**B. 4414.** There are 98 sticks lying on the table, their lengths are 1,2,3,...,98 units. Ann and Bill play the following game: With Ann starting the game, they take turns removing one stick of their choice. The game ends when there remain exactly three sticks on the table. Ann wins if the three sticks can form a triangle. Otherwise, Bill wins. Which player has a winning strategy?

(4 points)

**B. 4415.** Let *B* be an interior point of a line segment *AC*. The Thales circles of the line segments *AB*, *BC* and *AC* are *k*_{1}, *k*_{2} and *k*_{3}, respectively, with radii of *r*_{1}, *r*_{2} and *r*_{3}. A common exterior tangent of circles *k*_{1} and *k*_{2} cuts the circle *k*_{3} into two circular segments. The radius of the circle inscribed in the smaller circular segment is *r*_{4}. Prove that *r*_{1}^{.}*r*_{2}=*r*_{3}^{.}*r*_{4}.

(Suggested by *Sz. Miklós,* Herceghalom)

(4 points)

**B. 4418.** Regular triangles *ABD*, *BCE* and *CAF* are drawn over the sides of triangle *ABC *on the outside. Let *G*, *H* and *I* denote the midpoints of line segments *DE*, *EF* and *FD*, respectively. Prove that the sum of the angles *AHB*, *BIC* and *CGA *is 180^{o}.

(Suggested by *Sz. Miklós,* Herceghalom)

(5 points)

**B. 4419.** Jack Potter is an addicted gambler. Yesterday, he threw 20 000 forints (HUF, Hungarian currency) in a one-armed bandit machine without his family knowing about it. To make things worse, he took that money from the amount that the family reserved for food. To avoid the affair being revealed, today he is taking the remaining 40 000 forints of the family, too, and goes back to the casino to play roulette. Since he does not want to risk too much, he is betting 1000 forints on red or on black in each game. If he wins, which has a probability of 18/37, then he gets another 1000 forints. Otherwise he loses his bet. He stops playing either when he has gathered a total of 60 000 forints -- in that case he can put all the money back in its place -- or when he has lost all the money. What is the probability that Jack Potter will manage to gather the 60 000 forints?

(Based on the idea of *D. Pálvölgyi,* Budapest)

(5 points)

**K. 321.** We cut a regular hexagon shaped tabletop into two parts along his symmetry diagonal, like this we got two symmetric trapezoid tabletops. In the reading area of a library, there are such trapezoidal tables in three colours: green, blue and red. We compiled a bigger regular hexagon from eight such tabletops.

*a*) In how many different ways is it possible to make this arrangement if two tables of the same colour cannot have an edge in common? (Arrangements obtained from each other by rotation are not considered different.)

*b*) At the long side of a trapezoidal table there is room for two chairs, while at each of the other sides there is only room for one chair. Therefore, 12 people can be seated comfortably around this hexagonal arrangement of tables. How many tables are needed for a big hexagon around which 18 people can be comfortably seated?

(6 points)

This problem is for grade 9 students only.