**B. 3982.** 100 freshly graduated mathematicians are looking for jobs. They consult two headhunter companies that have the same 100 positions to offer. Each company makes an offer to each applicant, a different one to each of them. Each applicant chooses from the two offers and, luckily, all positions are filled. Three months later, however, each of them decides to change for the job offered by the other company. (If it is the same as his actual job, then he decides to stay.) Show that all positions will be filled again.

Suggested by *B. Csajbók* (Budapest)

(3 points)

**B. 3988.** The midpoints of the sides of a convex pentagon are *F*_{1}, *F*_{2}, *F*_{3}, *F*_{4}, *F*_{5}, in this order. (The vertex *A* is between *F*_{5} and *F*_{1}.) Let *P* be the point in the plane for which the quadrilateral *PF*_{2}*F*_{3}*F*_{4} is a parallelogram. Prove that the quadrilateral *PF*_{5}*AF*_{1} is also a parallelogram.

(3 points)

**K. 123.** Three bunnies are sitting in the grass. There is a heap of carrots in front of each of them, 36 carrots altogether. Suppose they simultaneously pass some carrots to one another: the first bunny passes one third of his carrots to the second one, the second one passes one fourth of his carrots to the third one, and the third bunny passes one fifth of his carrots to the first one. As a result, each bunny will have the same amount of carrots as initially. How many carrots does each of them have?

(6 points)

This problem is for grade 9 students only.

**K. 124.** ``Maci sajt'' is a brand name of processed cheese that comes in a round box containing eight individually wrapped sectors that just fit in the box. Stevie ate two sectors for breakfast, and arranged the remaining ones in the box as shown in the figure. His mom did not like that. She said that Stevie was squeezing the cheese pieces. Stevie insisted that there was no squeezing, the pieces just fit. Who was right?

Suggested by *G. Kós* (Budapest)

(6 points)

This problem is for grade 9 students only.

**K. 126.** There are one thousand rooms in the sultan's palace. In each room there is a switch that switches all the lamps in the room on or off. When the lamps were on in each room and the sultan was bored, he walked through all his rooms one by one and repeated his walk again and again, always starting with the first room. During the first walk, he turned all the switches. The second time he turned the switch in every second room. The third time he turned the switch in every third room, and so on. (He turned the light on if it was off, and he turned it off if it was on). When he had walked through his room 500 times, he got tired of the game and decided to go to bed. He needed a room in which the lights were off. Which rooms did he have to choose from?

Suggested by *G. Bohner* (Budapest)

(6 points)

This problem is for grade 9 students only.