**A. 458.** In space, *n*+1 points *P*_{1},*P*_{2},...,*P*_{n} and *Q* are given, *n*4, no four of which are in the same plane. It is known that for each triple of distinct points *P*_{i}, *P*_{j} and *P*_{k} one can find a point *P*_{l} such that *Q* is interior to the tetrahedron *P*_{i}*P*_{j}*P*_{k}*P*_{l}. Show that *n* must be even.

*Bulgarian competition problem*

(5 points)

**A. 459.** Denote by *F*_{n} the *n*th Fibonacci number (*F*_{0}=0, *F*_{1}=1, *F*_{k+1}=*F*_{k}+*F*_{k-1}). Show that one can find a positive integer *n*, having at least 1000 distinct prime divisors, such that *n* divides *F*_{n}.

Proposed by: *Péter Csikvári,* Budapest

(5 points)

**B. 4110.** The hexagon *ABCDEF* and the triangle *PQR* lie in the same plane. The quadrilaterals *ABRQ*, *CDPR*, *EFQP* are all rectangles.

*a*) Prove that the perpendicular bisectors of the sides *BC*, *DE*, *FA* are concurrent.

*b*) Show that there exists a triangle *P*'*Q*'*R*', too, for which the quadrilaterals *BCR*'*Q*', *DEP*'*R*', *FAQ*'*P*' are rectangles.

(5 points)

**K. 169.** Four boys and four girls went dancing. In the first four dances, each boy danced exactly once with each girl. (They remained with the same partner during each dance, and only switched between the dances.) Charles danced the Viennese waltz with Fanny and Bernard danced it with Helen. Albert danced tango with Gaby and David with Fanny. Gaby danced mambo with Charles and Emily with David. Who were the partners dancing the first dance, the English waltz together?

(6 points)

This problem is for grade 9 students only.