5. In an isosceles triangle $M N K$, the sides $M N = N K = 8$, and $M K = 4$. A point $F$ is chosen on side $N K$ such that the circles inscribed in triangles $M N F$ and $M K F$ touch each other. Find the areas of triangles $M N F$ and $M K F$.
Answer: $S_{M N F}=3 \sqrt{15}, S_{M K F}=\sqrt{15}$.

## Solution.

Let $R, L, E, P, Q$ be the points of tangency, and $x, y, u, v$ be the lengths of the tangent segments as indicated in the diagram.

## Then

$$
\left\{\begin{array}{c}
\begin{array}{r}
x+y=8 \\
y+u=4, \\
x+2 v+u=8
\end{array} \Rightarrow\left\{\begin{array}{r}
x-u=(x+v)-(u+v)=4 \\
(x+v)+(u+v)=8
\end{array}\right. \\
\Rightarrow\left\{\begin{array} { l } 
{ x + v = 6 , } \\
{ u + v = 2 }
\end{array} \Rightarrow \left\{\begin{array}{l}
N F=6 \\
F K=2
\end{array}\right.\right.
\end{array}\right.
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_67fb85ec2e8269040458g-2.jpg?height=520&width=352&top_left_y=1011&top_left_x=1383)

Therefore,

$$
\frac{S_{M N F}}{S_{M K F}}=\frac{N F}{F K}=3 \Rightarrow S_{M K F}=\frac{1}{4} S_{M N K}=\frac{1}{8} M K \sqrt{M N^{2}-\frac{(M K)^{2}}{4}}=\frac{1}{2} \sqrt{8^{2}-\frac{4^{2}}{4}}=\frac{1}{2} \sqrt{60}=\sqrt{15}
$$

Then $S_{M N F}=3 S_{M K F}=3 \sqrt{15}$.

## Evaluation criteria are provided in the table:

| Points | Evaluation Criteria |
| :---: | :--- |
| $\mathbf{7}$ | Complete and justified solution. |
| $\mathbf{6}$ | Justified solution with minor flaws. |
| $\mathbf{5 - 6}$ | The solution contains minor errors, gaps in justification, but is generally correct and can become fully correct after minor corrections or additions. |
| $\mathbf{4}$ | The problem is more solved than not, for example, one of the two (more complex) significant cases is correctly considered. |
| $\mathbf{2 - 3}$ | The problem is not solved, but formulas, diagrams, considerations, or auxiliary statements relevant to the solution are provided. |
| $\mathbf{1}$ | The problem is not solved, but an attempt at a solution has been made, for example, individual (partial) cases have been considered in the absence of a solution or in the case of an incorrect solution. |
| $\mathbf{0}$ | No solution, or the solution does not meet any of the criteria listed above. |

## Ministry of Science and Higher Education of the Russian Federation  Council of Rectors of Universities of Tomsk Region  Open Regional Inter-University Olympiad 2021-2022  MATHEMATICS (9th grade)  Final Stage  Variant 2