**B. 3979.** The vertices of triangle *ABC* are labelled in counterclockwise order. The angles at the vertices *A*, *B*, and *C* are , and , respectively. The vertex *B* is rotated about the point *A* through the angle in clockwise direction. The point *B*_{1} obtained is rotated about *B* through the angle in clockwise direction. Finally, the point *B*_{2} obtained is rotated about *C* in clockwise direction to the point *B*_{3}. Given the points *B*, *B*_{3} and *O* the middle point of the incentre of the triangle *ABC*, construct the triangle. Investigate how the solution depends on the positions of the three given points.

(Based on a problem from the *National Mathematics Competition for Secondary Schools *(OKTV))

(4 points)

**K. 115.** When more than one files are selected on the computer for copying at the same time, and the process of copying is started, there are two progress bars displayed. The upper one shows what percentage of the file just being copied is done and the lower one shows what percentage of the whole copying task is done (including the files copied before the current file). Suppose there are three files copied in this way. When the first file is being copied and the upper progress bar reads 35%, the lower bar reads 14%. When the upper bar reads 24% during the copying of the second file, the lower bar reads 46%. What does the upper progress bar read when the lower one reads 72%? (The given data are exact values, not approximations.)

(6 points)

This problem is for grade 9 students only.

**K. 116.** Let us make 3×3 Latin squares out of a deck of French cards. (There is a card in each field of the Latin square.) Number cards are worth the value printed on them; jacks, queens, kings and aces are worth 11, 12, 13 and 1, respectively. In a Latin square, the sum of the numbers is the same in each row, column and diagonal. Let us call this equal sum the ``magic number'' of the square.

*a*) What is the largest possible magic number that can be achieved if all the nine cards used are clubs?

*b*) Is there a square whose magic number is 37 if any 9 cards may be used?

(6 points)

This problem is for grade 9 students only.

**K. 118.** Find all five-digit numbers , such that there are exactly *a* zeros, *b *ones, *c* twos, *d* threes and *e* fours among the digits. (Different letters do not necessarily denote different digits.)

(6 points)

This problem is for grade 9 students only.