3. On the extensions of sides $\boldsymbol{A B}, \boldsymbol{B C}, \boldsymbol{C D}$ and $\boldsymbol{A}$ of the convex quadrilateral $\boldsymbol{A} \boldsymbol{B C D}$, points $\boldsymbol{B}_{1}, \boldsymbol{C}_{1}, \boldsymbol{D}_{1}$ and $\boldsymbol{A}_{1}$ are taken such that $\boldsymbol{B} \boldsymbol{B}_{1}=\boldsymbol{A B}, \boldsymbol{C} \boldsymbol{C}_{1}=\boldsymbol{B C}, \boldsymbol{D} \boldsymbol{D}_{1}=\boldsymbol{C D}$ and $\boldsymbol{A} \boldsymbol{A}_{1}=\boldsymbol{D} \boldsymbol{A}$. How many times larger is the area of quadrilateral $\boldsymbol{A}_{1} \boldsymbol{B}_{1} \boldsymbol{C}_{1} \boldsymbol{D}_{1}$ compared to the area of quadrilateral $A B C D .(10$ points)
# Solution:

Let the area of quadrilateral $\boldsymbol{A} \boldsymbol{B C D}$ be $\boldsymbol{S}$. The median divides the area of a triangle in half. Therefore, $S_{A B C}=S_{C B B 1}=S_{C B 1 C 1}$. Consequently, $S_{B B 1 C 1}=2 S_{A B C}$. Similarly, we have $S_{C C 1 D 1}=2 S_{B C D}$, $S_{D D 1 A 1}=2 S_{C D A}$, and $S_{A A 1 B 1}=2 S_{D A B}$. Adding these four equations, we get: $S_{B B 1 C 1}+S_{C C 1 D 1}+S_{D D 1 A 1}+S_{A A 1 B 1}=2\left(S_{A B C}+S_{B C D}+S_{C D A}+S_{D A B}\right)=4 S$. Therefore, $S_{A I B I C I D 1}=4 S+S=5 S$.

Answer: 5.

| Points | Criteria for evaluating the performance of task № 3 |
| :---: | :--- |
| $\mathbf{1 0}$ | 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. |
| $\mathbf{8}$ | A correct sequence of all steps of the solution is provided.  There are gaps in the justification of the area relationships of triangles 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. |
| $\mathbf{5}$ | 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 property of the median of a triangle is given or the areas of  right triangles are correctly calculated);  - another part is either not performed or performed incorrectly, possibly even with  logical errors (for example, the area relationships of triangles are obtained incorrectly or only a special case of a quadrilateral is considered).  The solution may be incomplete. |
| $\mathbf{0}$ | All cases of the solution that do not meet the criteria for scoring 5, 8, and 10 points. |