5. The distance between the centers $O_{1}$ and $O_{2}$ of circles $\omega_{1}$ and $\omega_{2}$ is $10 r$, and their radii are $5 r$ and $6 r$ respectively. A line intersects circle $\omega_{1}$ at points $M$ and $N$ and is tangent to circle $\omega_{2}$ at point $K$, such that $M N=2 N K$. Find the length of the chord $M N$.
Answer: $2 r \sqrt{21}$ or $6 r$.

Solution: a similar problem is present in variant 1 under the same number, the difference is that there are 2 cases of the chord and circle placement:

1) the point of tangency with circle $\omega_{2}$ lies outside circle $\omega_{1} ; 2$ ) the point of tangency with circle $\omega_{2}$ lies inside circle $\omega_{1}$.

Evaluation criteria are provided in the table:

| Points | Evaluation Criteria |
| :---: | :--- |
| $\mathbf{7}$ | Complete 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, drawings, considerations, or auxiliary statements relevant to the solution of the problem 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. |
| $\mathbf{0}$ | The solution is absent, or the solution does not correspond to any of the criteria listed above. |

10th grade. Problem 5.
![](https://cdn.mathpix.com/cropped/2024_05_06_4de35ed8ca5599d2cb91g-6.jpg?height=478&width=1400&top_left_y=548&top_left_x=435)

Fig. 5

Fig. 4

Fig. 6

11th grade. Problem 5.

![](https://cdn.mathpix.com/cropped/2024_05_06_4de35ed8ca5599d2cb91g-6.jpg?height=377&width=505&top_left_y=1533&top_left_x=524)

Fig. 7

![](https://cdn.mathpix.com/cropped/2024_05_06_4de35ed8ca5599d2cb91g-6.jpg?height=480&width=422&top_left_y=1459&top_left_x=1271)

Fig. 8