**A. 570.** Given a triangle *ABC*. For an arbitrary interior point *X* of the triangle denote by *A*_{1}(*X*) the point intersection of the lines *AX* and *BC*, denote by *B*_{1}(*X*) the point intersection of the lines *BX* and *CA*, and denote by *C*_{1}(*X*) the point intersection of the lines *CX* and *AB*. Construct such a point *P* in the interior of the triangle for which each of the quadrilaterals *AC*_{1}(*P*)*PB*_{1}(*P*), *BA*_{1}(*P*)*PC*_{1}(*P*) and *CB*_{1}(*P*)*PA*_{1}(*P*) has an inscribed circle.

Proposed by: *G. Holló,* Budapest

(5 points)

**B. 4474.** The points *K*, *L*, *M* and *N*, respectively, lie on sides *AB*, *BC*, *CD* and *DA* of a square *ABCD*. Given that *KLA*=*LAM*=*AMN*=45^{o}, prove that *KL*^{2}+*AM*^{2}=*LA*^{2}+*MN*^{2}.

(4 points)

**B. 4477.** In an acute triangle *ABC*, < (with conventional notations). Let *R *and *P* be the feet of the altitudes drawn from vertices *A* and *C*, respectively. Let *Q* denote a point of line *AB*, different from *P*, such that *AP*^{.}*BQ*=*AQ*^{.}*BP*. Prove that line *RB* bisects the angle *PRQ*.

Suggested by *J. Mészáros,* Jóka

(5 points)

**B. 4480.** The escribed circle drawn to side *AB* of a triangle *ABC* touches the lines of sides *AB*, *BC* and *CA* at the points *E*, *F*, *G*, respectively. The intersection of lines *AF* and *BG* is *H*. The inscribed circle of the triangle formed by the midlines of triangle *ABC* touches the side parallel to *AB* at point *N*. Prove that the points *E*, *H* and *N* are collinear.

Suggested by *Sz. Miklós,* Herceghalom

(5 points)

**C. 1135.** Six students qualified for the finals of the mathematics competition organized by a school: Alan (A), Bill (B), Cecilia (C), Diana (D), Eleanor (E) and Frank (F). When the first problem had been assessed, they were told that 3 students achieved 10 points and the others achieved 7. Everyone was instructed to guess who had the 10 points. The guesses were as follows: A, B, D; A, C, E; A, D, E; B, C, E; B, D, E; C, D, E. Three of them each guessed two right, two of them each guessed one right, and one had no right guess. It turned out that no one guessed all three right. Which students got 10 points?

(5 points)

**K. 343.** A box contains buttons, with four holes, two holes or one hole on them. There is at least one button of each kind, and there are 61 holes and 27 buttons altogether. Given that the number of buttons with one hole is the largest, find the possible numbers of the individual types of buttons.

(6 points)

This problem is for grade 9 students only.

**K. 344.** In a fun fair, adults and children are charged the same amount for a ride on the carousel, and the first few rides are for free. However, the number of free rides is greater for children. Two adults with one child paid 3200 forints (HUF) for 20 rides, three adults with two children only paid 2600 forints for the same number of rides, and two adults with three children paid 2400. What is the number of free rides for children and for adults, if everyone used up all their free rides?

(6 points)

This problem is for grade 9 students only.

**K. 346.** From the third term onwards, the terms of a sequence are obtained by subtracting from the previous term the term preceding that. (Thus the third term = the second term - the first term, the fourth term = the third term - the second term, and so on.) The first term is 2, the sum of the first 2012 terms is 2012. What is the second term?

(6 points)

This problem is for grade 9 students only.