4. In square $A B C D$ with side 2, point $A_{1}$ lies on $A B$, point $B_{1}$ lies on $B C$, point $C_{1}$ lies on $C D$, point $D_{1}$ lies on $D A$. Points $A_{1}, B_{1}, C_{1}, D_{1}$ are the vertices of the square of the smallest possible area. Find the area of triangle $A A_{1} D_{1} .(\mathbf{1 1}$ points)
# Solution:

We need to prove that the square of the smallest area is obtained under the condition that points $A_{1}, B_{1}, C_{1}, D_{1}$ are the midpoints of the sides of the original square. This is equivalent to proving that the minimum length of the segment $\boldsymbol{A}_{1} \boldsymbol{D}_{1}$ is obtained when points $\boldsymbol{A}_{1}$ and $\boldsymbol{D}_{1}$ are the midpoints of the sides of the original square. Let's prove this. Let $a$ be the distance from $A_{1}$ to $A$, and $b$ be the distance from $A$ to $D_{1}$. We need to prove that $\sqrt{a^{2}+b^{2}} \geq \sqrt{2}$. Since $a+b=2, (a+b)^{2}=4, (a-b)^{2} \geq 0$. Therefore, $(a+b)^{2}+(a-b)^{2}=2\left(a^{2}+\right.$ $\left.b^{2}\right) \geq 4, \sqrt{a^{2}+b^{2}} \geq \sqrt{2}$.

The desired area is $\frac{1}{2}$.

Answer: 0.5.

| Points | Criteria for evaluating the solution to problem № 4 |
| :---: | :---: |
| 11 | 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. |
| 9 | A correct sequence of all steps of the solution is provided. There are gaps in the justification of the solution 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. |
| 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, it is claimed that the square of the smallest area is obtained when points $A_{1}, B_{1}, c_{1}, D_{1}$ are the midpoints of the sides of the original square, and the area of triangle $A A_{1} D_{1}$ is correctly found); - the other part is either not performed or performed incorrectly, possibly even with logical errors (for example, this statement is not proven). In this case, the solution may be incomplete. |
| $\mathbf{0}$ | All cases of the solution that do not meet the criteria for scoring 5, 9, and 11 points. |