**A. 576.** Find all positive integers *n*, nonzero reals and real *t* for which there exists a finite, nonempty set of points in the plane *S* and a nonconstant function such that

for every similarity transformation of the plane *S*.

Proposed by: *Tamás Ágoston,* Budapest and *Márton Mester,* Cambridge

(5 points)

**B. 4493.** Let (*n*,*k*) denote the greatest common divisor of the positive integers *n* and *k*, and let [*n*,*k*] denote their least common multiple. Show that, for all positive integers *a*, *b*, *c*, the greatest common divisor of the numbers [*a*,*b*], [*b*,*c*], [*c*,*a*] equals the least common multiple of the numbers (*a*,*b*), (*b*,*c*), (*c*,*a*).

(4 points)

**B. 4494.** *F* is the midpoint of base *BC* of the isosceles triangle *ABC*. *D* is an interior point of the line segment *BF*. The perpendicular dropped from point *D *onto *BC* intersects side *AB* at *M*, and the line through point *D*, parallel to *AB* intersects side *AC* at *P*. Express the ratio of the areas of triangles *AMP* and *ABC* in terms of *k*=*BD*:*BC*.

*Matlap,* Cluj-Napoca - Kolozsvár, Romania

(3 points)

**B. 4495.** Given the parallelogram *ABCD* and points *F* and *G*, such that *AF*=*FC*, *BG*=*GD* and given that triangles *AFC* and *BGD* are similar, prove that the line *FG* is perpendicular to a side of the parallelogram.

Suggested by *Sz. Miklós,* Herceghalom

(5 points)

**B. 4496.** An absent minded professor is travelling in a train along a subway line having infinite length in both directions. He would like to get from Alpha station to Tau station. At every station, the probability that he looks up from his notes is the same *p*>0 (independently of each other). If he looks up and observes that he has reached his destination, then he gets off. If he can see that he has passed his destination, then he also gets off and gets on the train travelling in the opposite direction. If he looks up and sees that he has not reached his destination yet, he stays in his train. If he does not look up, he will stay on the train wherever he is. What is the probability that he will get off at Tau station after a while?

(4 points)

**B. 4499.** Let *P* be an interior point of an acute-angled triangle *A*_{1}*A*_{2}*A*_{3}. The points *T*_{i} are such that the line segment *A*_{i}*T*_{i} touches the circle of diameter *PA*_{i+1} at *T*_{i} (*i*=1,2,3, *A*_{4}=*A*_{1}). Show that .

Suggested by *Á. Péter,* Sepsiszentgyörgy (*Matlap,* Cluj-Napoca - Kolozsvár, Romania)

(4 points)

**K. 358.** In a warehouse, there are cubical boxes of the same mass and size stacked in the form of a large rectangular block. A worker is loading the boxes onto a truck. In each step, he transfers to the truck one vertical or horizontal layer of boxes, with a thickness of a single box. (He is able to approach the stack of boxes from any direction.) The mass of the boxes is a whole number of kilograms. The total mass of the boxes loaded in the first step is 60 kg, in the second step it is 84 kg, and in the third step it is 112 kg. What may be the total mass of all the boxes, in kilograms?

(6 points)

This problem is for grade 9 students only.

**K. 359.** Let the natural number *n* be at least 5. Prove that it is possible to divide a rectangle into *n* smaller rectangles, such that no two adjacent small rectangles form a larger rectangle together.

Based on a problem of the *Bolyai* Competition, 2012

(6 points)

This problem is for grade 9 students only.