overrightarrow{AD} + \frac{3}{2} \overrightarrow{CD}$

3. $\overrightarrow{BC}$ through $\overrightarrow{AD}$ and $\overrightarrow{AM}$: Since $M$ is the midpoint of $AB$ and $AB = 3 \cdot CD$, we can relate $\overrightarrow{BC}$ to $\overrightarrow{AD}$ and $\overrightarrow{AM}$ through vector addition in terms of parallel and proportional segments.

Let me know if you would like further clarification on any part of this solution, or if you need the entire solution expanded.

Here are five related questions to deepen your understanding:

1. How do we determine vector expressions in terms of other vectors in trapezoids?
2. Why is $M$ being the midpoint of $AB$ important for these vector calculations?
3. Can vector addition principles apply similarly in non-parallel quadrilaterals?
4. How do ratios between segment lengths, like $AB = 3 \cdot CD$, impact vector relationships?
5. What are other ways to express vectors in terms of points in different types of quadrilaterals?

Tip: When working with vector expressions, always look for symmetries or midpoints that can simplify calculations.