A. 458. In space, n+1 points P1,P2,...,Pn and Q are given, n4, no four of which are in the same plane. It is known that for each triple of distinct points Pi, Pj and Pk one can find a point Pl such that Q is interior to the tetrahedron PiPjPkPl. Show that n must be even.
Bulgarian competition problem
A. 459. Denote by Fn the nth Fibonacci number (F0=0, F1=1, Fk+1=Fk+Fk-1). Show that one can find a positive integer n, having at least 1000 distinct prime divisors, such that n divides Fn.
Proposed by: Péter Csikvári, Budapest
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.
Solution (in Hungarian)