8.2. Through the vertices $A$ and $C$ of triangle $ABC$, lines are drawn perpendicular to the bisector of angle $ABC$ and intersect the lines $CB$ and $BA$ at points $K$ and $M$ respectively. Find $AB$, if $BM=10, KC=2$.
Solution. Let point $K$ be located on side $BC$ of triangle $ABC$. Triangles $ABK$ and $MBC$ are isosceles (the bisector drawn from vertex $B$ is also an altitude), so $AB = BK = BC - CK = 10 - 2 = 8$.

If point $K$ is located on the extension of segment $BC$ beyond point $C$, then similarly we find that $AB = 12$.

Answer: 8 or 12.

## Criteria:

7 points - all cases considered, solution is complete;

4 points - only one case considered;

2 points - it is concluded that the triangles are isosceles, further progress is absent.

8.3. Two people write a 36-digit number using only the digits $1, 2, 3, 4, 5, 6, 7$. The first digit is written by the first person, the second by the second, the third by the first, and so on. Can the second person ensure that the resulting number is divisible by 9, if the first person tries to prevent it?

Solution. The strategy of the second person: write a digit so that the sum of it with the previous digit equals 8. Since $18 \cdot 8$ is divisible by 9, the resulting 36-digit number will be divisible by 9.

Answer: Yes.

## Criteria:

7 points - complete solution;

3 points - noted that after the first person's move, any addend can be added to 8;

1 point - noted that after the first person's move, any number except 1 can be added to 9, meaning the second person's strategy of adding to 9 does not work.