# Task 7. (14 points)

Given an equilateral triangle $A B C$ with side length 2. Point $K$ lies on the extension of side $A C$ beyond point $A$, point $N$ lies on the line parallel to line $A C$ and passing through point $B$, such that $|A K|=2,|B N|=1$. Consider broken lines $K L M N$ such that point $L$ lies on side $A B$, point $M$ lies on side $B C$, and segment $L M$ is parallel to side $A C$. Find the smallest possible value of the sum $|K L|+|M N|$, if $|A N|>|C N|$.
# Solution.

Rotate segment $B N$ by $60^{\circ}$ around point $B$ so that point $N$ moves to the midpoint of side $B C$, which is point $N^{\prime}$ in the diagram. Then segment $M N$ will transform into segment $L N^{\prime}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_5aa290d73f0e34665c68g-6.jpg?height=448&width=948&top_left_y=1963&top_left_x=197)

Thus, the sum $|K L| + |M N|$ is equal to the length of the broken line $K L N^{\prime}$. This broken line will have the minimum length if point $L$ lies on the line $K N^{\prime}$. By the cosine theorem, we find that $(|K L| + |M N|)_{\text{min}} = |K N^{\prime}| = \sqrt{4^2 + 1^2 - 2 \cdot 1 \cdot 4 \cdot \cos 60^{\circ}} = \sqrt{13}$.

Answer. $\sqrt{13}$

| Criterion | Evaluation | Points |
| :---: | :---: | :---: |
| Complete solution. | + | 14 |
| All main logical steps of the solution are presented. A computational error is made. | +. | 12 |
| A correct solution scheme is provided. The rotation of segment $B N$ is described but incorrectly implemented. An incorrect answer is obtained. | $\pm$ | 11 |
| The problem is reduced to finding the maximum value of a function: the sum $\|K L\| + \|M N\|$ is represented as a function of some variable. Errors are made in transformations and/or finding the derivative. An incorrect answer is obtained. |  |  |
| The problem is reduced to finding the maximum value of a function: the sum $\|K L\| + \|M N\|$ is represented as a function of some variable. The solution to the resulting problem is missing. | $+/ 2$ | 7 |
| The solution does not meet any of the criteria described above. | $-/ 0$ | 0 |
| Maximum score |  | 14 |