Bucharest, Romania - 16th July, 1999
A set S of points from the plane will be called completely symmetric if it has at least three elements and fulfils the condition: for every distinct points A,B from S, the perpendicular bisector of the segment AB is an axis of symmetry for S.
Prove that if a completely symmetric set is finite then it consists of the vertices of a regular polygon.
Let n> 1 be a fixed integer. Find the least constant C such that the inequality
holds for any x 1 , x 2 , ... , x n greater than or equal to 0 .
For this constant, characterize the instances of equality.
Let n be an even positive integer. We say that two different cells of a n x n board are neighboring if they have a comon side. Find the minimal number of cells on the n x n board that must be marked so that any cell (marked or not marked) has a marked neighboring cell.
Bucharest, Romania - 17th July, 1999
Find all pairs of positive integers (n,p) such that p is a prime, n <2p+1 ,and x p-1 is a divisor of (p-1) n +1 .
Two circles C 1 and C 2 touch internally the circle C in M and N and the center of C 2 is on C 1. The common chord of the circles C 1 and C 2 intersects C in A and B. MA and MB intersect C 1 in P and Q. Prove that C 2 is tangent to PQ.
Find all the functions f from real numbers to real numbers which satisfy
f(x-f(y))=f(f(y))+xf(y)+f(x)-1for all real numbers x,y .