**A. 538.** In the 3-dimensional hyperbolic space there are given a plane and four distinct lines *a*_{1}, *a*_{2}, *r*_{1}, *r*_{2} in such positions that *a*_{1} and *a*_{2} are perpendicular to , *r*_{1} is coplanar with *a*_{1}, *r*_{2} is coplanar with *a*_{2}, finally *r*_{1} and *r*_{2} intersect at the same angle. Rotate *r*_{1} around *a*_{1} and rotate *r*_{2} around *a*_{2}; denote by and the two surfaces of revolution they sweep out. Show that the common points of and lie in a plane.

(5 points)

**B. 4363.** The reciprocals of the natural numbers 2 to 2011 are written on a blackboard. In each step, two numbers *x* and *y* are erased and replaced with the number . This step is repeated 2009 times, until a single number remains. What may the remaining number be?

(Suggested by *B. Kovács,* Szatmárnémeti)

(4 points)

**B. 4366.** Let *M* denote the orthocentre of the acute-angled triangle *ABC*, and let *A*_{1}, *B*_{1}, *C*_{1}, respectively, denote the circumcentres of triangles *BCM*, *CAM*, *ABM*. Prove that the lines *AA*_{1}, *BB*_{1} és *CC*_{1} are concurrent.

(4 points)

**B. 4368.** Let *D*, *E* and *F*, respectively, be points on sides *AB*, *BC*, *CA* of a triangle *ABC *such that *AD*:*DB*=*BE*:*EC*=*CF*:*FA*1. The lines *AE*, *BF*, *CD* intersect one another at points *G*, *H*, *I*, respectively. Prove that the centroids of triangles *ABC* and *GHI* coincide.

(Suggested by *Sz. Miklós,* Herceghalom)

(3 points)

**B. 4369.** Each of the circles *k*_{1}, *k*_{2} and *k*_{3} passes through a point *P*, and the circles *k*_{i} and *k*_{j} also pass through the point *M*_{i,j}. Let *A* be an arbitrary point of circle *k*_{1}. Let *k*_{4} be an arbitrary circle passing through *A* and *M*_{1,2}, and let *k*_{5} be an arbitrary circle passing through *A* and *M*_{1,3}. Show that if the other intersections of the pairs of circles *k*_{4} and *k*_{2}, *k*_{5} and *k*_{3}, *k*_{4} and *k*_{5} are *B*, *C* and *D*, respectively, then the points *M*_{2,3}, *B*, *C*, *D* are either concyclic or collinear.

(4 points)

**B. 4370.** Let *a*, *b*, *c* denote the lengths of the sides of a triangle, and let *u*, *v*, *w*, respectively, be the distances of the centre of the incircle from the vertices opposite to the sides. Prove that .

(Suggested by *J. Mészáros,* Jóka)

(5 points)

**C. 1080.** A steerable airship has two engines and a given supply of fuel. If both engines are operated, the airship covers 88 kilometres per hour. If the first engine were used only, the fuel would last 25 hours longer, but the distance covered per hour would only be 45 km. If the second engine were used only, the fuel would last 16 hours longer than with two engines, and the distance covered per hour would only be 72 km. In which case can the airship travel the longest distance?

(5 points)