9.5. $A L$ and $B M$ are the angle bisectors of triangle $A B C$. The circumcircles of triangles $A L C$ and $B M C$ intersect again at point $K$, which lies on side $A B$. Find the measure of angle $A C B$.
Answer. $\angle A C B=60^{\circ}$.

Solution. Draw the segment $C K . \angle L C K=\angle L A K$ (these angles are inscribed in the circle and subtend the same arc). Similarly, $\angle M C K$ $=\angle M B K$. Since $\angle A C B=\angle L C K+\angle M C K$, the desired angle $A C B$ is one-third of the sum of the angles of triangle $A B C$, that is, it is equal to $60^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1a97ee453e4e6cef5ed6g-2.jpg?height=466&width=468&top_left_y=915&top_left_x=1459)

## Criteria and methodology for evaluating the performance of olympiad tasks

At the olympiad, a 7-point scale should be used: each problem is scored with an integer from 0 to 7. The final score is the sum of the points earned by the participant.

The main principles of evaluation are presented in the table.

| Points | Correctness (incorrectness) of the solution |
| :---: | :--- |
| 7 | Complete correct solution. |
| $6-7$ | Correct solution. There are minor flaws, which do not significantly affect the solution. |
| $5-6$ | The solution contains minor errors, gaps in reasoning, but is generally correct and can become fully correct after minor corrections or additions. |
| $2-3$ | Auxiliary statements that help in solving the problem have been proven. |
| $0-1$ | Important individual cases have been considered in the absence of a solution (or in the case of an incorrect solution). |
| 0 | The solution is incorrect, no progress has been made. |
| 0 | The solution is absent. |

a) any correct solution is scored 7 points. It is not permissible to deduct points for the solution being too long, or for the solution differing from the one provided in the methodological materials or from other solutions known to the jury; when checking the work, it is important to understand the logic of the participant's reasoning, assessing the degree of its correctness and completeness;

b) the olympiad work is not a participant's test, so any corrections in the work, including crossing out previously written text, are not grounds for deducting points; it is not permissible to deduct points for untidiness in writing solutions;

c) points are not awarded "for the participant's effort," including for writing a large volume of text that does not contain progress in solving the problem.